Linknot: Knot Theory by Computer (Series on Knots and Everything)

Lin Knot Knot Theory by Computer SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinoi...

Author: Slavik Jablan | Radmila Sazdanovic

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Lin Knot Knot Theory by Computer

SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago)

The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman’s Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas. Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise.

Published: Vol. 1:

Knots and Physics (3rd Edition) by L. H. Kauffman

Vol. 2:

How Surfaces Intersect in Space — An Introduction to Topology (2nd Edition) by J. S. Carter

Vol. 3:

Quantum Topology edited by L. H. Kauffman & R. A. Baadhio

Vol. 4:

Gauge Fields, Knots and Gravity by J. Baez & J. P. Muniain

Vol. 5:

Gems, Computers and Attractors for 3-Manifolds by S. Lins

Vol. 6:

Knots and Applications edited by L. H. Kauffman

Vol. 7:

Random Knotting and Linking edited by K. C. Millett & D. W. Sumners

Vol. 8:

Symmetric Bends: How to Join Two Lengths of Cord by R. E. Miles

Vol. 9:

Combinatorial Physics by T. Bastin & C. W. Kilmister

Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics by W. M. Honig Vol. 11: History and Science of Knots edited by J. C. Turner & P. van de Griend

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Vol. 12: Relativistic Reality: A Modern View edited by J. D. Edmonds, Jr. Vol. 13: Entropic Spacetime Theory by J. Armel Vol. 14: Diamond — A Paradox Logic by N. S. Hellerstein Vol. 15: Lectures at KNOTS ’96 by S. Suzuki Vol. 16: Delta — A Paradox Logic by N. S. Hellerstein Vol. 17: Hypercomplex Iterations — Distance Estimation and Higher Dimensional Fractals by Y. Dang, L. H. Kauffman & D. Sandin Vol. 19: Ideal Knots by A. Stasiak, V. Katritch & L. H. Kauffman Vol. 20: The Mystery of Knots — Computer Programming for Knot Tabulation by C. N. Aneziris Vol. 21: LINKNOT: Knot Theory by Computer by S. Jablan & R. Sazdanovic Vol. 24: Knots in HELLAS ’98 — Proceedings of the International Conference on Knot Theory and Its Ramifications edited by C. McA Gordon, V. F. R. Jones, L. Kauffman, S. Lambropoulou & J. H. Przytycki Vol. 25: Connections — The Geometric Bridge between Art and Science (2nd Edition) by J. Kappraff Vol. 26: Functorial Knot Theory — Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants by David N. Yetter Vol. 27: Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy by H. Pierre Noyes; edited by J. C. van den Berg Vol. 28: Beyond Measure: A Guided Tour Through Nature, Myth, and Number by J. Kappraff Vol. 29: Quantum Invariants — A Study of Knots, 3-Manifolds, and Their Sets by T. Ohtsuki Vol. 30: Symmetry, Ornament and Modularity by S. V. Jablan Vol. 31: Mindsteps to the Cosmos by G. S. Hawkins Vol. 32: Algebraic Invariants of Links by J. A. Hillman Vol. 33: Energy of Knots and Conformal Geometry by J. O'Hara

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Vol. 34: Woods Hole Mathematics — Perspectives in Mathematics and Physics edited by N. Tongring & R. C. Penner Vol. 35: BIOS — A Study of Creation by H. Sabelli Vol. 36: Physical and Numerical Models in Knot Theory edited by J. A. Calvo et al. Vol. 37: Geometry, Language, and Strategy by G. H. Thomas Vol. 38: Current Developments in Mathematical Biology edited by K. Mahdavi, R. Culshaw & J. Boucher Vol. 39: Topological Library Part 1: Cobordisms and Their Applications edited by S. P. Novikov and I. A. Taimanov Vol. 40: Intelligence of Low Dimensional Topology 2006 edited by J. Scott Carter et al. Vol. 41: Zero to Infinity: The Fountations of Physics by P. Rowlands

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K(gE Series on Knots and Everything - Vol. 21

LinKnot Knot Theory by Computer

Slavikjablan Radmila Sazdanovic The Mathematical Institute, Belgrade, Serbia

>World Scientific NEW J E R S E Y • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G K O N G • T A I P E I • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Series on Knots and Everything — Vol. 21 LINKNOT Knot Theory by Computer Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-277-223-7 ISBN-10 981-277-223-5

Printed in Singapore.

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Preface

Knot theory is a new and rich field of mathematics. Although “real” knots are familiar to everyone and many ideas in knot theory can be formulated in everyday language, it is an area abundant with open questions. One of the main ideas of this book is to avoid obvious classification of knots and links according to their number of components. For this reason knots and links are referred to as KLs and treated together whenever possible. KLs are denoted by Conway symbols, a geometrical-combinatorial way to describe and derive KLs. The same notation is used in the Mathematica based computer program LinKnot that represents an integral part of this book. LinKnot is not only a supplementary computer program, but the best and most efficient tool for obtaining almost all of the results presented in the book, that belong to the field of experimental mathematics. Hands-on computations using Mathematica or the webMathematica package LinKnot along with detailed illustrations facilitate better learning and understanding. The program LinKnot can be downloaded from the web address http://www.mi.sanu.ac.yu/vismath/linknot/ and used as a powerful educational and research tool for experimental mathematics– implementation of Caudron’s ideas and the Conway notation enables working with large families of knots and links. The electronic version of this book and the program LinKnot that provides webMathematica on-line computations are available at the address http://math.ict.edu.yu/. Each knot theory problem described in this book is accompanied with the corresponding LinKnot function that enables the reader to actively use the program LinKnot, not only for illustrating some problems, but for computations and experimentation. LinKnot is software open to future development: a reader can change it or add new functions. For the systematic v

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and exhaustive derivation of KLs we have accepted the concept proposed by J.H. Conway and A. Caudron, supported and used in a form adapted for computer implementation. As a prerequisite for the use of the Conway notation, the complete list of basic polyhedra up to 20 crossings is given in the program LinKnot. The key idea is the “vertical” classification of KLs into well-defined categories– worlds, subworlds, classes, and families, according to new sets of recursively computed invariants. Patterns obtained from computing KL invariants imply the existence of more general KL family invariants that agree with all proposed conjectures. We strongly believe that the concept of family invariants will be placed on a firm theoretical foundation in the future. New KL tables, organized according to KL families, are given in Appendix A that can be downloaded from the address http://www.mi.sanu.ac.yu/vismath/Appendix.pdf. After a short graph-theoretical introduction, we consider different notations for KLs: Gauss, Dowker, and Conway notation, along with their advantages and disadvantages. All basic KL invariants such as the minimum crossing number, minimum writhe, linking number, unknotting or unlinking number, cutting number, and KL properties such as chirality, periodicity, unlinking gap, and braid family representatives of KLs are discussed in Chapter 1. In Chapter 2 we address two important problems: recognition and generation of KLs. As recognition criteria we consider KL colorings, KL groups, and more powerful tools such as polynomial KL invariants. Again, we try to show that polynomial KL invariants can be recovered from the Conway notation and recursively computed for KL families. Chapter 3 contains a short excursion into the history of knot theory and places an emphasis on the beauty, universality, and diversity of knot theory through various non-standard applications such as mirror curves, fullerenes, self-referential systems, and KL automata. We wish to thank Wolfram Research and ICT for supporting our project, Professors Mitsuyuki Ochiai and Noriko Imafuji for their cooperation in the development of the program LinKnot and joint distribution of LinKnot and Knot2000 (K2K), Dror Bar Natan for joining program LinKnot with Mathematica package KnotTheory, and Professors Donald Crowe, Louis Kauffman, Jay Kappraff, Charles Livingston, Jozef Przytycki, and Thomas Gittings for their advice and suggestions. Slavik Jablan and Radmila Sazdanovi´c

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Contents

Preface

v

1. Notation of Knots and Links

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

1.12 1.13 1.14

Basic graph theory . . . . . . . . . . . . . . . . . . . . . . Shadows of KLs . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Gauss and Dowker code . . . . . . . . . . . . . . . KL diagrams . . . . . . . . . . . . . . . . . . . . . . . . . Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . Conway notation . . . . . . . . . . . . . . . . . . . . . . . Classification of KLs . . . . . . . . . . . . . . . . . . . . . LinKnot functions and KL notation . . . . . . . . . . . . Rational world and KL invariants . . . . . . . . . . . . . 1.8.1 Chirality of rational KLs . . . . . . . . . . . . . . Unlinking number and unlinking gap . . . . . . . . . . . . Prime and composite KLs . . . . . . . . . . . . . . . . . . Non-invertible KLs . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Tangle types . . . . . . . . . . . . . . . . . . . . . 1.11.2 Non-invertible pretzel knots . . . . . . . . . . . . 1.11.3 Non-invertible arborescent knots . . . . . . . . . . 1.11.4 Non-invertible polyhedral knots . . . . . . . . . . Reduction of R-tangles . . . . . . . . . . . . . . . . . . . . 1.12.1 KLs with unlinking number one . . . . . . . . . . Braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.1 KLs and braids . . . . . . . . . . . . . . . . . . . Braid family representatives . . . . . . . . . . . . . . . . . 1.14.1 Applications of minimum braids and braid family representatives . . . . . . . . . . . . . . . . . . . . vii

1 10 16 25 40 50 59 66 69 77 81 119 125 131 136 140 142 145 148 157 161 165 179

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1.15 1.16

More KL invariants . . . . . . . . . . . . . . . . . . . . . 182 Borromean links . . . . . . . . . . . . . . . . . . . . . . . 187

2. Recognition and Generation of Knots and Links 2.1 2.2 2.3 2.4 2.5 2.6 2.7

2.8 2.9 2.10 2.11

195

Recognition of KLs . . . . . . . . . . . . . . . . . . . . . 195 2.1.1 Group of KL . . . . . . . . . . . . . . . . . . . . . 201 Polynomial invariants . . . . . . . . . . . . . . . . . . . . 207 Vassiliev invariants . . . . . . . . . . . . . . . . . . . . . . 219 Experimenting with KLs . . . . . . . . . . . . . . . . . . 225 Derivation and classification of KLs . . . . . . . . . . . . 227 Basic polyhedra and polyhedral KLs . . . . . . . . . . . . 241 Basic polyhedra and non-algebraic tangles . . . . . . . . . 268 2.7.1 Generalized tangles . . . . . . . . . . . . . . . . . 282 2.7.2 n-tangles and basic polyhedra . . . . . . . . . . . 283 2.7.3 Non-algebraic tangle compositions and component algebra . . . . . . . . . . . . . . . . . . . . . . . . 295 KL tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 2.8.1 Non-alternating and almost alternating KLs . . . 307 Projections of KLs and chirality . . . . . . . . . . . . . . 311 Families of undetectable KLs . . . . . . . . . . . . . . . . 341 2.10.1 Detecting chirality of KLs by polynomial invariants 356 A dream— new KL tables . . . . . . . . . . . . . . . . . . 363

3. History of Knot Theory and Applications of Knots and Links 3.1 3.2

3.3

History of knot theory . . . . . . . . . . . . . . . Mirror curves . . . . . . . . . . . . . . . . . . . . 3.2.1 Tamil treshold designs . . . . . . . . . . 3.2.2 Tchokwe sand drawings . . . . . . . . . . 3.2.3 Construction of mirror curves . . . . . . 3.2.4 Enumeration of mirror curves . . . . . . 3.2.5 Lunda designs . . . . . . . . . . . . . . . 3.2.6 Polyominoes . . . . . . . . . . . . . . . . 3.2.7 KLs and mirror curves . . . . . . . . . . 3.2.8 Mirror curves on different surfaces . . . . 3.2.9 Mirror curves in art . . . . . . . . . . . . 3.2.10 KLs and self-avoiding curves . . . . . . . KLs and fullerenes . . . . . . . . . . . . . . . . . 3.3.1 General fullerenes, graphs, symmetry and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . isomers

375 375 383 384 385 388 393 395 395 399 400 401 416 426 427

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Contents

3.3.2 3.3.3 3.3.4

3.4 3.5 3.6

5/6 fullerenes . . . . . . . . . . . Knot theory and fullerenes . . . . Nanotubes, conical and biconical their symmetry . . . . . . . . . . 3.3.5 Fullerenes on other surfaces . . . KLs and logic . . . . . . . . . . . . . . . . Waveforms . . . . . . . . . . . . . . . . . Knot automata . . . . . . . . . . . . . . .

. . . . . . . . . . . . fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . and . . . . . . . . . . . . . . .

428 430 436 441 443 449 453

Bibliography

459

Index

475

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Chapter 1

Notation of Knots and Links

1.1

Basic graph theory

This basic introduction to graph theory is written according to the books by R.A. Wilson (2002), N.D. Gilbert and T. Porter (1994), with some changes in definitions, and certain additions. We will start with the definition of a graph: Definition 1.1. A graph G consists of a set of vertices V (G) and a set of edges E(G), such that each edge is incident with two (not necessarily distinct) vertices. A graph G can be denoted by G = G(V, E). Two vertices are adjacent if there exists an edge join them, and they are the endpoints of the edge. Two edges are adjacent if they have a common endpoint. An edge which joins a vertex to itself is called a loop, k edges which join the same pair of vertices are called k-multiple edges, and the corresponding graph is called a multigraph. If a multigraph contains only single and 2-multiple (or double) edges, it is called a 2-graph. A graph is simple if it contains no loops and multiple edges. A graph without loops is called proper, or reduced graph. If we distinguish the order of the endpoints of edges, treating them as ordered pairs of vertices, we obtain oriented graphs (or digraphs). As usual, we will draw graphs with enlarged (labelled) dots for the vertices, and straight or curved lines for edges, in such a way that a vertex and an edge are incident iff they meet in the diagram. The placement of points in the diagram, and whether the lines representing edges are straight, curved or have to cross one another in any point other then vertex, is irrelevant. Definition 1.2. The valence (or degree) of a vertex is the number of edges 1

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which are incident to it (in a graph with loops, we usually count a loop-edge twice). The valence of a vertex v will be denoted d(v). A vertex of a graph G is single or isolated if its valence is 0. Usually, a non-oriented graph without isolated vertices will be given by a list of unordered pairs representing edges. In the case of digraphs, ordered pairs will represent oriented edges. A graph can also be given by its adjacency list whose entries are lists, each starting with a vertex followed by vertices adjacent to it, where the order of adjacent vertices is irrelevant. Definition 1.3. A graph in which all vertices are k-valent is called a kvalent graph (or k-regular graph). Since knots are 1-component links (c = 1), we will use the term “link” or KL (knot or link) for both knots and links, unless we need to talk about properties specific to knots only. A graph is (3, 4)-valent if it contains vertices of valences only 3 or 4. Four-valent graphs will be extremely important for study of KLs as they represent KL shadows. Among the graphs corresponding to five Platonic regular polyhedra, the tetrahedron, cube and dodecahedron graphs are 3-valent, the octahedron graph is 4-valent, and the icosahedron graph is 5-valent (Fig. 1.1). Definition 1.4. A graph is complete if every pair of vertices is adjacent. A graph is bipartite if the vertices can be partitioned into two disjoint sets X and Y such that all the edges join a vertex in X to a vertex in Y . A graph is complete bipartite if it contains all possible edges from a vertex in X to a vertex in Y . The complete graph on n vertices is usually denoted Kn , while the complete bipartite graph on two sets of m and n vertices is denoted Km,n . For example, K5 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}}, K3,3 = {{1, 4}, {1, 5}, {1, 6}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}}.

Definition 1.5. A walk of length n is a sequence v1 e1 v2 e2 ...vn en vn+1 of vertices vi (1 ≤ i ≤ n + 1) and edges ej (1 ≤ j ≤ n) such that each is incident to the next.

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Fig. 1.1 graph.

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(a) Tetrahedron, (b) cube, (c) dodecahedron, (d) octahedron, (e) icosahedron

A walk is closed if v1 = vn+1 , and open otherwise. A trail is a walk in which all edges are distinct, a circuit is a closed trail with at least one edge. A path is a trail in which all vertices are distinct (except v1 and vn+1 if a trail is closed). A cycle is a circuit with all distinct vertices (except v1 and vn+1 ). An Euler’s circuit is a walk that uses each edge of a graph exactly once. Definition 1.6. Two vertices are connected if there is a walk from one to the other. The relation of vertex connectivity is an equivalence relation (reflexive, symmetric and transitive) which partitions a set of vertices V (G) into equivalence classes called (connected) components of G. Definition 1.7. A graph is connected if every pair of vertices is connected (i.e., if all vertices belong to one component).

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A tree is a connected graph with no cycles. The vertex connectivity k(G) of a graph G is the minimum number of vertices that need to be removed together with their incident edges in order to obtain disconnected or 1-vertex graph (in the case when G can not be disconnected by removing vertices). Definition 1.8. A graph is called k-vertex connected (or just k-connected) if at least k vertices need to be removed in order to disconnect the graph, or to obtain a 1-vertex graph. Definition 1.9. A connected graph G is k-edge connected, if at least k edges need to be removed in order to disconnect the graph. The edge connectivity of a graph G is the minimum number of edges which need to be deleted in order to disconnect the graph. A subgraph G′ = G′ (V ′ , E ′ ) of a graph G = G(V, E) is a graph such that V ′ ⊂ V , E ′ ⊂ E, and both endpoints of each edge from E ′ belong to V ′. Definition 1.10. Two graphs G and G′ are isomorphic (G ≃ G′ ) if there is a one-to-one correspondence between their vertices and one-to-one correspondence between their edges, which preserves incidence. If the edge e in G corresponds to the edge e′ in G′ , then the endpoints of e correspond to the endpoints of e′ . Definition 1.11. A graph G is plane if it is drawn in plane (or on the sphere) with no two edges crossing each other, and it is planar if it is isomorphic to a plane graph. A simple planar graph can be embedded in ℜ2 so that each edge is a straight line (for the proof see, e.g., Cromwell, 2004). Stereographic projection caries plane embeddings to embeddings on a sphere and vice versa. Definition 1.12. An embedding of a graph G is a drawing of G on a certain surface in which the edges do not intersect. A non-planar graph can be always embedded on some surface, other then the plane (or sphere). For example, all graphs of polyhedra (Fig. 1.1) are planar, and the graphs K5 and K3,3 are non-planar. Definition 1.13. An automorphism of a graph G is any isomorphism of G to itself.

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All automorphisms of a graph G make its automorphism group denoted as Aut(G). In every graph drawing, edges are drawn by broken and/or smooth lines, where the break points or nugatory edge crossings in graphs which are not plane should not be perceived as vertices. Usually, graph vertices are labelled and/or denoted by dots. Definition 1.14. An embedding of a graph induces a map M : the division of the unbounded surface on which the graph is embedded into disjoint simply-connected regions called faces. A face with two vertices and two edges will be called a bigonal face (or just bigon). The dual D(M ) of a given map M can be constructed in the following way: in the map M you draw a vertex of D(M ) in the interior of each region of M (including the exterior region), and you join them by edges, one edge of D(M ) crossing each edge of M . The graph D(M ) is called the dual of M . In the case of polyhedra and their corresponding graphs, you can join up the points in the interior of adjacent faces of a polyhedron P to obtain the dual polyhedron D(P ). Doing this a second time gets you back to a polyhedron D(D(P )) isomorphic to P . Different planar embeddings of a planar graph G may give different dual graphs D(G), so there exist isomorphic graphs with non-isomorphic duals (Fig. 1.2). A component of graph G is its maximal connected subgraph, and a component of map M is its maximal connected submap. Every embedding can be described by an embedding adjacency matrix. For each entry, the first vertex is followed by a sequence of its adjacent vertices given in the same (left or right) cyclic order. For the end points, only the cyclic permutation corresponding to them is important, and not their particular position in the permutation. For example, a planar embedding of the octahedron graph (Fig. 1.3) O = {{1, 2}, {1, 3}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {3, 5}, {4, 5}, {4, 6}, {5, 6}}

is {{1, 2, 6, 5, 3}, {2, 3, 4, 6, 1}, {3, 1, 5, 4, 2}, {4, 3, 5, 6, 2}, {5, 3, 1, 6, 4}, {6, 2, 4, 5, 1}}.

After drawing the first vertex 1, we draw its incident edges in the right cyclic order: {1, 2}, {1, 6}, {1, 5}, {1, 3}, then we continue with the vertex 2 and its adjacent edges in the same right cyclic order: {2, 3}, {2, 4}, {2, 6},

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Fig. 1.2

Isomorphic graphs (a), (b), and their non-isomorphic duals (a′ ), (b′ ).

Fig. 1.3

The planar embedding of octahedron graph.

{2, 1}, having in mind that {2, 1} is already drawn as {1, 2}, etc., until using all edges of the graph. LinKnot function fPlanarEmbGraph gives the planar embedding of a 3-connected planar graph given by a list of unordered pairs. The output

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is a list that consists of the input graph, its planar embedding, and the faces of the planar embedded graph. The basis of this program is the external program planarity.exe written by J.M. Boyer (Boyer and Myrvold, 2005). The LinKnot function DrawPlanarEmbGraph draws a planar embedding of a graph given by a list of unordered pairs, and the function DrawPlanarEmbKL draws a planar embedding of a KL without multiple edges. The basis of these functions is the program 3-Dimensional Convex Drawings of 3-Connected Planar Graphs by M. Ochiai, N. Imafuji and N. Morimura. In every plane graph drawn in the plane ℜ2 we visually distinguish an external face and internal faces placed inside it. In the octahedron plane graph (Fig. 1.3) the external face is {1, 2, 3}, and other (internal) faces are placed inside it. Sometimes, especially for the plane graphs obtained from symmetric polyhedra, it is useful to imagine them on a sphere S 3 . Theorem 1.1. (Euler’s formula) Every planar map M with v vertices, e edges, f faces and c components satisfies Euler’s formula: v − e + f = c + 1. The term χ = v − e + f obtained from a map on any surface is called the Euler characteristic of the surface. Euler’s formula for polyhedra (c = 1) was discovered around 1750 by L. Euler and first proven by A.M. Legendre in 1794. The interested reader can find 19 different proofs of Euler’s formula in D. Eppstein’s The Geometry Junkyard (http://www.ics.uci.edu/∼ eppstein/junkyard/euler/). A few of them are based on Jordan Curve Theorem, proved by O. Veblen in 1905: Theorem 1.2. (Jordan Curve Theorem) If c is a simple closed curve in ℜ2 , then ℜ2 \ c has two components (an “inside” and “outside”), with the c boundary of each (Jordan, 1887; Veblen, 1905; Hatcher, 2002; Grabowski, 2005). The complete proof of Jordan Curve Theorem is given in Algebraic Topology by A. Hatcher (2002), and the computer proof in the proof checker Mizar required 200 000 lines (Grabowski, 2005). The most celebrated result about the planarity of graphs is PontryaginKuratowski’s Theorem. Two graphs G and G′ are isomorphic modulo ver′′ tices of degree 2 if G is isomorphic to a graph G obtained from G′ by the addition or deletion of vertices with just two incident edges:

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Theorem 1.3. (Pontryagin-Kuratowski’s Theorem) Let G be a finite graph. G is planar iff it contains no subgraph isomorphic modulo vertices of degree 2 to K5 or K3,3 (Kuratowski, 1930). Short proof of the sufficiency part of this theorem is given by Y. Makarychev (1997), and the complete proof can be found, e.g., in Graphs on Surfaces by B. Mohar and C. Thomassen (2001). The transformations described above are a subdivision and a contraction of a graph edge. A subdivision of a graph G is a graph obtained from G by a finite number of the following operations. Let v, w be the vertices of G which are connected by the edge vw. Introduce a new vertex x and replace the edge vw by two edges vx and xw, i.e., insert a vertex x in the middle of an existing edge vw. Definition 1.15. Replacing two adjacent vertices by a single vertex of a graph is an operation called elementary contraction. The new vertex is joined to every other vertex which was joined to one or both original two vertices. A contraction of G is any graph that can be obtained from G by a finite sequence of elementary contractions. The same operations, subdivision and contraction, can be applied on any line segment AB in ℜ3 replacing it by two line segments AC and CB or vice versa. In the language of contraction, Pontryagin-Kuratowski’s Theorem can be formulated as: Theorem 1.4. A graph G is planar iff it contains no subgraph which has K5 or K3,3 as a contraction. In considering KLs, a special kind of contraction where edges forming a bigon (Definition 1.14) are contracted simultaneously will play an important role. We will call such contraction a bigon collapse. As well as a plane or sphere, we may consider other smooth surfaces, which can be orientable, and like a sphere have an inside and an outside, or can be non-orientable, such as the projective plane or Klein bottle. An orientable surface can be thought of as a sphere with g handles (g = 0, 1, 2, . . .), and the number of handles g is the genus of the surface. For a torus or

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“sphere with a handle”, g = 1, for a double torus g = 2, etc. The simplest non-orientable surface is the projective plane, which may be regarded as a sphere with antipodes identified, or as a hemisphere with opposite peripheral points identified, or as a sphere with a cross-cap. For a non-orientable surface without boundary, the genus g is given by the formula g = 2 − χ, and for an orientable surface without boundary by the formula g = 2−χ 2 (see, e.g., Coxeter and Moser, 1980). Instead of embedding a graph into a plane or sphere, we may try any any other smooth (orientable or non-orientable) surface. An embedding of a graph G in a surface can be constructed by the method known as the Edmonds algorithm, named after J. Edmonds who described it in 1960 (Edmonds, 1960; Gilbert and Porter, 1994). As an input, the LinKnot function fEdmonds uses unoriented graph given by the list of unordered pairs (edges) and calculates its embeddings (given by labelled polygons), the Euler characteristic of the surface and its genus. From the output, we can draw the corresponding embedding. For example, for K3,3 given by the list of unordered pairs, the function fEdmonds gives the result {{{1, 4, 2, 5, 3, 6}, {1, 4, 3, 6, 2, 5}, {1, 5, 3, 4, 2, 6}}, 0, 1}. Since the surface has Euler characteristic 0 and genus 1, according to the classification of surfaces, the graph G is embedded on a torus. Hence, the embedding of the non-planar graph K3,3 on a torus (Fig. 1.4) is given by {{1, 6, 4, 5}, {2, 6, 4, 5}, {3, 6, 4, 5}, {4, 1, 2, 3}, {5, 1, 2, 3}, {6, 1, 2, 3}}. In a similar way we obtain the embedding {{1, 2, 4, 3}, {1, 2, 5, 4}, {1, 3, 2, 5}, {1, 4, 3, 5}, {2, 3, 5, 4}}, of the non-planar graph K5 on a torus. We can also consider colored (or weighted) graphs: Definition 1.16. A vertex k-coloring of a graph is a coloring of the vertices by k colors, and an edge k-coloring is a coloring of the edges by k colors. A coloring with two colors (usually black and white) will be called a (vertex or edge) bicoloring. If the colors are treated as weights assigned to vertices or edges of a graph, such a graph is a weighted graph.

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Fig. 1.4

1.2

Embedding of the graph K3,3 on a torus.

Shadows of KLs

Knots have been used for various purposes from prehistoric times till today, even serving as the basis for mathematical recording systems (e.g., for Inca quipu). Examples of knots can be found in all ancient civilizations, in Chinese art, Celtic art, ethnic Tamil and Tchokwe art, in Arabian, Greek or Smyrnian laces... In contemporary science and art, KLs can be found in DNA, physics, chemistry, sculpture, etc. Knots in design are the example of modular structures (Jablan, 2002), since they can be composed from only five basic pieces (modules) (Fig. 1.5). Before giving a precise definition of KLs, we can start from an intuitive description of knotting. Given a piece of string (like shoe laces or yarn) we tangle it any way we want. A “mathematical” KL is different from the “real” one since it is closed. A string ends should be glued together. Moreover, mathematical KLs are made of string with no thickness– just a closed curve in 3D-space with no self-intersections. A link is a set of several disjoint tangled knots. Definition 1.17. A knot is a smooth embedding of a circle S 1 into Euclidean 3-dimensional space ℜ3 (or the 3-dimensional sphere S 3 ), and a ccomponent link is a smooth embedding of c disjoint copies of a circle S 1 into ℜ3 (or S 3 ), where the embeddings of circles Si1 are its components (i = 1, 2, . . . , c).

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Fig. 1.5

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KLs as modular structures.

Requiring an embedding to be smooth insures that wild knots or links (Fig. 1.6b) are excluded from the definition, i.e., we work only with tame KLs. Two more examples of wild embeddings are Alexander’s horned sphere and Antoine’s necklace. Another way of avoiding problems with wild knots is using polygonal (piecewise linear) KLs that are finite by nature. R.H. Crowell and R.H. Fox (1965) proved the equivalence of the smooth-curve and piecewise linear approach. The main topic of knot theory is classifying KLs. Therefore we introduce the relation of ambient isotopy. Two links L and L1 are ambient isotopic iff there is a continuous movement (or deformation) of space S 3 that transforms L into L1 . More precisely, L and L1 are ambient isotopic if one can be transformed to the other by a diffeomorphism of the ambient space onto itself, where a diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse. In other words, transformation of L to L1 has to be smooth with smooth inverse, i.e., tearing a thread and regluing, as well as shrinking one part of a link to a point is not allowed. If we imagine that the curves defining a link are made of flexible and elastic thread, then the ambient isotopy is equivalent to allowing the threads to be twisted and moved continuously in space (cutting and gluing back together is not allowed). In order to give a precise definition of ambient isotopy we need the following mathematical background:

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Definition 1.18. Let X be a non-empty set. A topology on X is a collection of subsets of X, called open sets, satisfying: (1) the empty set and X itself are open sets; (2) an arbitrary union of open sets is an open set; (3) the intersection of finitely many open sets is an open set. The set X together with a collection of open sets satisfying the rules (1), (2), (3) is a topological space. Definition 1.19. Let X and Y be topological spaces. A function f : X → Y is continuous if, for every open set V in Y , its origin f −1 (V ) is open in X. A function f : X → Y which is bijective, continuous, and has a continuous inverse f −1 : Y → X is called a homeomorphism. A function f : X → Y which is injective, continuous, and such that the bijection f : X → f (X) has a continuous inverse is called an embedding. Definition 1.20. Knots K and K1 are ambient-isotopic if there exists a continuous function H: ℜ3 × [0, 1] → ℜ3 such that: (1) h0 = H((x, y, z), 0) is the identity ℜ3 → ℜ3 ,

(2) for all t ∈ [0, 1], ht = H((x, y, z), t) is a homeomorphism ℜ3 → ℜ3 ,

(3) if h1 = H((x, y, z), 1), then h1 (K) = K1 .

Definition 1.20 provides a continuous sequence of homeomorphisms of ℜ3 from time t = 0 to t = 1. If K and K1 are ambient-isotopic, then knots ht (K) (t ∈ [0, 1]) represent a continuous deformation from K into K1 . This definition can be extended to links in a natural way (Gilbert and Porter, 1994). In order to distinguish KLs we need to know what are the embeddings of KLs in ℜ3 or S 3 , (i.e., embeddings of homeomorphic images of circles representing them). Hence, we can try something different: to compare their complements with regard to ℜ3 (or S 3 ). It is easy to see that if two links L1 and L2 are ambient isotopic, their complements are homeomorphic. Also, a link and its mirror image have homeomorphic complements. Does inverse statement hold? Theorem 1.5. Two knots K1 and K2 are ambient isotopic iff their complements are homeomorphic (Gordon and Luecke, 1989).

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However, this statement is not true for links; according to the famous Whitehead example, two links that are not ambient isotopic can have homeomorphic complements (Fig. 1.6a). Whitehead proved that there are infinitely many links with the same complement as the Whitehead link (Fig. 1.6a, left) (Whitehead, 1937; Gordon, 2002). In order to avoid the necessity of introducing differentiable or smooth curves, as well to avoid some peculiar cases such as wild KLs (Fig. 1.6b), we can think about KLs as piecewise linear. Definition 1.21. A link consisting of c closed polygonal lines in ℜ3 is called a polygonal link. A link L is called tame if it is ambient isotopic to a polygonal link, and wild otherwise. All smooth KLs are tame, and the polygonal (piece-wise linear) knot theory approach is equivalent to the smooth-curve approach (Crowel and Fox, 1965; Burde and Zieschang, 1985). For polygonal KLs a elementary planar isotopy is achieved either by subdividing an edge AB by the vertex C, or by applying a contraction on AC and CB. An ambient isotopy for a polygonal KL is a finite sequence of elementary isotopies.

Fig. 1.6

(a) Different links with homeomorphic complements; (b) wild knot.

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Having this in mind, we can define a link, or more precisely, a representation of a link as a collection of c disjoint closed polygonal curves tangled together in space. Hence, a link is a class of its equivalent representations, where the equivalence relation is ambient isotopy. The wild knot (Fig. 1.6b) has a single isolated pathological point, toward which a succession of smaller and smaller knots converges. In a similar way, wild knots with a finite number of pathological points can be constructed, but one can go further and construct a wild knot with an infinite, and even uncountable set of pathological points that represents the Cantor continuum (Sossinsky, 2002). Perhaps the play of shadows on the wall of a cave was the first movie in the history of mankind– let us do the same with KLs and their ambient isotopies, i.e., take their shadows. A link shadow L′ is the orthogonal projection of a link L onto the plane ℜ2 .

Fig. 1.7

Knot shadow.

Let p denote the projection of R3 onto a plane, and L′ = p(L) the projection of an arbitrary link L. Except in special cases, when L is an unknot or unlink in general position, curve representing L′ will have at least finite number intersection points, i.e., double points. The word “at least” deserves special attention: for every link L we need to obtain its regular (or generic) shadow– a shadow without catastrophes, or degenerations (Sossinsky, 2002). An intersection of a generic projection should be transverse (not tangent).

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Definition 1.22. Let L be a polygonal link. Its shadow L′ is regular if the following conditions are satisfied: (1) the shadow is a 4-valent (or 4-regular) graph; (2) if Q′ is a point in L′ , then the inverse image p−1 (Q′ ) ∩ L has at most two points. If it has two points, they must be the interior points of two distinct edges of L, and Q′ is the vertex (or crossing) in L′ . In other words, Q′ can have only one inverse image. This means that every vertex of a shadow will be 4-valent, two or more vertices of the polygonal link L can not be projected to the same point, vertices of L′ can not belong to an interior point of any edge, three or more points can not be projected to the same point, and projections of different edges of L or their parts can not coincide in L′ . The forbidden situations can be removed by a suitable choice of projection plane and/or by slightly displacing the vertices of the polygonal link L. As with any other graph, a shadow of a KL can be given by a list of unordered pairs, or by an adjacency list, but the information from which you can draw regular shadow is given by the code of planar embedding of its graph. From the graph theory point of view, link shadow L′ is a 4-valent (or 4-regular) plane graph. The most important question is how much information about link L can be obtained from its shadow L′ . As an example, we will describe how to determine number of components of a link L from its shadow L′ , according to the following rules: Component Algorithm • Choose an arbitrary oriented edge (x, y) in L′ . Choose the middle of the three remaining edges incident with y and orient it so that its beginning point is y. Repeat the same rule until closing a component of L′ . • Choose a new edge and repeat the same procedure until using all edges of the graph L′ . Using all edges means that we have traced all components of L′ and obtained one of the simplest KL invariants– the component number c. From every 4-valent plane graph L′ there is a link L, such that L′ is its shadow, and the number of circuits c in L′ obtained by the Component Algorithm is the number of components of L. The LinKnot function fComponentNo calculates the number of components of any KL.

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So far, all knot and link tables have been organized according to the number of components of KLs. In order to make classification of KLs and find common properties of knots and links, we will try to avoid this approach as long as possible. 1.2.1

Gauss and Dowker code

C.F. Gauss was the first to consider knots as mathematical entities. Gauss made codes of immersed curves by assigning letters to the crossing points of a self-intersecting curve and determined “words” defining a closed curve. Gauss problem of characterizing codes with planar immersions and representing projections of knots has an extensive literature and many solutions (Dehn, 1936; Treybig, 1968; Marx, 1969; Lov´asz and Marx, 1976; Read and Rosenstiehl, 1976; de Fraysseix and Ossona de Mendez, 1999). Giving a labelled plane graph L′ and applying Component Algorithm, while keeping trace of vertices that were visited, we obtain the Gauss code of L′ : the list of vertex sequences divided into circuits. For example, from the graph given by the plane embedding adjacency list {{1, 2, 2, 3, 4}, {2, 1, 1, 4, 3}, {3, 1, 2, 4, 4}, {4, 3, 3, 2, 1}},

starting from the vertex 1 and the second oriented edge (1,2), we obtain the Gauss code {{1, 2, 4, 3, 2, 1, 3, 4}}. All edges of the graph are exhausted in one circuit, so our graph is a shadow of a knot, namely the shadow of a figure-eight knot (knot 41 in the classical notation, where a symbol of the form nji denotes ith KL with n crossings and j components, and for knots the upper index j = 1 is omitted; for knots, this notation is known as the Alexander-Briggs notation) (Alexander and Briggs, 1926-27; Rolfsen, 1976). Starting from the same vertex by the first oriented edge (1, 2) we obtain the Gauss code {{1, 2, 3, 4, 2, 1, 4, 3}}, and so on. Notice that Gauss code is not unique and depends on the choice of the beginning point and oriented edge incident to it (Fig. 1.8). From the octahedron graph {{1, 2, 6, 5, 3}, {2, 3, 4, 6, 1}, {3, 1, 5, 4, 2}, {4, 3, 5, 6, 2}, {5, 3, 1, 6, 4}, {6, 2, 4, 5, 1}},

beginning from 1 and (1, 2), then from 1 and (1, 6), and finally from 2 and (2, 6) we obtain the Gauss code {{1, 2, 4, 5}, {1, 6, 4, 3}, {2, 6, 5, 3}},

so it is a shadow of well known 3-component link– Borromean rings 632 (Fig. 1.9).

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Fig. 1.8

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The Gauss code of figure-eight knot.

The other way round, graph of a KL can be recovered from its Gauss code. Unoriented edges of KL graph are given by unordered pairs of adjacent numbers in each component of the Gauss code (where the first and the last number in every component are also adjacent) (see Fig. 1.10). The Gauss code of a KL is invariant with regard to a change of the order of components, cyclic rotation and reversing components (if we are not interested in the orientation of components). According to the Component Algorithm, for every component we choose a beginning point, and a first edge which induces the orientation of a whole component. For a knot shadow with n crossings we have n choices for the beginning point and 4 choices for direction, total of 4n possibilities. In the case of links, this number grows with the number of components. At the beginning of every classification the natural problem arises: to enumerate all possible objects which can be obtained. Inspired by William

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Fig. 1.9

Fig. 1.10

The Gauss code of Borromean rings.

Drawing figure-eight shadow from its Gauss code.

Thomson’s (Lord Kelvin’s) vortex theory (1867), his friend P.G. Tait started with knot tabulation. He established cooperation with the Reverend Thomas Penyngton Kirkman, who spend about 30 years considering combinatorial problems in graph theory. Kirkman translated knot enumeration to a problem of enumerating 4-regular planar graphs– shadows of knots. In his first paper sent to Tait in May 1884 Kirkman enumerated all knot projections with n ≤ 10 crossings. Tait used these tables to extract different alternating knots with n ≤ 10 crossings. Before publishing their tables, Tait received an enumeration of knots up to 10 crossings from C.N. Little, the material from his Ph.D. thesis On Knots, with a Census for Order 10. After correcting one duplication in his own list, and a duplication and omission in Little’s, Tait sent the paper to press. In the meantime, he received Kirkman’s list of 1581 knot projections with n ≤ 11 crossings but decided that determining different knots is too demanding and retired

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from knot tabulation in 1885 (Keller, 2004). T.P. Kirkman (1885a,b) and P.G. Tait (1876-1885) completed the first knot tables of alternating knots with n ≤ 10 crossings. Independently, after six years of work, C.N. Little classified in 1889 non-alternating knots with n ≤ 10 crossings. Tait’s, Kirkman’s and Little’s tables of alternating knots are confirmed as complete, and the only duplicate among 43 non-alternating knots with n = 10 crossings is identified in 1974 (Perko pair). At the time when they finished their work, after the wide recognition of D. Mendeleev’s periodic table of elements by scientific community, almost none was interested in knot tables. After 20 years, mathematicians recovered the subject. Before trying to repeat and extend Tait’s, Kirkman’s and Little’s results (now using computers), it will be useful to try to reduce the number of possibilities by minimizing the number of crossings, and to make the codes of KLs as concise as possible. Similar to the definition of a proper (or reduced) graph, a KL shadow is called proper, or reduced if it has no loops. The Gauss code {{1, 2, 3, 4, 2, 3, 4, 1}} represents a possible knot shadow (you can recognize the shadow of a trefoil with a loop) (Fig. 1.11). In order to delete loops from KL codes, first we need to recognize them. In a Gauss code, the appearance of the same numbers in adjacent places indicates a loop (where the first and the last number in any component are treated as adjacent as well).

Fig. 1.11

The shadow of a trefoil with loop.

From the Gauss code of a KL we can extract another, more concise code– Dowker code of the KL. In the case of knots, we only need to find the positions which the same numbers occupy in the Gauss code. More precisely, information that Gauss code of a knot carries can be written in a more concise form, obtained in the following way:

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• each number appears twice in a Gauss code– extract both positions in an unordered pair; • each pair contains odd and even number– place odd number in the first place; • sort the list of pairs D in the increasing order; • Dowker code of a knot consists of second elements of all pairs. As a consequence of the Jordan Curve Theorem (Theorem 1.2), every crossing in Dowker code is denoted by one odd and one even number. The list D is sorted according to the first members of ordered pairs, this means, according to odd numbers. Hence, we don’t need to work with the whole list, but only with a half of it. For example, in the Gauss code of the figure-eight knot {{1, 2, 4, 3, 2, 1, 3, 4}} the positions of the number 1 are {1, 6}, the positions of the number 2 are {2, 5}, the positions of the number 3 are {4, 7}, and the positions of the number 4 are {3, 8}. In every pair we put an odd number at the first position ({1, 6}, {5, 2}, {7, 4}, {3, 8}), and then sort the list of the ordered pairs obtained, so the result is D = {{1, 6}, {3, 8}, {5, 2}, {7, 4}}. By reading every second member of the ordered pairs from the list D we obtain the Dowker code {{6, 8, 2, 4}}. Since figure-eight knot has only one component and 4 crossings, we write it’s Dowker code in the form Dow = {{4}, {6, 8, 2, 4}}, where the first part is the number of vertices of the component. With a minor modification, this construction can be extended to links. If pairs of numbers of the same oddity appear in some component, rotate cyclically this component for one place. When parity is fixed for all components, after prepending a list with lengths of all components (where the length of each component is a half of its length in the Gauss code), we obtain Dowker code of a link. For example, from the Gauss code of Borromean rings {{1, 2, 4, 5}, {1, 6, 4, 3}, {2, 6, 5, 3}}, from the unordered pair {1, 5} we immediately conclude that the second component must be rotated one place to the right, so we obtain {{1, 2, 4, 5}, {3, 1, 6, 4}, {2, 6, 5, 3}}. The parity is now fixed, so D = {{1, 6}, {3, 8}, {5, 12}, {7, 10}, {9, 2}, {11, 4}}, and the Dowker code is Dow = {{2, 2, 2}, {6, 8, 12, 10, 2, 4}} (Fig. 1.12). The same result can be obtained directly from a plane graph of a knot if we enumerate vertices visited in Component Algorithm by 1, 2,. . ., 2n. Then, every point of a knot shadow will be denoted by two numbers: one even and the other odd. After that, we work in the same way as before. In the case of links we enumerate first one component, then we continue to

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Fig. 1.12

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The Dowker code of Borromean rings.

enumerate the second, beginning from its point that is visited at the most once. If the parity is disturbed for some component, we continue from the next number (that is the same as a cyclic rotation of a component used before). Applying the same rule until two numbers, one odd and the other even, are assigned to every crossing, we obtain a Dowker code of a KL (Fig. 1.13a). Dowker code of a KL is sufficient for drawing its corresponding shadow (Fig. 1.13b).

Fig. 1.13 (a) Obtaining the Dowker code of figure-eight knot from its shadow; (b) drawing the shadow from the Dowker code of the same knot.

In order to recognize loops in Dowker codes of knots with n crossings we need to go one step back and look at the list of ordered pairs: loops are represented by pairs of successive numbers (where the numbers 1 and 2n are treated as successive as well). However, this criterion can not be

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applied to links. For example, the Dowker code {{1, 1}, {4, 2}} denotes a shadow of a Hopf link 221 (2 in Conway notation) without loops. Now we have all necessary definitions and notions to pursue Kirkman’s approach to systematic classification and enumeration of all alternating knot diagrams with n crossings. In solving this problem, Dowker and Gauss codes will be used. In order to classify all KLs with n crossings first we create all distinct permutations of even numbers 2,4, . . ., 2n and all their partitions into 1, 2, . . ., n parts and obtain all possible Dowker codes with n crossings. Then we delete from them non-proper codes. The result obtained are all possible potential Dowker codes of KL shadows with n crossings. We emphasize “potential” because not all of them are necessarily realizable. For example, it is impossible to draw the potential Dowker code {{5}, {8, 10, 2, 4, 6}} (Fig. 1.14a), since it is not planar graph. In fact, it is K5 . Moreover, a Dowker code can be non-realizable even if its corresponding graph is planar. For example, to the non-realizable Dowker code {{6}, {4, 6, 8, 10, 12, 2}} corresponds the (non-realizable) Gauss code {1, 2, 3, 1, 4, 3, 5, 4, 6, 5, 2, 6}, and the planar graph

{{1, 2}, {1, 3}, {1, 4}, {1, 6}, {2, 3}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 5}, {4, 6}, {5, 6}}

which is realizable as the link {{2, 2, 2}, {6, 8, 10, 12, 2, 4}}. A bit strange, but a natural question is: where can a non-realizable Dowker code be realized? The answer is: on some surfaces other than plane ℜ2 (or sphere S 2 ). The graph K5 corresponding to the code {{5}, {8, 10, 2, 4, 6}} can be embedded on a torus, and represents a shadow of a 3-component link (Fig. 1.14b). One of its Gauss codes is {{1, 5, 3}, {2, 5, 4}, {1, 2, 3, 4}}, but its Dowker code does not exist (because Jordan Curve Theorem holds only in ℜ2 or S 2 ). In fact, this Gauss code represents Borromean rings. The virtual knot theory introduced by L. Kauffman (Kauffman, 1997, 1999, 2000, 2001; Green, 2004; Manturov, 2002, 2003, 2004; Zin-Justin and Zuber, 2004; Zinn-Justin, 2006) is a “non-realizable” part of the knot theory and gives the alternative answer to the question about realizability of Dowker codes. By projecting four-valent graphs onto ℜ2 or S 2 , virtual crossings are intersection points in the projection which are not vertices of the original graph. For example, graph on a torus with one vertex corresponds to the

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Fig. 1.14 (a) An attempt to draw a knot shadow defined by the potential Dowker code {{5}, {8, 10, 2, 4, 6}}; (b) the result obtained by applying the Component Algorithm on the embedding of non-planar graph K5 on a torus (that we obtain by identifying opposite edges of the square); (c) virtual links obtained from Hopf link and Borromean rings.

Hopf link, and Borromean rings can be represented as a five-vertex graph on a torus. Projection of a Hopf link onto ℜ2 has two vertices, where one vertex is the image of the vertex of the original graph, and the other is the “new” virtual vertex. The vertex of the projection corresponding to the vertex of original graph is called classical, and the other vertices are virtual. By introducing in classical vertices the relation “over-under”, we obtain virtual KL-diagrams: Definition 1.23. A virtual link diagram is a 4-valent plane graph of the following structure: each vertex has an overcrossing or undercrossing, or is marked by a virtual crossing. The equivalence of virtual KLs will be described later, after introducing generalized Reidemeister moves for virtual KLs (Definition 1.40). In the spirit of E.A. Abbott’s Flatland: A Romance of Many Dimensions (1884) we wonder how would knot and link tables look like if they were made by people living on torus? Hopf link will be the only one-crossing link, Borromean rings will belong to five-crossing (and not six-crossing) links, and many “virtual knots” will be totaly “real” for “Torus-landers”– they will have only “classical” crossings (Fig. 1.14c).

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Fig. 1.15

(a) The graph of figure-eight knot; (b) the graph of Borromean rings.

The LinKnot function fDowCodes calculates all Dowker codes of KLs realizable in the plane ℜ2 (or on the sphere S 2 ). The first step is deriving all distinct permutations of the set {2, 4, . . . , 2n}. Next, to decide length and number of components, it makes all partitions of each permutation. Then it creates the corresponding Dowker codes, and checks their realizability using Dowker-Thistlethwaite algorithm (Dowker and Thistlethwaite, 1983), extended to links by H. Doll and J. Hoste (1991). From a shadow of a KL we can create another plane graph– a graph of a link, in the following way: first color every other region of the KL shadow black or white, so that the infinite outermost region is black. In the checker-board coloring (or Tait coloring) of the plane obtained, put a vertex at the center of each white region. Then connect any two vertices that are in regions which share a crossing with an edge containing this crossing. The result obtained is the graph of the KL corresponding to a

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particular KL shadow (Fig. 1.15). There is a one-to-one correspondence between KL shadows and graphs of KLs. The LinKnot function fGraphKL calculates and draws a graph of any given KL. The output is a graph of a KL given by the list of unordered pairs and its drawing. 1.3

KL diagrams

In order to obtain a KL shadow, we project a 3-D link L onto the plane ℜ2 . To avoid the loss of the information on three-dimensionality of L, we need to introduce in each crossing the relation “over-under” and draw KL shadows. The pictures obtained in this way are link diagrams or link projections. The notion of a proper shadow can be directly transferred to a proper diagram (or reduced diagram)– a KL diagram without loops. Let us now take an arbitrary set of points P1 , P2 ,. . ., Pc belonging to shadows of distinct components on a diagram L′ of a link L, and move them each along the corresponding component shadow. If every point Pi (i = 1, . . . , c), travelling around the component in a fixed direction meets crossings that alternate between over and under, the diagram L′ is called an alternating diagram. Definition 1.24. A link L that has at least one alternating diagram is called an alternating link (Fig. 1.16). Otherwise, it is a non-alternating link. At the beginning of knot theory, all knots were thought to be alternating. The simplest non-alternating knots occur among 8-crossing knots, and simplest non-alternating link is the 6-crossing link 633 (2, 2, −2) (Figs. 1.17-1.18). If we want to prove that a KL is alternating, we need to find its alternating diagram. It is, by no means, trivial to prove that this can be achieved, since there is infinite number of diagrams representing each link. Among all diagrams of a link L we can distinguish those with a minimal number of crossings. Definition 1.25. If n is the least number of crossings in any projection of L, it is called minimum crossing number (or just crossing number) of L. Regular (generic) KL-shadows (without splitting points) are already defined (Definition 1.22) (see also, e.g., Murasugi, 1996, page 26; Sosinsky, 2002, pages 38-39). We will consider only regular proper KL diagrams

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Fig. 1.16

Fig. 1.17

The projection of Borromean rings.

The first non-alternating knot 819 (3, 3, −2).

(regular KL diagrams without loops): if there is no explicit remark that some KL diagram is non-proper, the term “diagram of a KL” means “a regular proper diagram”. The most important property of alternating diagrams is that they are minimal: each alternating diagram of a link L is reduced to a minimal number of crossings. In 1986, L. Kauffman, K. Murasugi and M. Thistlethwaite used properties of the Jones and Kauffman polynomial to independently prove the famous Tait’s First Conjecture: Theorem 1.6. An alternating KL in a reduced alternating projection of

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The first non-alternating link 633 (2, 2, −2).

n crossings has crossing number n (Kauffman, 1987b; Murasugi, 1987a,b; Thistlethwaite, 1987, 1988). This theorem is now known as Kauffman-Murasugi Theorem. Its proof is given also in the book Knot Theory and its Applications (Murasugi, 1996, Theorem 11.5.5), and a new proof based on the notion of atom is given by V. Manturov (2004, Chapter 15). In general, it is very difficult to determine the crossing number of a given KL. Given a non-alternating projection of a KL with k crossings, can we even hope to prove that this KL has a projection with fewer than k crossings? How to find the crossing number n and prove that it is really the minimal number of crossings for a given link L? In fact, we are able to answer this question only for a few special special KL classes (such as alternating, torus, and stellar KLs). The relation “over-under” can be included in a Gauss code by using overlined and underlined numbers. In a Gauss code of a KL every number appears twice. Now it will appear once as overlined, and once as underlined. In the case of an alternating KL, we will have an alternating sequence “overunder” (or “under-over”) for each component. For example, to the trefoil knot diagram corresponds the Gauss code {1, 2, 3, 1, 2, 3} or {1, 2, 3, 1, 2, 3}, where one of the alternating KL diagrams is always a mirror image of the other in a mirror reflection plane coinciding with the projection plane ℜ2 (Fig 1.19). Definition 1.26. A KL is achiral (or amphicheiral) if it is ambient isotopic to its mirror image. Otherwise it is chiral.

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Fig. 1.19

Fig. 1.20

Trefoil knot diagram.

The mirror image of the trefoil knot diagram from Fig. 1.19.

Trefoil knot is an example of a chiral knot, and the figure-eight knot and Borromean rings are achiral. Gauss code of Borromean rings is {{1, 2, 4, 5}, {3, 1, 6, 4}, {2, 6, 5, 3}} or {{1, 2, 4, 5}, {3, 1, 6, 4}, {2, 6, 5, 3}}. In translating them to Dowker codes, for a trefoil knot we obtain from the first Gauss code the list of ordered pairs ({1, 4},{2, 5},{3, 6}}, the sorted list D = {{1, 4}, {3, 6}, {5, 2}}, and the corresponding Dowker code {{3}, {4, 6, 2}}. From the second Gauss code we obtain the list of ordered pairs ({1, 4},{2, 5},{3, 6}}, the sorted list D = {{1, 4}, {3, 6}, {5, 2}}, and the Dowker code {{3}, {4, 6, 2}}. In the same way, from the Gauss codes of Borromean rings we obtain the first sorted list divided into components D = {{{1, 6}, {3, 8}}, {{5, 12}, {7, 10}}, {{9, 2}, {11, 4}}},

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and the second divided sorted list D = {{{1, 6}, {3, 8}}, {{5, 12}, {7, 10}}, {{9, 2}, {11, 4}}},

so the corresponding Dowker codes are {{2, 2, 2}, {6, 8, 12, 10, 2, 4}} and {{2, 2, 2}, {6, 8, 12, 10, 2, 4}}, respectively. Notice that for alternating diagrams all Dowker codes are “lower” or “upper”, meaning that all the numbers in the code are underlined or overlined, in contrast to non-alternating diagrams when some of them will be underlined, and others overlined. Unless indicated explicitly, all KL projections are considered on a sphere 3 S , and not on the plane. Definition 1.27. Two KL projections L′ and L′′ are sense-preserving isomorphic iff there is an isomorphism of their corresponding graphs preserving the relation “over-under”. If the handedness of the projections (“left” or “right”) is irrelevant, we do not care whether the isomorphism preserves or reverses the relation “over-under” (Figs. 1.21-1.22). For example, every KL projection and its mirror image will be isomorphic.

Fig. 1.21

Two non-isomorphic projections of the same knot 75 (Figs. 1.21, 1.22).

The Knot 2000 (K2K) function GetMirrorImageKnot generates P data of the mirror image of any KL. The first tables of knots were made experimentally by Tait, Kirkman and Little in the second half of XIX century (Tait, 1876/77a,b,c, 1883/84, 1884/85; Kirkman, 1885a,b; Little, 1885, 1890, 1892, 1900). The first book which contains knot tables is Knotentheorie by K. Reidemeister (1932)

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Fig. 1.22

Another 7-crossing projection of the knot 75 from Fig. 1.21.

(Fig. 1.23). These tables list knots classified according to their crossing number. Every knot is presented by one minimal projection and denoted by ordering number (in the Alexander-Briggs notation, 1926-27). It’s unclear how Reidemeister selected minimal projections of those knots which have more then one such. On the list of n-crossing knots, for each n, alternating knots precede non-alternating knots. Most of the knot theory books closely follow Reidemeister tables: knot projections are just redrawn, sometimes turned upside down (e.g., 76 ), and never changed into a different minimal projection of the same knot. The most influential book containing extensive tables of KLs, and the first to use Conway notation from his seminal paper (1970), is D. Rolfsen’s Knots and Links (1976). The only duplicate in his tables was found by K. Perko (the famous Perko pair, Fig. 1.26), who also corrected Conway’s eleven crossing knot tables, where four knots were omitted (Perko, 1974, 1982). Definition 1.28. The KL notation that follows Alexander-Briggs and Reidemeister’s concept will be called the classical notation of KLs, where a symbol of the form nji denotes ith knot or link with n crossings and j components, and for knots the upper index j = 1 is omitted. Although the classical notation does not give algebraic information about a link, except for the number of crossings and the number of components, it is the most commonly used. The first non-alternating knot diagram, labelled 819 , has 8 crossings. A possible pair of Gauss codes for this diagram and its mirror image is {{8}, {1, 2, 3, 4, 8, 6, 2, 3, 5, 8, 7, 1, 4, 5, 6, 7}},

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Fig. 1.23

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A part of Reidemeister’s tables.

and {{8}, {1, 2, 3, 4, 8, 6, 2, 3, 5, 8, 7, 1, 4, 5, 6, 7}}, respectively. In the same way, for the first non-alternating link, denoted 633 , Gauss codes can be {{1, 2, 3, 4}, {4, 3, 5, 6}, {2, 5, 6, 1}} and {{1, 2, 3, 4}, {4, 3, 5, 6}, {2, 5, 6, 1}}. In order to simplify notation, instead of labelling over- and undercrossings in a Dowker code we will mark only crossings that differ from a pattern corresponding to the alternating KL with the same numerical code, and mark such crossings with a minus sign. In this way, if we don’t need to distinguish a KL projection from its mirror image, alternating KLs will

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be denoted by a Dowker code with only positive numbers, and for nonalternating KLs some entries will be negative. For example, the alternating projection of the figure-eight knot and its mirror image will have the same code {{4}, {6, 8, 2, 4}}; the non-alternating projection of the knot 819 (Fig. 1.17) and its mirror image has the code {{8}, {12, 14, 10, −16, 4, 6, 2, −8}}. Likewise, the code {{2, 2, 2}, {8, −10, 2, 12, 6, −4}} is assigned to the nonalternating projection of the link 633 (Fig. 1.18) and its mirror image. The codes obtained will be called DT-codes (where DT comes from DowkerThistlethwaite), or Dowker codes in Knotscape notation (according to the computer program Knotscape where that notation is used). It is important to underline that DT-codes essentially differ from a Dowker code with signs that we will introduce now. Definition 1.29. A link projection is called oriented if an orientation is assigned to each component. Let L′ denote an oriented diagram of a link L. A vertex is negative (“left”) if it appears in the form shown in Fig. 1.24a, and positive (“right”) if it appears in the form shown in Fig. 1.24b. Every vertex of L′ belongs to one of two types mentioned, so we can obtain a vertex-colored KL diagram by setting “left”-“right” = “black”-“white”. If we label white (positive) vertices by 1, and black (negative) by −1, we obtain a labelled diagram from which we can directly read a Gauss code with signs (or simply, Gauss code), and a Dowker code with signs (or simply, Dowker code). From here and in the sequel, if there is no other explicit remark, the terms Gauss code and Dowker code will be used only in the sense: Gauss code and Dowker code with signs. After introducing signs we have a one-to-one correspondence between oriented KL projections and their Dowker codes. For example, it enables us to distinguish knots from Fig. 1.25. If we orient them, the Dowker code of the first will be {{6}, {4, 6, 2, 10, 12, 8}}, and the Dowker code of the other {{6}, {4, 6, 2, −10, −12, −8}}. Definition 1.30. A link is called prime if each of its shadows represents a graph which is at least 3-edge connected. A link which is not prime is called a composite link. In other words, a prime KL can not be represented by some 2-edge connected shadow. The LinKnot function fPrimeKL checks if a KL is prime, giving as the output 1 for prime, and 0 for composite KLs. In a similar way, the function fPrimeGraph tests whether the alternating

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(a) A vertex with the sign −1; (b) a vertex with the sign +1.

KL obtained from a graph given by a list of unordered pairs is prime or composite (the output is 1 for prime, and 0 for a composite KL). For prime knots, the information contained in a DT-code is sufficient to draw the corresponding KL projection (or its mirror image). However, ambiguity occurs in the case of alternating KLs that are not prime. For example, the two different composite knots in Fig. 1.25 have the same DT-code {{6}, {4, 6, 2, 10, 12, 8}}.

Fig. 1.25

Two different composite knots with the same Dowker codes.

Introducing signs gives rise to a new invariant of alternating KLs– the writhe.

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Definition 1.31. The sum of the signs of the crossing points of a KL diagram L′ of a link L is called the writhe, and denoted as w(L′ ). The Second Tait’s conjecture holds for alternating KLs: Theorem 1.7. If L′ and L′′ are two reduced alternating KL diagrams of the same link L, then w(L′ ) = w(L′′ ) (Murasugi, 1996, Theorem 11.5.6). The proof follows directly from the proof of the Kauffman-Murasugi Theorem (Theorem 1.6), or the Tait’s Flyping Theorem (Theorem 1.11). The following theorem is the consequence of the Theorem 1.7. Theorem 1.8. For every achiral alternating knot K, w(K) = 0. Proof. If D = D(K) is a reduced alternating diagram of achiral alternating knot K, and D∗ = D(K ∗ ) is reduced alternating diagram of its mirror-image K ∗ , then w(D∗ ) = −w(D). Since K is ambient isotopic to K ∗ , w(K) = w(D) = w(K ∗ ) = w(D∗ ) = −w(D), so w(D) = −w(D), and w(D) = w(K) = 0. Corollary 1.1. Every alternating odd crossing number knot is achiral. Writhe can be computed with Knot 2000 (K2K) function Writhe KnotFromPdata. For example, the writhe of the “right” trefoil knot is 3, the writhe of the left trefoil is −3, and the writhe of the composite knots in Fig. 1.25 is 6 and 0, respectively, which makes it easy to distinguish them. In the case of non-alternating knots, two different minimal projections of the same knot can have a different writhe. The first such example is the Perko pair (Fig. 1.26). Writhe is not an invariant of an oriented link, since changing orientation of one component changes signs of all its crossings with other components, while other crossings remain the same. Hence, we need a different invariant for links– the linking number. Definition 1.32. Let c1 and c2 be two components of a link L. The linking number of these two components is the absolute value of the sum of signs of their crossings divided by 2. The linking number of a link L is the sum of the linking numbers of all its components. Theorem 1.9. Linking number of a link is invariant under ambient isotopy (see, e.g., Murasugi, 1996, Theorem 4.5.1).

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Fig. 1.26 Perko pair of knots 3 : −2 0 : −2 0 = 2 1 : −2 0 : −2 0, first with the writhe w = 8, and the other with w = 10.

Usually, the invariance is checked on link diagrams using combinatorial moves, called Reidemeister moves (see page 42). The idea of the proof is the following: a linking number is a link invariant since it is preserved under Reidemeister moves: Ω1 affects only one component, for Ω2 the sum of signs of crossings of two different components is 0, so linking number remains unaffected, and for Ω3 we need to consider all possible orientations of components and show that the linking number remains preserved. For example, the linking number of Borromean rings is 0. The LinKnot function LinkingNo calculates the linking number for any given link. Here is a brief description of the algorithm. Take a KL shadow with n crossings. Then sprinkle signs onto vertices in all possible ways and obtain 2n states of the KL shadow. Each of them represents a projection of some KL. What will happen if we leave some vertices of a KL shadow unsigned? In this case one must consider singular links that differ from true KLs since they contain double points where one part of the KL cuts another part transversally. Singular KLs are KLs with intersections. Their projections are called special projections and play a great role in the construction of Vassiliev invariants. The number of all special projections which can be obtained from a KL shadow with n vertices is 3n . For any KL given by its Conway symbol, the LinKnot function fGen Sign computes the signs of the crossing points in the order corresponding to the Dowker code or P -data of a given KL. The function fGaussExtSigns calculates the Gauss code with signs for a KL projection given by its Conway symbol, Dowker code, or P -data. The function fSignsKL calculates

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the Dowker code with signs of a KL given by its Dowker code in Knotscape form (or DT-code). For an alternating KL, an input is the Dowker code without signs, and for a non-alternating KL, an input is the Dowker code containing only signs of crossings with signs changed with regard to the corresponding alternating KL. The output is a Dowker code with signs. The function fKnotscapeDow calculates from a Conway symbol of a KL its Dowker code in the Knotscape format: Dowker code without signs for an alternating KL, or Dowker code with signs of changed crossings for a non-alternating KL. The most complete, and at the same time, the most concise code is P-data, which has the same form as a Dowker code. P data gives a numerical code, signs of crossing points, and for a nonalternating KL the information about crossings with relation “over-under” changed with regard to the corresponding alternating KL. The program Knot 2000 (K2K) uses P -data to internally represent a KL. Let us suppose that we have already computed for some KL its Dowker code with signs Dow, and its DT-code. For example, for the nonalternating knot 819 (Fig. 1.17), Dow={{8}, {12, 14, 10, 16, 4, 6, 2, 8}}, and DT={{8}, {12, 14, 10, −16, 4, 6, 2, −8}}. From Dow we can obtain a list of ordered pairs, where each crossing is labelled by two numbers: odd and even. In our example, that list is: D = {{1, 12}, {3, 14}, {5, 10}, {7, 16}, {9, 4}, {11, 6}, {13, 2}, {15, 8}}. If in DT the k th number (k = 1, . . ., n) is negative, we need to reverse the order of the numbers of the k th ordered pair in D, leaving signs in D at their places. Because the positions of negative numbers in DT are {4, 8}, after reversing the 4th and 8th ordered pair in D, we obtain the list {{1, 12}, {3, 14}, {5, 10}, {16, 7}, {9, 4}, {11, 6}, {13, 2}, {8, 15}}.

After sorting this list according to the first members of ordered pairs, we obtain the list D1 = {{1, 12}, {3, 14}, {5, 10}, {8, 15}, {9, 4}, {11, 6}, {13, 2}, {16, 7}}.

The first part of the P-data is the same as the first part of Dow, and the second part, called P-word, is the list of the second members of the ordered pairs from D1 , so P-data={{8}, {12, 14, 10, 15, 4, 6, 2, 7}}. In fact, P-data is very similar to a Dowker code with signs. The only difference is that P -data contains odd instead of the corresponding even numbers in

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all crossings where the relation “over-under” is changed with regard to the corresponding alternating KL. Dowker codes contain all relevant information about a KL diagram, so can be used as an input for computer programs. Morwen Thistlethwaite used the Dowker notation to list all prime knots with up to 16 crossings (Hoste, Thistlethwaite, and Weeks, 1998), and Jim Hoste derived the list of all links with at the most nine crossings, representing each KL by its minimal Dowker code (Doll and Hoste, 1991; Cerf, 1998). Any regular diagram of a KL has a finite number of crossings, and this number is called the crossing number of the diagram. Given all regular diagrams of a KL, the crossing number of the regular diagram with the fewest number of crossings is the crossing number of a KL (Definition 1.25). As we already mentioned, according to Kauffman-Murasugi Theorem (Theorem 1.6) crossing number of an alternating KL is the crossing number of its reduced alternating diagram. Since every KL diagram can be presented with many different sequences, there are more sequences then KLs. If we are working with minimal diagrams of a KL (where the number of crossings coincides with the crossing number of the KL) this ratio is finitely many to one. Otherwise, we have infinitely many sequences for one KL. The first attempt to minimize the amount of data is a minimization of Dowker codes. We already mentioned that a Dowker code of a KL projection is dependent on the choice of the beginning point of each component and on its orientation (i.e., on the choice of the first oriented edge). Among all Dowker codes which correspond to a specific KL projection we can choose the minimal one. In the case of knots, this means choosing the minimal permutation among all possible Dowker codes taken without signs. In the case of links, in order to obtain the minimal Dowker code we use two criteria in the following order: the length of components (where shorter components have the priority), and the minimal code (i.e., minimal permutation criterion). The simplest, but certainly the slowest minimization algorithm creates all possible Dowker codes for a given knot projection, sorts them and chooses the first: the minimal Dowker code of the given projection. In a similar way, for a link projection one can make all possible choices for beginning points of components and all their orientations, calculate all Dowker codes and take the minimal one. A more smarter algorithm first sorts components according their lengths, then finds an optimal beginning point for each of them, permute components equivalent with regard to the two cri-

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teria mentioned above, calculates Dowker codes and chooses the minimal one. In this way, we can compute the minimal Dowker code for every KL projection. Sometimes it is useful to work with weighted graphs of KLs given by a list of unordered pairs and list of vertex signs instead of Gauss or Dowker codes. The LinKnot function fGraphInc calculates from a Conway symbol, Dowker code, or P -data of any KL its corresponding graph. An output is the graph given by edges (as a list of unordered pairs) and by the list of vertex signs. The LinKnot function fPlanarEmb calculates the planar embedding of a prime KL given by a Conway symbol, P -data or Dowker code. An output is the list that consists of the graph of the input KL, its planar embedding given by vertex cycles, and the faces of the planar embedded graph. The basis of this program is the external program planarity.exe written by J.M. Boyer (Boyer and Myrvold, 2005). As we already mentioned, every KL shadow is a 4-valent graph. If we have any polyhedral graph G, we can obtain its corresponding midedge graph M (G) defined by mid-edge points of G by connecting mid-edge points belonging to adjacent edges of G. Clearly, the result M (G) is always a 4-valent graph. For example, for the tetrahedron graph {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}} the result is the octahedron graph {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}}

(Fig. 1.27a). The LinKnot function fMidEdgeGraph gives a mid-edge graph for any polyhedral graph G given by a list of unordered pairs of vertices. Every 4-regular graph represents a shadow of a KL, and can be used to find the corresponding alternating KL diagram given by Dowker code. The function fKLfromGraph gives the Dowker code in the DT form (in Knotscape format) of a KL defined by a given 4-regular graph G. The corresponding Dowker code with signs can be obtained from it using the function fSignsKL. From a signed graph of KL we can recover KL from which it originated by constructing mid-edge graph, where there exists one-to-one correspondence between bigons in the graph of KL and bigons of the mid-edge graph (Fig. 1.27b).

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Fig. 1.27 (a) Mid-edge graph obtained from tetrahedron graph; (b) figure-eight knot reconstructed from its graph.

Two other graph functions work with graphs that are not necessarily 4-valent. The function fKLinGraph gives all non-isomorphic KL projections contained in a given graph G. For example, the graph (Fig. 1.28a)

{{1, 2}, {1, 2}, {1, 4}, {1, 4}, {1, 5}, {2, 3}, {2, 3}, {2, 5}, {3, 4}, {3, 4}, {3, 5}, {4, 5}}

contains two KL graphs: the graph {{1, 2}, {1, 2}, {1, 4}, {1, 4}, {2, 3}, {2, 3}, {3, 4}, {3, 4}}

corresponding to a link 421 (4) and the graph {{1, 2}, {1, 2}, {1, 4}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {3, 4}, {3, 5}, {4, 5}}

corresponding to the Whitehead link 521 (2 1 2) (Fig. 1.28b). The function fAddDig takes a given graph G as an input, and produces all 4-regular non-isomorphic graphs by replacing single edges by double (bigonal) edges. For example, from 3-valent graph (Fig. 1.29a) {{1, 2}, {1, 3}, {1, 6}, {2, 4}, {2, 6}, {3, 4}, {3, 5}, {4, 5}, {5, 6}}

we obtain two non-isomorphic 4-valent graphs (Fig. 1.29b)

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Fig. 1.28

KLs in graph.

{{1, 2}, {1, 2}, {1, 3}, {1, 6}, {2, 4}, {2, 6}, {3, 4}, {3, 4}, {3, 5}, {4, 5}, {5, 6}, {5, 6}}

and {{1, 2}, {1, 3}, {1, 3}, {1, 6}, {2, 4}, {2, 4}, {2, 6}, {3, 4}, {3, 5}, {4, 5}, {5, 6}, {5, 6}}.

Fig. 1.29 Graphs derived from 3-valent graph (a) by replacing some single edges by double edges (b).

1.4

Reidemeister moves

The next step in derivation of all KLs with a given number of crossings is finding all different (non-isomorphic) minimal projections of a given KL, i.e., all its different projections with the number of vertices equal to the crossing number. In the case of alternating KLs it is sufficient to find all proper non-isomorphic alternating projections of a given KL with a fixed

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number of crossings. Non-alternating KLs can be derived from alternating by crossing changes.

Fig. 1.30

Reidemeister moves for polygonal KLs.

KLs are usually given by their non-minimal projections, so it is necessary to minimize them. There is a finite algorithm based on Haken-Hemion method (Haken, 1961; Hemion, 1979), described in detail by S. Matveev (2003), that guarantees a minimization and solves the recognition problem of KLs. However, it is impossible to implement because of its complexity. Even its special case, the unknot recognition problem, is NP-hard (where NP means “non-deterministic polynomial time”) (Hass and Lagarias, 2001), and the upper bound for the number of Reidemeister moves needed to unknot an n-crossing unknot diagram is 2cn , where c = 211 . However, there are two computer programs attempting to produce best possible minimizations:

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• knotfind.c, the part of Knotscape, based on a heuristic algorithm and restricted to knots; • the function Reduction KnotLink, written by M. Ochiai and N. Imafuji, and included in the Mathematica-based program Knot 2000 (K2K).

The authors emphasize that this function fails for some classes of links, but most of its reductions are the optimal ones1 . Before a more detailed discussion of this problem, we will describe the elementary steps of a reduction process: Reidemeister moves2 . So far we had only static images: KL shadows and projections. Now we can make a movie with ambient isotopy playing the main role. All KLs will be represented as polygonal KLs, and all the moves that consist of a finite series of elementary isotopies will be expressed as finite compositions of Reidemeister moves. The move Ω0 was already introduced as a planar isotopy. Recall that for a polygonal link a planar isotopy Ω0 is achieved either by subdividing an edge AB by the vertex C, or contracting AC and CB. The move Ω0 can be introduced as the elementary planar isotopy (Fig. 1.30). An ambient isotopy for a polygonal KL is a finite sequence of elementary isotopies. Reidemeister moves, Ω1 , Ω2 , and Ω3 are illustrated in Fig. 1.30. We represent Reidemeister moves as polygonal moves, and the piecewise-linear and the smooth knot theory give the same classification of KLs (Crowel and Fox, 1965; Burde and Zieschang, 1985). The equivalent of ambient isotopy of KLs, for knot and link diagrams are Reidemeister moves. Hence, Reidemeister moves are planar isotopies. Theorem 1.10. Two diagrams D′ and D′′ of polygonal KLs correspond to ambient isotopic KLs iff D′ can be transformed into D′′ using a finite sequence of Reidemeister moves Ω1 , Ω2 , Ω3 (Reidemeister, 1932). For the elegant proof of this theorem see Kauffman (2004) or Manturov (2004, Theorem 2.1). The last book (appendix A) contains the proof of the independence of Reidemeister moves. 1 This holds even for much better heuristic program for reduction of knots knotfind.c written by M. Thistlethwaite and used as a part of the program Knotscape. Its first known unsuccessful reduction is for a 40-crossing knot. 2 J.C. Maxwell determined all regions bounded with fewer then four arcs corresponding to Reidemeister moves. The proof that they suffice to pass between equivalent diagrams was published by both Reidemeister (1926) and Alexander and Briggs (1926-27) (Hoste, 2006).

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Definition 1.33. A link is an equivalence class of link diagrams modulo Reidemeister moves. In general, Reidemeister moves can be used for obtaining various diagrams of a given KL. In the case of alternating KLs they are not necessary for a transformation of one minimal diagram to another. Instead, we can use flypes, moves introduced by P.G. Tait. They are well known from his Flyping Conjecture (1876/77) (or Tait’s Third conjecture), which became the Tait’s Flyping Theorem in 1990, when it was proven by W. Menasco and M. Thistlethwaite (Menasco and Thistlethwaite, 1991, 1993). A natural way of expressing a flype as a sequence of Reidemeister moves is still unknown. Before explaining what a flype is, we need to define a tangle, one of the fundamental notions in knot theory, introduced by J.H. Conway in 1967 (Conway, 1970).

Fig. 1.31 (a) A tangle, (b) flype; (c) mutation; (d) vertical mutation; (e) horizontal mutation.

Definition 1.34. A 1-dimensional manifold properly embedded in 3dimensional disk is called a tangle (or 2-tangle) if it is composed of two arcs and any number of circles.

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The same name, tangle, will be used to denote a projection of a tangle into 2-dimensional disk. Intuitively, a tangle (or more precisely, 2-tangle) in a KL projection is a region in the projection plane ℜ2 (or on the sphere S 3 ) surrounded with a circle such that the KL projection intersects with the circle exactly four times. From the intersections, four arcs emerge pointing in the compass directions NW, NE, SW, SE (Fig. 1.31a). Definition 1.35. Two tangles are equivalent if one can be transformed into the other by a sequence of Reidemeister moves provided that four endpoints of the strings are fixed and that the strings belonging to the tangle remain inside of the circle. Suppose an alternating KL diagram contains a tangle, as shown in a Fig. 1.31b. Let us fix four ends a, b, c, d and then rotate this tangle by a 2-fold rotation (half-turn). The twist on the left in Fig. 1.31b is moved to the right. Such an operation is called a flype. Theorem 1.11. (Tait’s Flyping Theorem) Suppose that L′ and L′′ are two reduced alternating diagrams of an alternating link L on the sphere S 3 . Then we can change L′ into L′′ by performing a finite number of flypes. According to Theorem 1.7, writhe is an invariant of alternating KL diagrams. It is clear that Reidemeister moves Ω2 and Ω3 does not change a writhe. However, in transition from one minimal diagram of a nonalternating KL to another sometimes we must use Reidemeister move Ω1 that changes the writhe, increase the number of crossings, and then reduce the diagram in some of next steps, so writhe can be changed. An example of two minimal diagrams of the same non-alternating knot with a different writhe is Perko pair 3 : −2 0 : −2 0 and 2 1 : −2 0 : −2 0 (Fig. 1.26). This example is easy to generalize to KL (sub)families (Definition 1.49) called Perko families: Conway symbols (2k + 1) : −2 0 : −2 0 and (2k) 1 : −2 0 : −2 0 are two families of minimal diagrams of the same non-alternating knot with a different writhe. In this family Perko pair is obtained for k = 1, for k = 2 two diagrams of the knot 12n850 , for k = 3 two diagrams of the knot 14n26229 , and for k = 4 two diagrams of the knot 16n965076 given in Knotscape notation. The same holds for the diagrams 2 (2k) : −2 0 : −2 0 and 2 (2k − 1) 1 : −2 0 : −2 0 of the knots 11n135 , 13n3546 , and 15n114094 obtained for k = 1, 2, 3, respectively. Hence, for every n ≥ 10 there exists at least one non-alternating knot which has two minimal diagrams with different writhe. Moreover, if t is any rational

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tangle, minimal diagrams t (k + 1) : −2 0 : −2 0 and t k 1 : −2 0 : −2 0 of the same KL have this property. In all these cases, the writhe of the diagrams differs by 2. There are following open questions: (1) for a given number of crossings n find all non-alternating knots having at least two minimal diagrams with a different writhe; (2) how many different values can writhe of an non-alternating knot take on its minimal diagrams; (3) is it possible to find two minimal diagrams of the same KL whose writhes differ by an arbitrary number; (4) can we transform one minimal diagram of a prime knot to another minimal diagram with the same writhe using only Reidemeister moves Ω2 and Ω3 which does not change a writhe? Definition 1.36. The winding number or Whitney degree of a knot diagram is total turn of the tangent vector to the curve as one traverses it in the given direction (Kauffman, 1995a). In the case of polygonal oriented KL diagrams in each vertex there is an exterior angle: the angle one must turn to continue to traverse the diagram. The sum of the oriented exterior angles (where the counter-clockwise orientation is taken as positive) divided by 2π is the winding number of the diagram. Only Reidemeister move Ω1 changes winding number. Theorem 1.12. Two diagrams of a link L are related by a finite sequence of Reidemeister moves Ω2 and Ω3 (without Ω1 ) iff they have the same writhe and winding number (Trace, 1983; Cromwell, 2004). Conjecture 1.1. All minimal diagrams of an oriented prime link L that have the same writhe have the same winding number and vice versa. In the case of non-minimal diagrams of composite knots this is not true: there are two non-minimal diagrams of the same composite knot with the same writhe and different winding numbers. For transforming one of them to the other, all Reidemeister moves are necessary, so this is an alternative proof of the independence of Reidemeister moves (Fig. 1.32) (Hagge, 2005). Theorem 1.13. The winding number is the invariant of KL subfamilies.

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This conclusion follows immediately, since winding number remains unchanged by adding two bigons to any single bigon or chain of bigons. For example, the winding number of the knot family (2k + 1) (31 , 51 , 71 ,...) is 2, of the family (2k) (2l) it is 0, etc.

Fig. 1.32 number.

Two diagrams of the knot 2 2#8∗ with the same writhe and different winding

Defined in a standard way (Definition 1.31), writhe is the invariant only of reduced alternating knot diagrams (Theorem 1.7). Hence, Definition 1.31 is not consistent with the other similar definitions of KL invariants (e.g., crossing number, Definition 1.25; unlinking number, Definition 1.56), where they are defined as the minimum numbers taken over all KL diagrams. This is a good enough reason to try to give a different definition of writhe. However, an attempt to define the minimum writhe of a link L as the minimum of the absolute value of writhe, min |w(D(L))| taken over all diagrams D(L) of the link L results in a trivial invariant: such minimum writhe equals zero for all links. Using Reidemeister move Ω1 that changes writhe by +1 or −1 for every link L we can obtain a diagram with w = 0. Even the restriction to proper diagrams is not helpful, because by Reidemeister move Ω2 we can eliminate (or, better to say “hide”) loops, and preserve w = 0. Definition 1.37. The minimum writhe of a link L, denoted by wm (L), is the minimum of the absolute value of writhe, min |w(DL )| taken over all minimal diagrams Dm (L) of L. The minimum writhe will be well-defined KL invariant, easy to calculate for alternating, but hard to calculate for non-alternating KLs. The Knot 2000 (K2K) function MutationOfTangle calculates the re-

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sult of a flype, mutation, vertical, and horizontal mutation applied to some tangle (Fig. 1.31c,d,e).

Fig. 1.33

Fig. 1.34

2-pass.

Reidemeister moves translated into signed KL graphs.

In the case of non-alternating KLs, in addition to flypes we can introduce a 2-pass: KL transformation where a string is pulled over a tangle (Fig. 1.33). The Reidemeister move Ω3 is a special case of a 2-pass. However, flypes and 2-passes are not sufficient to go between all minimal diagrams of a non-alternating KL (Hoste, Thistlethwaite and Weeks, 1998). Using signs of crossings, we can introduce a signed link graph, where each edge takes the sign of a vertex it passes through. In this way we have established one-to-one correspondence between edge-weighted KL graphs

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and signed KL diagrams. Reidemeister moves Ω1 , Ω2 , Ω3 , can be represented as local moves I′ , II′ , III′ on signed KL graphs shown on Fig. 1.34. Two signed plane KL graphs G1 and G2 represent the same link L iff G1 can be transformed into G2 by some finite sequence of the moves I′ , II′ , III′ and their inverses. If G is any signed plane KL graph and G′ is its plane dual with the signs of the edges multiplied by −1, then the links L(G) and L(G′ ) are ambient isotopic. It is easy to notice that Reidemeister moves Ω1 and Ω2 decrease the number of crossings. At the first glance, it seems that there is a finite minimization algorithm for non-alternating projections: application of Ω1 move makes them proper, and Ω2 decreases the number of crossings (in every step by 2). Unfortunately, things are not so simple and straightforward. Sometimes, in order to minimize a KL projection, it is necessary to increase the number of crossings and then reduce it by the application of Ω1 and Ω2 . Examples of unknot diagrams that can not be minimized without increasing the number of crossings are the Nasty unknot ((−1, 3, 1), −1, −1) (Adams, 1994), Goeritz’s unknot ((1, 1, 3, −1, −1), −1, −1) (Goeritz, 1934), or the Monster unknot 3 2 − 1 1 1 2 from R. Sharein’s program KnotPlot (Fig. 1.35). Another problem is that we do not know the order in which Reidemeister moves should be applied: the sequence of moves producing the final result– a minimal projection. Usually, we prefer to consider spherical then plane KL diagrams, thinking of the sphere as the 1-point compatification of the plane. Without loss of generality, one can assume that a shadow of KL does not contain this point. In this case, there appears one more “elementary isotopy” when some edge of the shadow passes through the infinity. This operation is called the infinity change (see, e.g., Manturov, 2004). By the infinity change operation we can obtain many plane diagrams from spherical diagrams. The smallest plane diagram that can not be unknotted without increasing the number of crossings is the Nasty unknot (Fig. 1.35a) with n = 7 crossings. However, considered as the diagram on a sphere, after the infinity change it can be reduced by the Reidemeister move Ω2 (Fig. 1.35b). Definition 1.38. An unknot (unlink) diagram on a sphere is called hard diagram if it has the following properties (Kauffman and Lambropoulou, 2006):

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Fig. 1.35

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(a), (b) Nasty unknot; (c) Goeritz’s unknot; (d) Monster unknot.

(1) there are no simplifying Reidemeister moves Ω1 and Ω2 on the diagram; (2) there are no Reidemeister moves Ω3 on the diagram. In a diagram, there is a Reidemeister move Ω3 if it contains a triangle with one edge forming two overcrossings or two undercrossings. Hence, the Monster unknot with n = 10 crossings and Goeritz’s unknot with n = 11 crossings are hard unknot diagrams. Up to signs of crossings, the smallest hard unknot and unlink diagrams have n = 9 crossings. For n = 9 there are two hard unknot diagrams and one hard unlink diagram (Fig. 1.36); for n = 10 we recognized six hard unknot diagrams and five hard unlink diagrams (Fig. 1.37), etc. For virtual KL diagrams (Definition 1.23) generalized Reidemeister moves can be introduced. Definition 1.39. Generalized Reidemeister moves consist of:

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Fig. 1.36

Hard (a,b) unknot and (c) unlink diagrams with n = 9 crossings.

(1) classical Reidemeister moves related to classical vertices; (2) virtual versions Ω′1 , Ω′2 , Ω′3 of Reidemeister moves (Fig. 1.38a); (3) the “semivirtual” version Ω′′3 of the third Reidemeister move (Fig. 1.38a) (Manturov, 2006). Two virtual diagrams are equivalent if there exists a sequence of generalized Reidemeister moves transforming one diagram to the other one. Definition 1.40. A virtual link is an equivalence class of virtual diagrams modulo generalized Reidemeister moves. The remaining two versions of the third move (Fig. 1.38b) are forbidden. Actually, the forbidden move is a very strong one: each virtual knot can be transformed to another one using all generalized Reidemeister moves and the forbidden move (Nelson, 2001). 1.5

Conway notation

The Conway notation of KLs, based on the notion of a tangle is introduced in 1967 (Conway, 1970). The main advantage of Conway symbols is the amount of important KL properties that are almost directly visible or can be derived from the codes, like symmetry, recognition of the worlds (Caudron, 1982) to which particular KLs belong, the proof of the equality of rational KLs using their corresponding continued fractions, etc. For example, all rational KLs with an even number of crossings and with symmetric

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Fig. 1.37

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Hard (a) unknot and (b) unlink diagrams with n = 10 crossings.

(palindromic) Conway symbol are achiral (Theorem 1.20, Theorem 1.22). J.H. Conway used his approach to derive KLs up to 11 crossings, but then, unfortunately, divided them according to the number of components. We would like to bring to light work of A. Caudron (1982) and the po-

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Fig. 1.38

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(a) Generalized Reidemeister moves Ω′1 , Ω′2 , Ω′3 , Ω′′ 3 ; (b) forbidden move.

tential of the Conway notation in the search for the universal classification principle– “periodic table” of KLs. It is very surprising that the Conway notation, the only geometrical-topological notation that gives complete, interpretable and understandable information on KLs is still not widely accepted. Maybe the main reason why most of knot theory books have only the classical notation is the non-uniqueness of the Conway notation. Hence, we need to make a choice of the “standard” Conway symbol of a KL, according to the notation introduced in the original Conway’s paper (Conway, 1970) and in the papers and books following it (Caudron, 1982; Rolfsen, 1976; Adams, 1994). Notation becomes more complicated for polyhedral KLs. We need to specify the symbol of the particular basic polyhedron and the particular order and orientation of its vertices (Figs. 2.43-2.47). For example, the same link .2 can be denoted as : 2, : .2, :: 2, :: .2, or even as 6∗ 2, 6∗ .2, 6∗ : .2, 6∗ :: 2, and 6∗ :: .2, where as the standard symbol we choose the first of them. The LinKnot function fClassicToCon gives Conway symbol of a KL given in the classical notation. Elementary tangles are shown in Fig. 1.39 and denoted by 0, 1 and −1, where for alternating KLs 0 and 1 are sufficient. Any tangle can be obtained

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from elementary tangles using operations: sum, product, and ramification defined below (Figs. 1.40-1.41). Given tangles a and b, image of a under reflection with mirror line NW-SE is denoted by −a, and sum is denoted by a + b. Product a b is defined as a b = −a + b, and ramification is defined as (a, b) = −a − b.

Fig. 1.39

Fig. 1.40

The elementary tangles.

A sum and product of tangles.

Fig. 1.41

A ramification of tangles.

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The tangles 2, 2 1 2, and 2, 2 obtained by applying the operations mentioned above are illustrated in Fig. 1.42.

Fig. 1.42

Tangles (a) 2 = 1 + 1; (b) 2 1 2 = (1 + 1) 1 (1 + 1); (c) 2, 2 = (1 + 1), (1 + 1).

Closed tangle can be obtained from a tangle in two ways (without introducing additional crossings): joining in pairs NE and NW, and SE and SW ends of a tangle we obtain a numerator closure; joining in pairs NE and SE, and NW and SW ends we obtain a denominator closure (Fig. 1.43a,b).

Fig. 1.43

(a) Numerator closure; (b) denominator closure; (c) basic polyhedron 1∗ .

Definition 1.41. A rational tangle is any finite product of elementary tangles. A rational KL is a numerator closure of a rational tangle. Rational knots are also known as 2-bridge knots, Viergeflechte, or 4-

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plats. O. Simony (1882, 1884) was the first who considered rational KLs from the mathematical point of view, while their complete classification was given by H. Schubert (1956). J. Conway noticed the beautiful relation between rational tangles and continued fractions (Conway, 1970; Kauffman and Lambropoulou, 2002) (Theorem 1.18). Definition 1.42. A tangle is algebraic if it can be obtained from elementary tangles using the operations of sum and product. KL is algebraic if it is a numerator closure of an algebraic tangle. For every KL shadow, its basic polyhedron can be identified by collapsing bigons until none of them remains (Fig. 1.44). In fact, most of the basic polyhedra are geometrical polyhedra– 3-vertex connected 4-regular graphs, but in the list of basic polyhedra some 2-vertex connected graphs also are included (e.g., the first of them is 12E). More precisely, Definition 1.43. A 4-regular, 4-edge-connected, at least 2-vertex connected plane graph is called basic polyhedron. The basic polyhedron 1∗ is illustrated in Fig. 1.43c, and the other basic polyhedra with n ≤ 12 crossings in Figs. 2.43-2.47. Definition 1.44. A link L is algebraic link or 1∗ -link if there exists at least one shadow of L which can be reduced to the basic polyhedron 1∗ by a finite sequence of bigon collapses. Otherwise it is a non-algebraic or polyhedral link. Definition 1.45. A KL with single bigons, or equivalently, a KL given by Conway symbol containing only tangles 1, −1, 2, or −2 is called a source link. Three operations with tangles: sum, product, and ramification (page 52) are sufficient for the notation of algebraic KLs. Polyhedral KLs require special notation. In addition to the operations used for algebraic KLs we need to know a symbol of a basic polyhedron P ∗ = nO∗ , where n denotes the number of vertices, and O the ordering number of a particular basic polyhedron among those with the same number of vertices. For example, 123∗ is the third basic polyhedron with n = 12 vertices. A knot or link obtained from a basic polyhedron P ∗ =nO∗ by substituting tangles t1 , . . ., tk in appropriate places is denoted by P ∗ t1 . . . tk , where the number of dots between two successive tangles shows the number of omitted substituents of value 1.

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For example, 6∗ 2 : 2 : 2 0 means 6∗ 2.1.2.1.2 0.1, and 6∗ 2 1.2.3 2 : −2 2 0 means 6∗ 2 1.2.3 2.1. − 2 2 0.1 (Fig. 1.45).

Fig. 1.44

Fig. 1.45

A link shadow collapsing into the basic polyhedron 6∗ – an octahedron.

Basic polyhedron 6∗ and the knots 6∗ 2.1.2.1.2 0.1 and 6∗ 2 1.2.3 2 : −2, 2 0.

Conway symbols are used for the first time as an alternative notation in the book Knots and Links by D. Rolfsen (1976), and then by some other authors (e.g., C.C. Adams, 1994). Notice that in Rolfsen’s tables some drawings of KLs do not correspond to their Conway symbols. For example, in the case of the knot 915 , it is clear that from its Conway symbol 2 3 2 2 we obtain the projection with 5, and not with 4 bigons, pretzel knot 819 with the Conway symbol 3, 3, −2 is drawn as 6∗ 2 0. − 2 0. − 1. − 1, etc. It is interesting that in Conway symbols of all non-alternating polyhedral KLs with n ≤ 10 crossings the symbol . − 1, i.e., a single vertex with

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a changed sign never appears, except in the case of the 4-component link 103∗ − 1. − 1. − 1. − 1 :: . − 1 (or 10−∗∗∗ according to Conway’s paper). The first polyhedral non-alternating knots that can not be expressed by Conway symbols without . − 1-s are 12-crossing knots 8∗ 2 0.−2.−1.−2 0.2 0, 8∗ 2 0.−2 0.−1.−2 0.2 0, 1212∗ −1.−1.−1.−1.−1.−1 (12n801 , 12n835 , and 12n837 in the Knotscape notation, respectively). In the program Knot 2000 (K2K), the function GetPdatabyTracking is used for entering KLs by drawing them in the mouse-tracking window. The output is P -data of the KL. Instead of the graphical input, LinKnot function fCreatePData uses a Conway symbol of KL (given as a Mathematica string) and computes P -data. For example, the figure-eight knot 41 is denoted by ”2 2”, knot 95 by ”5 1 3”, link 521 is denoted by ”2 1 2”, link 9221 by ”3 1, 3, 2” (for all of them a space between tangles denotes a product of tangles), etc. A sequence of k pluses at the end of the Conway symbol is denoted by +k, and a sequence of k minuses by + − k (e.g., knot 1076 given in Conway notation as 3, 3, 2 + + is denoted by ”3, 3, 2 + 2”, and the link 9317 given in Conway notation as 3, 2, 2, 2 − − by ”3, 2, 2, 2 + −2”). The space denoting a product of tangles is used in the same way in all other symbols. For example, the knot 10133 is denoted by ”2 3, 2 1, 2 + −1”, and the knot 10154 by ”(2 1, 2) − (2 1, 2)” (with spaces). The program LinKnot contains the database of basic polyhedra with at most n = 20 crossings, where every basic polyhedron is represented by its corresponding alternating KL diagram. For the basic polyhedra with n < 10 crossings, the standard notation is used (.1, 6∗ , 8∗ , 9∗ , where symbols of polyhedral KLs beginning with a dot correspond to Conway’s basic polyhedron 6∗∗ or .1). For example, the knot 1095 is denoted by ”.2 1 0.2.2”, and 10101 by ”2 1..2..2”. For n ≥ 10 in each symbol the first two digits represent the number of crossings, and the next the ordering number of the polyhedron (e.g., 101∗, 102∗, 103∗ for n = 10 denoting 10∗ , 10∗∗ , 10∗∗∗ , respectively, and 111∗ , 112∗ , 113∗ for n = 11 denoting 11∗ , 11∗∗ , 11∗∗∗ , respectively, etc.). For n = 12 basic polyhedra are ordered according to their list made by A. Caudron (1982), so polyhedra originally denoted with 12A-12L are 121∗ -1212∗. For n > 12 the database of basic polyhedra is derived from the list of simple 4-regular, 4-edge-connected, but not necessarily 3-vertex connected plane graphs generated by Brendan McKay using the program plantri written by Gunnar Brinkmann and Brendan McKay (http://cs.anu.edu.au/∼bdm/plantri/).

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The data base PolyBase.m contains basic polyhedra up to 16 crossings and downloads automatically when LinKnot is loaded. For n = 12 it contains 12 basic polyhedra beginning from 121∗ to 1212∗, for n = 13 it contains 19 basic polyhedra from 131∗ to 1319∗, for n = 14 it contains 64 basic polyhedra from 141∗ to 1464∗, for n = 15 this file contains 155 basic polyhedra from 151∗ to 15155∗, and for n = 16 it contains 510 basic polyhedra, beginning from 161∗ to 16510∗. In order to work with the basic polyhedra from n = 17 to n = 20 vertices, you need to open an additional database PolyBaseN.m, for n = 17 to n = 20 (by writing, e.g. <
Fig. 1.46

Alternating links (a) 11∗∗∗ 2 ∼ 125∗ ; (b) 136∗ ∼ 1318∗ .

For n ≤ 11 the correspondence between basic polyhedra (source links) and their corresponding alternating KLs is one-to-one: every alternating

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KL corresponding to a basic polyhedron (source link) has exactly one minimal projection. However, for n = 12 the 4-component link 11∗∗∗ 2 has as another minimal projection the basic polyhedron 125∗ (or 12E according to Caudron’s list) (Fig. 1.46a), and for n = 13 the 4-component link (Fig. 1.46b) has the basic polyhedra 136∗ and 1318∗ as its minimal projections. In both cases, one projection can be obtained from the other by a single flype, so the corresponding alternating links are equivalent, i.e., 11∗∗∗ 2 ∼ 125∗ and 136∗ ∼ 1318∗, where the symbol ∼ is used to denote flyping equivalence. Hence, for n ≥ 12 there is no one-to-one correspondence between basic polyhedra and their corresponding alternating KLs. 1.6

Classification of KLs

Classifications are always given with respect to one or several invariants. We can use the Component Algorithm to classify KLs according to the number of components– now we will try to avoid this classification as long as possible and use other graph-theoretical or combinatorial properties of KLs. For example, the most universal classification of KLs that contains two classes of KLs: algebraic and polyhedral KLs is based on bigon collapse. We will consider only prime KLs given in the Conway notation and several ways to classify them. One of the most obvious classifications is into alternating and non-alternating KLs. Each source link (Definition 1.45) induces an alternating KL and every alternating KL can be obtained from some source link by substituting single bigons by chains of bigons. All links that can be derived from one source KL by such replacements form a family (Definition 1.48). KLs will be distributed into disjoint sets, named worlds (page 63) by A. Caudron: algebraic and polyhedral, and their subworlds: rational, stellar, etc. Non-alternating KLs will be obtained from the alternating ones by the corresponding crossing changes. According to a result of complete bigon collapse, alternating KLs can be divided into two main worlds: algebraic and polyhedral, but even this depends from the definition of an algebraic KL. There are two possibilities for defining algebraic KLs (in Conway sense): Definition 1.46. (1) a KL is algebraic if it has a minimal algebraic representation (projection); (2) a KL is algebraic if it has an algebraic representation (projection).

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For example, alternating link .3.2.3.2 with n = 12 crossings has algebraic projection (4, −3 1) (3, −2 1) with n = 14 crossings, so it will be considered as polyhedral according to the first, and as algebraic according to the second definition. From now on, we will work with the first definition, so it will be considered as polyhedral. With non-alternating KLs situation is even more complicated: without introducing additional criteria, we are even not able to distinguish algebraic non-alternating KLs from polyhedral ones, because the same nonalternating KL can have both algebraic and polyhedral minimal representation. For example, the same non-alternating link can be represented as (4, −2) (2, 2), or 2 0. − 2. − 2 0. − 2 0. Therefore it will be algebraic, even though it has polyhedral minimal representation. Depending from the type of their corresponding graphs (Caudron, 1982), i.e., depending from the tangle operations used for their derivation, algebraic KLs can be rational, stellar, or arborescent. Rational tangles and rational KLs are obtained using the operation of product, and stellar (pretzel) KLs are obtained from rational tangles using the operation of ramification. Finally, composing rational and stellar tangles by the operations of product and ramification, arborescent KLs are obtained, so we have the complete stratification of algebraic alternating KLs. Unfortunately, polyhedral world does not admit similar stratification, since the basic polyhedra themselves are not ordered. For example, there are two different (non-isomorphic) basic polyhedra 136∗ and 1318∗ with n = 13 crossings that are two non-isomorphic diagrams of the same alternating link. Crazy Spider Algorithm (page 286) offers some perspective for a classification of basic polyhedra, giving possibility to derive families of basic polyhedra, establish their generic notation and the order based on the family principle. If bigons in KL shadows are denoted by colored (bold) lines, we can distinguish vertices of different valences: 2-valent (where in each vertex are two colored edges), 3-valent (where in each vertex there is one colored edge), and 4-valent. A link shadow is called basic if its edge-bicolored graph is regular. If it is 4-regular, such a graph is a basic polyhedron. Definition 1.47. Two basic polyhedra or source links are equal iff they are isomorphic. It is interesting that the fundamental term “family” (Definition 1.48) is very hard to find in knot theory books or papers. According to the book Knots and Surfaces (Farmer and Stanford, 1996), “a family of knots is an

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informal term used to describe a list of knots where each successive knot is obtained from the previous one by a simple process”. The source link 2 (i.e. the Hopf link 221 ) and its family are shown in Fig. 1.47.

Fig. 1.47

Hopf link and its family.

We can immediately recognize a simple pattern: regular periodic distribution of KLs in the family p (p = 2, 3, . . .), where knots (1-component links) are obtained for odd p, and 2-component links for even p. In a similar way, if we consider KLs given by the general Conway symbol p q (i.e., the family derived from a figure-eight knot 41 , whose Conway symbol is 2 2), we will obtain links for p = q = 1 (mod 2), and knots in all the other cases. Every Conway symbol of an alternating KL can be reduced by replacing all even numbers denoting chains of bigons by 2, all odd numbers greater or equal to 3 by 3, and all single vertices 1 remain unchanged. KL obtained in this way is called a generating link. Definition 1.48. For a link or knot L given in an unreduced3 Conway notation C(L) denote by S a set of numbers in the Conway symbol excluding numbers denoting basic polyhedra and zeros (denoting the position of tangles in the vertices of polyhedra). For C(L) and an arbitrary (non-empty) subset S˜ of S the family FS˜ (L) of knots or links derived from L is constructed by substituting each a ∈ Sf , a 6= 1, by sgn(a)(|a| + ka ) for ka ∈ N . 3 The Conway notation is called unreduced if in symbols of polyhedral KLs elementary tangles 1 in single vertices are not omitted.

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This means that all KLs generated from a source link S by substituting single bigons by chains of bigons make a family generated from S. Definition 1.49. All KLs generated from a generating link G by adding an even number to each number which denotes a chain of bigons make a subfamily generated from G. This means that a subfamily will be obtained if all ka from the Definition 1.48 are even. Theorem 1.14. KLs belonging to the same subfamily have the same number of components. The proof of this theorem is trivial, because the addition of any chain of bigons of an even length to a bigon or chain of bigons does not change the number of components. If there is no chance for a confusion, in order to simplify, we will use the common term “family” for both families and subfamilies. According to experimental results, various KL invariants will be related to subfamilies, and will represent some combination of numbers denoting chains of bigons (integer tangles n or −n) from a general Conway symbol of a subfamily. Definition 1.50. Any KL invariant which can be expressed in terms of chains of bigons (i.e., integer tangles n or −n) belonging to a Conway symbol of the subfamily will be called subfamily-dependent invariant. Subfamily-dependent invariant is linear if it can be expressed by a firstdegree polynomial depending from parameters denoting integer tangles in the Conway symbol of KL. Theorem 1.15. Minimum writhe is a linear subfamily-dependent invariant of alternating KLs, and linking number is a linear subfamily-dependent link invariant. The proof of this theorem is trivial as well. In the case of writhe, every addition of two bigons to a chain of bigons changes writhe by +2 in the case of positive, or by −2 in the case of negative chain, so knowing the writhe of the generating alternating knot given by its (minimal) Conway symbol, it is possible to calculate writhe of any knot belonging to a subfamily derived from it. In the case of linking number, we need to take in account only chains of bigons belonging to different components.

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Experimental results imply that, at least for certain classes of KLs, many KL invariants are are subfamily-dependent invariants, and not only invariants of particular KLs. According to experimental data obtained using the program LinKnot, we propose the following general conjecture: Conjecture 1.2. Crossing number, minimum writhe, unlinking number, u∞ -unlinking number, signature, and genus of a link L are linear subfamilydependent KL invariants. The combinatorial formula for a signature of alternating knot diagrams, derived by P. Traczyk (2004), supports this conjecture. Since alternating KLs are easier to work with, we will first consider alternating KLs, and then non-alternating ones. One of the most powerful tools in our consideration will be symmetry. Symmetry can be visualized representing KL diagrams as graphs on a sphere. For every KL diagram we can distinguish the symmetry group G of its edge-bicolored graph where double edges are colored (bold) and where the relation “over-under” is not taken into consideration, and its subgroup G′ of index 2 (or antisymmetry subgroup G/G′ ), obtained by introducing the relation “over-under”, that represents the actual symmetry of the KL diagram. Symmetry groups of alternating KL diagrams, i.e., groups of isometries that map a KL diagram onto itself, are considered by B. Gr¨ unbaum and 3 G.C. Shephard (1985). From 14 kinds of point groups in ℜ (Coxeter and Moser, 1980), the 8 groups [q], [q]+, [2, q]+, [2+, 2q], [2+, 2q], [2, ∞], [2, q] and [2, q+], where q is a positive integer, can appear as the symmetry groups of knot diagrams (Fig. 1.48). In the case of links, all 14 kinds of point symmetry groups are permitted. Symmetries followed by a change of some bivalent property (e.g., sign +1 ↔ −1, “black” ↔ “white”) are called antisymmetries, and corresponding groups are called antisymmetry groups. In his fundamental work Classification des nœuds et des enlancements, 371 pages published only as a preprint in 1982, A. Caudron introduced the new classification of KLs. Every link L is completely determined by a weighted graph GL , and all KLs are divided into several classes: “worlds”, according to the types of corresponding graphs. A. Caudron introduced complete set of graph transformations necessary for exhaustive derivation of KLs and recognition of equivalent KLs (Fig. 1.49d,e). Definition 1.51. A link L is rational if its graph GL is a line graph, it belongs to the “stellar world” if its graph contains no cycles and has only

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Fig. 1.48 A knot shadow with the symmetry group G = [2, 8] and its antisymmetry subgroup G/G′ = [2, 8]/[2, 8]+ obtained by introducing the relation “over-under”.

one vertex of valence greater than 2, or to the “arborescent world” if its graph is a tree (Fig. 1.49a,b,c). In a certain sense, the origin of J. Conway’s and A. Caudron’s concept are the ideas of T.P. Kirkman, used for the exhaustive derivation and classification of proper KL shadows: systematical introduction of bigons (that he called “flaps”) and classification of the obtained graphs into solid, subsolid, and unsolid ones. According to T.P. Kirkman, a solid KL (or a basic polyhedron in Conway terms) is a diagram containing no bigons, a subsolid KL (or a source KL in our work) is a diagram which admits no simple closed curve in the plane cutting it transversely in just two points (either crossing points or distinct edges), except through the two crossings of a bigon. Any other diagram is unsolid. In the existing knot tables only one memory remained from T.P. Kirkman’s classification principle: solid knots (or basic polyhedra) are at the end of the tables. Even this rule is sometimes disturbed: basic polyhedron 9∗ , “solid” knot 940 , is placed between “subsolid” knots 939 (2 : 2 : 2 0) and 941 (2 0 : 2 0 : 2 0) derived from the basic polyhedron 6∗ (Fig. 1.50). As a principal transformation in the reduction process, T.P. Kirkman used the removal of bigons (Fig. 1.44): an inverse of a systematical adding of bigons used for creating families or subfamilies (Fig. 1.47). In this way, every KL collapses into an irreducible diagram: a solid KL. J. Conway used the same idea, and A. Caudron improved it, so his “worlds” correspond to the different “levels of collapsing”, connected with the mentioned graph properties. Without using Component Algorithm (or some of its equivalents), from

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Fig. 1.49 Graphs corresponding to (a) rational; (b) stellar; (c) arborescent KLs; (d) four graphs of the same non-alternating knot 2 2, 3, −2; (e) graph transformation rules (Caudron, 1982).

weighted or any other graphs corresponding to KLs, it is not possible to directly get the number of components. In every subdivision of KLs according to component number, the ordering principle that follows from graph-theoretical approach will be unavoidably lost. This is partly the case

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Fig. 1.50

The basic polyhedron 9∗ placed between “subsolid” knots.

even with A. Caudron’s paper, where possibilities for some combinatorial results based on partition theory remained unexplored. 1.7

LinKnot functions and KL notation

The program Knot 2000 (K2K) was created by Mitsuyuki Ochiai and Noriko Imafuji from the Graduate School of human culture, Nara Women’s University, Nara, Japan. In 2003 the Mathematica-based program LinKnot written by Slavik Jablan and Radmila Sazdanovi´c from the Mathematical Institute, Belgrade, Serbia, was combined with it. LinKnot is compatible with Knot 2000 and provides tools for working with KLs given in Conway notation with no restriction on the number of crossings. The webMathemathica program LinKnot providing on-line computations and the electronic version of this book is available at the address http://www.math.ict.edu/. An input for the program K2K is a KL diagram drawn by mouse on the mouse tracking window (the function GetPdatabyTracking). The LinKnot function fCreatePData gives the possibility to use a Conway symbol of a KL as an input. For example, from the Conway symbol of the non-alternating link K=”111 ∗ 2.2.2. − 2 0.2 0. − 2 0” it calculates the corresponding P -data {{14, 3}, {−16, −20, −12, 23, 29, −6, −30, 32, −2, −24, −4, 7, −18, 14, −28, 26, 9}}.

P -data is a list having two entries. The first is a list of the numbers of crossings in each component, and the second is a list of numbers derived from the KL, called P-word. P -data are the basic input for all K2K functions.

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For example, the function ShowKnotfromPdata draws KL diagrams as Mathematica graphics (Fig. 1.51 shows the drawing of the nonalternating 2-component link given by its P −data). Function ShowKnot byOpengl gives picture rendered by OpenGL, and provides possibility of rotating it in 3D. The LinKnot function fCreateGraphics exports data to the program KnotPlot written by Robert Sharein (http://www.pims.math.ca/knotplot/ download.html). As an input for this function you can use the Conway notation of KLs as a Mathematica string. The function fCreateGraphics creates the file graphics.txt that can be loaded in KnotPlot by writing “load graphics.txt” in the KnotPlot command line. After that, you can work with it in KnotPlot in the same way as with any KnotPlot image file. Fig. 1.52 shows the PostScript drawing of the alternating 6-component link K = ”101∗2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.2 1.2 1 0” created by the function fCreateGraphics and processed in KnotPlot, before (Fig. 1.52) and after relaxation and 3D-rotation (Fig. 1.53). The LinKnot function DowfromPD produces Dowker code of a KL from P -data. For every KL diagram one can compute its writhe using the function WritheKnotFromPdata, and calculate the P -data of its mirror image using the function GetMirrorImageKnot. For example, for the knot K = ”2 3 4 5” the writhe of K is w = −10, and its mirror image is given by the following P -data:

{{14}, {15, −21, −23, −27, −25, −19, −17, 1, −13, −3, −5, −11, −9, −7}}.

Every notation has its advantages and disadvantages. A minimal Dowker code uniquely defines a KL. Dowker codes or P -data are indispensable as an input into computer programs. However, this type of code is not convenient for us: we prefer a symbolic notation, containing some geometrical-topological contents, like Conway notation. We designed the program LinKnot in attempt to use Conway notation as an input which enables us to work with infinite classes of KLs: subfamilies, families or worlds. This is crucial for understanding general properties of KLs extendable to these classes (families, worlds), computed in a general form, and even expressed by general formulas.

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Fig. 1.51 Drawing of the link 111∗ 2.2.2. − 2 0.2 0. − 2 0 by using the functions fCre atePData and ShowKnotfromPdata.

Fig. 1.52

Drawing of the link 101∗ 2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.2 1.2 1 0.

The LinKnot function Dow calculates a Dowker code with signs for every KL given by its Conway symbol. For example, for the link K=”111∗2.2.2. − 2 0.2 0. − 2 0” it calculates its Dowker code {{14, 3}, {16, 20, 12, −22, −34, 6, 30, −32, 2, 24, 4, −8, 18, −14, −10, 28, −26}}.

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Fig. 1.53

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The same link after a relaxation and rotation.

In addition to LinKnot function fCreatePData we provide the following K2K and LinKnot conversion functions: • K2K conversion functions KnotbyDT and KnotbyDowkerThist lethwaiteNotation converting DT -notation of knots (in Knotscape format) into P -data; • LinKnot conversion functions fPDataFromDow which calculates P data for links given in DT -notation and function fKnotscapeDow which calculates a DT -code for any KL given by its Conway symbol.

1.8

Rational world and KL invariants

The origin (or O-world) is the basic polyhedron 1∗ , the 4-valent graph with one vertex, a usual symbol of infinity (∞) (Fig. 1.43c). The first, linear world (or L-world) contains only one source link: the Hopf link 2 (221 in the classical notation). We use it to derive the family p of alternating KLs, with KL shadows represented by p-gons with bigonal edges (p ≥ 2). For odd p we have the infinite series of alternating knots 3, 5, 7, . . . (or 31 , 51 , 71 , . . .), and for even p the infinite series of alternating 2-component links

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2, 4, 6, . . . (or 221 , 421 , 621 , . . .) (Fig. 1.47). For n > 1 the members of the linear world can be included in the next, rational world. Rational world (or R-world) consists of rational links. Definition 1.52. A rational KL in Conway notation is given as any sequence of natural numbers n1 n2 . . . nk not beginning or ending with 1, where each sequence is identified with its inverse. From this definition we can compute the number of rational KLs with n crossings. Theorem 1.16. The number of rational KLs with n crossings is given by the formula n

2n−4 + 2[ 2 ]−2 which holds for every n ≥ 4. Proof. Think of n as a linearly ordered set of, say, stars; then choosing a composition amounts to choosing a subset of the set of n−1 spaces between the stars. For example, ∗∗ | ∗ | ∗ ∗ ∗ | ∗ (choices of spaces indicated by bars) is the composition 2 1 3 1 of 7. Suppose that n ≥ 3, and let bn denote the number of compositions of n with no 1 in either first or last position, and where a composition is identified with its reverse. Compositions of n not having 1 as either first or last part correspond to sets of spaces between n ordered dots not containing either the first or last space, so there are an = 2n−3 of these. If sn of these are symmetric (i.e. equal to their reverses) then we have an + sn an − sn + sn = . bn = 2 2 Choosing a symmetric composition of n without 1 in first or last place corresponds to choosing a subset of the set of spaces up to and including the middle space (if there is one) but excluding the first space. (The rest of the spaces are determined by symmetry). There are [ n−2 2 ] such spaces, [ n−2 ] 2 and thus sn = 2 . Hence 2n−3 + 2[ bn = 2

n−2 2 ]

= 2n−4 + 2[

n−2 2 −1]

n

= 2n−4 + 2[ 2 ]−2 .

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This simple formula was first derived by C. Ernst and D.W. Sumners (1987) in another form, and later independently by S. Jablan (1999a, 1999b). For n ≥ 4 we can compute the first 20 numbers of this sequence. For n = 3 we have one knot, so the sequence is: 1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, 8256, 16512, 32896, 65792, 131328, 262656, 524800, . . . This sequence is included in On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/∼njas/sequences/) as the sequence A005418. The number of rational knots with n crossings (n ≥ 3) is given by the formula (n−1)

n

2n−3 + 2[ 2 ]−2

(mod 2)

n

+ (−1)(n−1)[ 2 ]

(mod 2)

3 (giving the sequence A090596), so we can derive the formula for the number of rational links with n crossings as well. The LinKnot function RationalKL calculates the number and Conway symbols of all rational KLs for a given number of crossings n. The results are given in the following tables:

n

No.

2 3 4 5 6 7

1 1 2 3 7

8

12

Knots 3 22 5 42 7 313 62 413 3212 2312

32 312 52 2212 512 4112 3113 2222

No.

2112 43 21112 44 332 31112 22112

Links

1

2

1 1 3 3

4 212 6 412

33 3112

222 232

8

8 323 21212

53 3122 211112

422 242

n=9 9 423 3222 2322

72 4212 3213 23112

612 32112

5112 3132

24 Knots 63 54 4122 41112 31212 31122 22122 21312 432 31113

12 Links 414 252

522 342 311112 212112

513 333 2412 2111112

4113 22212

3312 221112

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n = 10 82 532 433 4123 3412 32113 31132 24112 221212

712 5212 4312 4114 3313 321112 311122 2332 2211112

73 442 343 31213 231112 2112112

622 424 3322 311212 22222 10

45 Knots 64 514 4222 41122 33112 31312 311113 23122 212212 27 Links 55 4213 3223 3111112 221122

613 5113 42112 41113 3232 31222 2512 22312 2121112 52 412 321 26 214

3 12 22 2 12

6112 51112 4132 352 32212 312112 2422 222112 21111112 5122 411112 3142 23212 213112

The LinKnot functions RK, R, and RL calculate the number of rational knots, rational KLs, and rational links for a given number of crossings n, respectively. The LinKnot function RatSourceKLNo calculates the number of rational source KLs with n crossings according to the general recursive formula: b[0] = 1, b[1] = 1, b[2n − 2] + b[2n − 1] = b[2n], b[2n] + b[2n − 1] − f [n − 1] = b[2n + 1], where f is the Fibonacci sequence given by the recursion f [0] = 1, f [1] = 1, f [n − 2] + f [n − 1] = f [n]. For n ≥ 4 we obtain the sequence 1, 1, 2, 2, 4, 5, 9, 12, 21, 30, 51, 76, 127, 195, 322, 504, 826, 1309, 2135, 3410 . . ., known as the sequence A102526. Both of these sequences, A005418 and A001224, have been discovered before, but in a different context, related to “Binary grids” and “Packing a box with n dominoes”. Rational source KLs are given in the following table:

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n 2 4 5 6 7 8 9

No. 1 1 2 2 2 4 5

10

9

KLs 2 22 212 222 2212 2222 2212 2111 2222 2221 2111

2 112 2 12 1112

2112 21112 21212 22212

22112 212112

211112 221112

212212 2112112

221122 2121112

221212 2211112

We also have rational generating KLs:

n 2 3 4 5 6 7 8

No. 1 1 1 2 4 6 11

9

18

KLs 2 3 2 3 3 2 2 3 3 3 3 2

2 2 3 3 1 1 3 3 1 2

2 1 1 3 1 1 1

12 3 2 13 112

2 2 3 3

1 2 1 2

2 2 3 3

2 2 3 3

3 1 1 1

2 3 1 1

312 322 332 2 12 22 112

3 2 3 2

1 2 1 1

3 1 2 1

2 22 12 1112

2112 2212 2222

3112 2312

3213 22212 32112

3222 23112 212112

n = 10 2332 22222 31213 32212 221122 311212 2211112

3223 22312 31222 33112 222112 312112 3111112

3232 23122 31312 213112 231112 321112 21111112 No. of KLs: 33

3313 23212 32113 212212 311113 2112112

3322 31132 32122 221212 311122 2121112

Results in the tables above were computed with the LinKnot function RatGenSourKL which gives the number and Conway symbols of all rational source KLs and rational generating KLs with n crossings. We can recognize a regular distribution of the source links and their corresponding

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alternating KLs (e.g., of the type 2 . . . 2, where 2 occurs l times, abbreviated as 2l ; they are knots for even l and 2-component links for odd l) (Figs. 1.54-1.55). In particular, the number of components of rational KLs is always 1 or 2. The LinKnot function fComponentNo calculates the number of components of an arbitrary KL, but there are some more elegant solutions.

Fig. 1.54

Shadows of source links (where exponent l denotes l-fold repetition).

The first solution for rational KLs is given below: take a rational link in Conway notation and reduce all numbers mod 2. Then apply the following cancellation rules: (1) for every sequence of the form xa0 (a ∈ {0, 1}), xa0 = x; (2) for every sequence of the form xa1 (a ∈ {0, 1}), a1 = 1 − a. Theorem 1.17. If the Boolean function f satisfies the conditions f (0) = 0, f (1) = 1, f (xy) = 1 − f (x)f (y), then L is a knot if f (L) = 1, and 2component link if f (L) = 0 (Caudron, 1982).

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Fig. 1.55

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The corresponding alternating source links.

For example, we conclude that 6 2 1 4 4 is a knot, because 6 2 1 4 4 = 0 0 1 0 0 (mod 2), 0 0 1 0 0 = 0 0 1 (Rule 1), 0 0 1 = 0 1 (Rule 2) and f (0 1) = 1 − f (0)f (1) = 1. On the other hand, 4 5 1 2 3 is a link, since 4 5 1 2 3 = 0 1 1 0 1 (mod 2), 0 1 1 0 1 = 0 1 1 1 = 0 1 0 (Rule 2), 0 1 0 = 0 (Rule 1), and f (0) = 0. The second solution is actually a trick. Knowing that a generating KL and all KLs derived from it, which make a subfamily, have the same number of components (Theorem 1.14), we can calculate the number of components only for generating KLs and extend this result to all KLs derived from them. For example, 2 2 is a generating 1-component KL (i.e., a figure-eight knot), and all KLs belonging to the subfamily (2p) (2q) (p ≥ 2, q ≥ 2, p ≥ q) will be knots. On the other hand, the generating KL 3 3 is a 2-component link, and all KLs in its subfamily (2p + 1) (2q + 1) (p ≥ 1, q ≥ 1, p ≥ q) will be 2-component links. This is just the first and

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simplest property which can be generalized from generating KLs to their subfamilies. A link (projection) is oriented if each of its components has an orientation (Definition 1.29). The LinKnot function fOrientedLink calculates Gauss codes of a link projection given by the Conway symbol, Dowker code, or P -data. The Gauss codes obtained correspond to different orientations of components (the first part of the data obtained). The second part gives the orientations of components (where + is denoted by 1, and − by 0), and the third part is the (signed) linking number of the corresponding oriented link. As we can conclude from their description, all rational KLs are alternating, because their Conway symbols do not contain negative entries. The product of tangles is the only operation used for obtaining rational KLs. Including negative elementary tangles into the Conway notation of rational KLs gives nothing new, because all rational KLs with mixed signs can be reduced to Conway symbols containing only positive or negative entries. For this reduction we use continued fractions. In particular, we have integer tangles n0 = 1 + . . . + 1 – bigons or chains of bigons, and rational tangles n1 . . . nk . To a rational tangle n1 n2 . . . nk corresponds the continued fraction 1 p r = = nk + 1 q nk−1 + 1 ··· + 1 n2 + n1 where the rational number r = pq (GCD(p, q) = 1) is the slope of the rational tangle. A rational knot or link L( pq ) is a knot if p is odd, and link if p is even. For rational tangles and rational KLs the following theorem holds: Theorem 1.18. Two rational tangles are equivalent iff their continued fractions yield the same rational number (Conway, 1970). ′ Unoriented rational links L( pq ) and L( pq′ ) are ambient isotopic iff: (1) p = p′ and (2) either q ≡ q ′ (mod p) or qq ′ ≡ 1 (mod p) (Schubert, 1956). The proof of the Theorem 1.18 can be found in J.M. Montesinos (1984), G. Burde and H. Zieschang (1985), or in the papers by J.R. Goldman and L. Kauffman (1997), and the direct combinatorial proof is given by

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L. Kauffman and S. Lambropoulou (2003). For further reading on rational tangles and rational KLs we recommend papers by L. Kauffman and S. Lambropoulou (2002, 2002a). The LinKnot function RatReduce reduces a rational KL by using continued fractions. In this way, from any Conway symbol of a rational KL with entries containing mixed signs (that is a non-alternating rational representation of a rational KL) we can obtain its minimal alternating representation. For example, from a non-alternating representation of a rational link given by 2 − 3 1 2 − 7 we obtain its minimal alternating representation 6 1 2 1 2. The LinKnot function MSigRat calculates the value of a continued fraction and Murasugi signature (see: Murasugi, 1996, pp. 192) for every rational KL given by its Conway symbol. 1.8.1

Chirality of rational KLs

Rational KLs are the main class of KLs for which we are able to analyze various general properties and construct large (infinite) subclasses of KLs satisfying these properties. Such a property is chirality: a KL is achiral (or amphicheiral) if its “left” and “right” forms are equivalent, meaning that one can be transformed to the other by an ambient isotopy (Liang and Mislow, 1994a, 1994b, 1995). We distinguish chirality of non-oriented and oriented KLs. Definition 1.53. An oriented KL is achiral if there is an orientation preserving ambient isotopy transforming the oriented link L into the oriented mirror image of the link L. For example, Hopf link 222 (2) is achiral as non-oriented and chiral as an oriented link. Theorem 1.19. An oriented knot K is achiral iff it can be represented by an antisymmetric vertex-bicolored graph on a sphere (Liang and Mislow, 1994b). For an oriented achiral knot K there exist a symmetry transposing orientations of crossings, i.e., mutually exchanging crossings with the signs +1 and −1. Its antisymmetries (this means sign-changing symmetries) can be sense-reversing– rotational antireflection and anti-inversion, or sensepreserving– 2-antirotation.

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Fig. 1.56 tation).

Antisymmetric 3-D-presentation of the figure-eight knot (2 2 in Conway no-

For the knot 2 2, the graph symmetry group is G = [2+, 4], and the knot symmetry group G′ = [2+, 4+] is generated by the rotational reflection, with the axis defined by the midpoints of colored (i.e., double) edges of the tetrahedron (Fig. 1.56). If we take into account the signs of crossings, it is a rotational antireflection. Its effect is preserved in all rational knots with an even number of crossings which have a mirror-symmetric (palindromic) Conway symbol. Theorem 1.20. A rational knot is achiral iff its Conway symbol is mirrorsymmetric and has an even number of crossings (Siebenmann, 1975; Caudron, 1982). For 4 ≤ n ≤ 12, achiral rational knots (Fig. 1.57) are: n=4 22 n=6 2112 44

3113

n=8 2222

4114

311113

n = 10 2332

212212

21111112

5115 312213

n = 12 4224 222222

3333 22111122

2442 21211212

66 321123 2111111112

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Fig. 1.57

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Centro-antisymmetric presentations of achiral rational knots (n ≤ 10).

Definition 1.54. An oriented KL is invertible if it remains the same after reversing orientations of all its components. Theorem 1.21. All oriented rational KLs are invertible (Kauffman and Lambropoulou, 2002a). No general technique is known for determining if a knot is invertible. Definition 1.55. A period of a knot projection is the order of rotation that transforms it to itself. A knot K has a period p if it has a projection with the period p. According to this, a knot can have several different periods. For example, a trefoil knot has periods 2 and 3. The period of the knot 2 2 is 2, and the same holds for all rational achiral knots. If the antisymmetry group of the vertex-bicolored graph corresponding to an achiral oriented projection

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of a knot K contains at least one sense-reversing antisymmetry, K is invertible achiral knot; otherwise, K is non-invertible. For the invertibility of achiral knots we have an additional criterion: if a period of achiral knot is 2, it is invertible; otherwise, it is non-invertible. The LinKnot function RationalAmphiK calculates the number and Conway symbols of all rational achiral knots for a given number of crossings n. The number of achiral knots for n = 2k (k = 1, 2, 3, . . .) yields the Jacobsthal sequence 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, . . . defined recursively with a[1] = 0, a[2] = 1, 2a[k] + a[k + 1] = a[k + 2] and given by the general formula 2n − (−1)n . 3 All rational links are 2-component. Since all oriented alternating links with an even number of components are chiral (Cerf, 1997), there are no oriented rational achiral links. For 2 ≤ n ≤ 12, achiral non-oriented rational links are: a[n] =

n=2 2 n=6 33 n=8 211112 n = 10 55

3223

221122 n = 12

411114 21122112 77 322223 22122122

31111113

231132

n = 14 5225 421124 31211213 241142 2121111212 211111111112

213312

3443 223322

Theorem 1.22. A rational non-oriented link is achiral iff its Conway symbol is mirror-symmetric (palindromic) and has an even number of crossings (Kauffman and Lambropoulou, 2002). The LinKnot function RationalAmphiL calculates the number and Conway symbols of all rational achiral non-oriented links for a given number

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of crossings n. For n = 2 (k = 1) there is one achiral non-oriented link: Hopf link 2 (221 ). Calculating the number of achiral non-oriented rational links for n = 2k (k = 2, 3, 4, . . .) we again obtain the Jacobsthal sequence.

1.9

Unlinking number and unlinking gap

The question of unknotting and unlinking numbers, or Gordian numbers is one of the most difficult in knot theory (Wendt, 1937; Nakanishi, 1981; Kawauchi, 1996; Kohn, 1991, 1993). General formulas for unknotting number are known only for some special classes of knots. For example, according to the famous Milnor conjecture (1968) proved by P.B. Kronheimer and T.S. Mrowka (1993, 1995), the unknotting number of a torus knot [p, q] is . In order to calculate unknotting and unlinking numbers, u = (p−1)(q−1) 2 we need link surgery. In every crossing of a KL it is possible to make a crossing change: switch an overcrossing to undercrossing or vice versa (Fig. 1.58a).

Fig. 1.58 A crossing change (a) and the Nakanishi-Bleiler example: (b) the minimal projection of the knot 5 1 4 that requires at least three crossing changes to be unknotted; (c) the minimal projection of the knot 3 1 2 with the unknotting number 1; (d) nonminimal projection of the knot 5 1 4 from which we obtain the correct unknotting number u(5 1 4) = 2.

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Definition 1.56. The unlinking number u(L) of a link L is the minimal number of crossing changes required to obtain an unlink from the link L, where the minimum is taken over all diagrams of L. The first estimation of the unlinking number follows from the theorem: Theorem 1.23. Let D(L) be a diagram of link L with n crossings. Making at most [ n2 ] crossing changes transforms D(L) into a diagram of an unlink (see, e.g., Adams, 1994; Gilbert and Porter, 1994). Corollary 1.2. For an n-crossing link L and its unlinking number u(L) the following inequality holds: u(L) ≤ [ n2 ]. The estimation given in this corollary is a very rough, but still useful. Definition 1.57. A link L will be called difficult if u(L) = [ n2 ]. Difficult knots are extremely rare, and the question of finding all difficult KLs with a given number of crossings n is open. Conjecture 1.3. The only difficult knots belong to the family 2p + 1 in Conway notation (knots 31 , 51 , 71 ,...). However, there is more then one family of difficult links, e.g., the family 2p in Conway notation (the links 221 , 421 , 621 ,...), all pretzel links of the form 2k1 , 2k2 , ..., 2kl (l ≥ 3), etc. There are two different, yet equivalent approaches for obtaining the unlinking number of a link L (see, e.g., C. Adams, 1994): (1) according to the classical definition, one is allowed to make a planar isotopy after each crossing change and then continue the unlinking process with the newly obtained projection, until an unlink is obtained; (2) the standard definition requires all crossing changes to be done simultaneously in a fixed projection. Unfortunately, both definitions are unsuitable for calculations, since there are infinitely many projections of any KL. From the well known example of the knot 108 (or 5 1 4 in Conway notation), given by Y. Nakanishi (1983) and S. Bleiler (1984), the restriction to minimal projections is not allowed. The rational knot 5 1 4 has only one minimal projection (Fig. 1.58b). According to the standard definition, unknotting takes at least three crossing changes (in the crossings denoted by circles).

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On the other hand, making a crossing change in the middle point of the diagram (Fig. 1.58b) followed by the reduction 5 −1 4 = 3 1 2, we obtain the minimal projection of the knot 3 1 2 (Fig. 1.58c) that can be unknotted by one crossing change. Hence, in this case unknotting according the classical definition, using only minimal projections, gives correct unknotting number 2. The unknotting number can also be obtained from the non-minimal projection of the knot 5 1 4 (Fig. 1.58d) using the standard definition. As it was shown by J. Bernhard (1994), the same property holds for the whole knot family (2k + 1) 1 (2k), k ≥ 2. Nakanishi-Bleiler example motivated us to define BJ-unlinking number which is computable due to the algorithmic nature of its definition. Definition 1.58. For a given crossing v of a diagram D representing link L, let Dv denote the link diagram obtained from D by switching crossing v (Fig. 1.58a). a) The unlinking number u(D) of a link diagram D is the minimal number of crossing changes on the diagram required to obtain an unlink. b) The classical unlinking number of a link L, denoted by u(L) can be defined by u(L) = min u(D) where the minimum is taken over all D

minimal diagrams D representing L. c) The BJ-unlinking number uBJ (D) of a diagram D is defined recursively in the following manner: (1) uBJ (D) = 0 iff D represents an unlink. (2) uBJ (D) = 1 + min uBJ (Dv ) where the minimum is taken over all Dv

minimal diagrams of a link represented by Dv for which the value is already defined. d) uM (L) = min u(D) where the minimum is taken over all minimal D

diagrams D representing L. e) The BJ-unlinking number uBJ (L) of a link L uBJ (L) = min uBJ (D) D

where the minimum is taken over all minimal diagrams D representing L. J.A. Bernhard (1994) and independently S. Jablan in 1995 (Jablan, 1998), conjectured: Bernhard-Jablan Conjecture For every link L we have that u(L) = uBJ (L).

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This means that we take all minimal projections of a KL, make a crossing change in every crossing, and then minimize all the projections obtained. The same algorithm is applied to the first, second, . . . k th generation of the KLs obtained. The BJ-unlinking number is the minimal number of steps k in this recursive unlinking process. If BJ-conjecture does not hold for all KLs, it may be true for some restricted class, e.g., for all rational KLs. Notice that, even if BJ-conjecture does not hold for all links, BJ-unlinking number is the best computable upper bound for the unlinking number, since δBJ (L) ≤ δ(L). Going further in the same direction, we conjecture that unlinking number is a linear subfamily-dependent KL invariant (Conjecture 1.2). We illustrate the importance of the conjecture by the following example. Consider alternating pretzel knots a, b, c where 1 < a ≤ b ≤ c and a, b, c are all odd numbers. We show in the following proposition that uBJ (a, b, c) = a+b 2 . However, the unknotting number of these knots is still unknown, except for the smallest knots, e.g., 3, 3, 3 with unknotting number 3, computed recently by B. Owens (2005). Proposition 1.1. a) for rational knots with the Conway symbol (2m+ 1) 1 (2n+ 1), (m ≥ n) BJ-unknotting number is uBJ = n + 1; b) for pretzel knots a, b, c (1 < a ≤ b ≤ c)4 , where a, b, c are odd numbers: uBJ (a, b, c) =

a+b 2

For mutant KLs, the following lemma holds: Lemma 1.1. If m(D) is a mutation of a diagram D, then u(D) = u(m(D)). Since a flype can be viewed as a special case of mutation, according to the Tait’s Flyping Theorem (Theorem 1.11) (Menasco and Thistlethwaite, 1991, 1993) and Lemma 1.1, all reduced minimal diagrams of an alternating KL have the same BJ-unlinking number, so it is sufficient to use only one minimal diagram. On the contrary, we need to consider all minimal diagrams of nonalternating KLs, since two minimal diagrams of the same non-alternating KL can have different BJ-unknotting numbers, as in the example by 4 Proposition

1.1 also holds for rational knots obtained for a = 1.

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A. Stoimenow (2001). Two minimal diagrams of the same 14-crossing knot 14n36750 (in the Knotscape notation), given by the Dowker codes 4 14 18 12 28 26 − 22 2 − 24 10 8 − 16 6 20 and 6 − 10 26 20 − 2 − 18 − 22 28 − 24 8 − 14 − 12 4 16 (Fig. 1.59), give two different BJ-unknotting numbers, 1 and 2, respectively.

Fig. 1.59

Two projections of the knot 1436750 .

(Murasugi, 1965; K. Murasugi proved the inequality u(K) ≥ |σ(K)| 2 Cromwell, 2004). Because uBJ (L) ≥ u(L), it follows that if BJ-unknotting number of a knot K is equal to half of the signature, σ(K) 2 , then it is equal to the unknotting number. In the case of links the same holds if BJ-unlinking (Murasugi, 1965). Additional criteria number of a link L is equal to σ(L)+1 2 for checking estimated unlinking numbers are given by P. Kohn (1993). Unknotting numbers computed according to the BJ-conjecture coincide with all unknotting numbers (n ≤ 10) from the book A Survey of Knot Theory by A. Kawauchi (Appendix F) (1996) and Kawauchi’s table updated with reference to recent unknotting number results (Livingston and Cha, 2005), if in all ambiguous cases (A=1 or 2; B=2 or 3) we take the bigger number. Hence, if any of these unknotting numbers is smaller than its maximal estimated value, this will be a counterexample for the BJ-conjecture. BJ-conjecture holds for all two-component links whose unlinking numbers were computed by P. Kohn (1993). The complete list of

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BJ-unknotting numbers for knots with n = 11 and n = 12 crossings computed by the authors, using the program LinKnot, is included in the Table of Knot Invariants by C. Livingston and J.C. Cha (2005). The most recent results by B. Owens (2005) and Y. Nakanishi (2005) confirmed the unknotting number u = 3 for the knot 935 , and the unknotting number u = 2 for the knots 1083 , 1097 , 10105 , 10108 , 10109 , and 10121 computed according to the BJ-conjecture. The BJ-conjecture was first introduced by J.A. Bernhard in 1994 and then independently proposed by S. Jablan in 1995, when it was effectively used for the calculation of BJ-unknotting numbers of the knots with n ≤ 10 crossings. LinKnot function UnR calculates BJ-unknotting and BJ-unlinking numbers of rational KLs, and UnKnotLink calculates BJ-unknotting (unlinking) numbers of all KLs. The output of both functions is a sequence of integers between lower bound coming from signature and upper bound uBJ . If they coincide, we obtain the value for the unlinking number. Y. Nakanishi (1996) and A. Stoimenow (2004) proved that the BJconjecture holds for one subclass of rational KLs: Theorem 1.24. All rational KLs with unlinking number one have an unlinking number one minimal diagram. This theorem is the corollary of the Theorem 1.28 and 1.29. Definition 1.59. The diagram unlinking number uD (L) is the minimal number of simultaneous crossing changes on D required to obtain a diagram of an unlink.

Definition 1.60. The BJ-unlinking gap of a diagram D, denoted by δBJ (D), is the difference δBJ (D) = u(D) − uBJ (D). Definition 1.61. The BJ-unlinking gap of a link L, denoted by δBJ (L), is defined by δBJ (L) = uM (L) − uBJ (L) or equivalently δBJ (L) = min δ(D) D

where D denotes minimal diagram of L.

According to Tait’s First Conjecture (Murasugi 1996, Theorem 11.5.5), proved independently in 1986 by L. Kauffman, K. Murasugi and

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M. Thistlethwaite (Kauffman, 1987b; Murasugi, 1987a,b; Thistlethwaite, 1987, 1988) and known after that as Kauffman-Murasugi Theorem (Theorem 1.6), a reduced alternating projection is the minimal projection of a prime alternating KL. In the case of alternating KLs, from this theorem and Lemma 1.1 it follows that all reduced minimal diagrams of a link L will have the same unlinking gap δM (L), so a particular value δM (L) will be the BJ-unlinking gap δBJ (L) of a link L. According to the experimental results, BJ-unlinking number can be viewed as the property of a whole family. For example, for all knots p 2, p 1 1 p, (p + 2) 1 1 p, p 2 1 2, p 1 1 1 2, uBJ = u = 1 and does not depend on p, while knots of the family p (p = 2k + 1) have uBJ = u = k. Unlinking number can be generalized to a metric on space of links, where the distance between links is defined as follows: Definition 1.62. A distance of a link projection L′1 from a link projection L′2 is the minimal number of crossing changes in L′1 required to obtain L′2 . A distance of links L1 and L2 (or Gordian distance), denoted by d(L1 , L2 ), is the minimal number of crossing changes in L1 required to obtain L2 , the minimum taken over all projections of L1 and L2 . Distance of links satisfies all properties of a metric (Darcy and Sumners, 1998): (1) d(L1 , L2 ) = 0 iff L1 = L2 ; (2) d(L1 , L2 ) = d(L2 , L1 ); (3) d(L1 , l2 ) ≤ d(L1 , L) + d(L, L2 ). In particular, for unlinking numbers the following inequality holds: d(L1 , L2 ) ≤ u(L1 ) + u(L2 ). Theorem 1.25. For the distance of knots K1 and K2 , the following in2 )| (Murakami, 1985; Kawauchi, equality holds: d(K1 , K2 ) ≥ |σ(K1 )−σ(K 2 1996). Computing distances of KLs is a recursive process whose basic step is finding pairs of KLs with the Gordian distance 1, i.e., all pairs of KLs where one of them can be obtained from a diagram of the other by one crossing change. In this case, both diagrams considered are not necessarily minimal. Two rational links L1 and L2 have the Gordian distance 1 iff they can be expressed as L1 = t1 , t2 , ±2, L2 = t1 , t2 , where t1 and t2 are rational

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tangles with the slopes r1 , r2 , respectively (Dazey and Sumners, 1997; Torisu, 1998). I. Darcy and D.W. Sumners (1998) computed distances of oriented knots with at most n = 8 crossings. We have computed all pairs of mutually different non-oriented knots with the Gordian distance 1 for n ≤ 9. For n ≤ 9 there are (including the unknot) 85 prime knots, ordered according to the classical notation (Rolfsen, 1976). In the following table every knot with n ≤ 9 crossings is given by its ordering number, classical, and Conway symbol. (1, 11 , 1) (2, 31 , 3) (3, 41 , 2 2) (4, 51 , 5)

(5, 52 , 3 2)

(6, 61 , 4 2) (9, 71 , 7) (13, 75 , 3 2 2)

(7, 62 , 3 1 2)

(8, 63 , 2 1 1 2)

(10, 72 , 5 2) (14, 76 , 2 2 1 2)

(16, 81 , 6 2) (20, 85 , 3, 3, 2) (24, 89 , 3 1 1 3) (28, 813 , 3 1 1 1 2) (32, 817 , .2.2) (36, 821 , 2 1, 2 1, −2)

(11, 73 , 4 3) (15, 77 , 2 1 1 1 2)

(17, 82 , 5 1 2) (21, 86 , 3 3 2) (25, 810 , 2 1, 3, 2) (29, 814 , 2 2 1 1 2) (33, 818 , 8∗ )

(37, 91 , 9) (40, 94 , 5 4) (43, 97 , 3 4 2) (46, 910 , 3 3 3) (49, 913 , 3 2 1 3) (52, 916 , 3, 3, 2+) (55, 919 , 2 3 1 1 2) (58, 922 , 2 1 1, 3, 2) (61, 925 , 2 2, 2 1, 2) (64, 928 , 2 1, 2 1, 2+) (67, 931 , 2 1 1 1 1 1 2) (70, 934 , 8∗ 2 0) (73, 937 , 2 1, 2 1, 3) (76, 940 , 9∗ ) (79, 943 , 2 1 1, 3, −2) (82, 946 , 3, 3, −3) (85, 949 , −2 0 : −2 0 : −2 0)

(12, 74 , 3 1 3)

(18, 83 , 4 4) (22, 87 , 4 1 1 2) (26, 811 , 3 2 1 2) (30, 815 , 2 1, 2 1, 2) (34, 819 , 3, 3, −2)

(38, 92 , 7 2) (41, 95 , 5 1 3) (44, 98 , 2 4 1 2) (47, 911 , 4 1 2 2) (50, 914 , 4 1 1 1 2) (53, 917 , 2 1 3 1 2) (56, 920 , 3 1 2 1 2) (59, 923 , 2 2 1 2 2) (62, 926 , 3 1 1 1 1 2) (65, 929 , .2.2 0.2) (68, 932 , .2 1.2 0) (71, 935 , 3, 3, 3) (74, 938 , .2.2.2) (77, 941 , 2 0 : 2 0 : 2 0) (80, 944 , 2 2, 2 1, −2) (83, 947 , 8∗ − 2 0)

(19, 84 , 4 1 3) (23, 88 , 2 3 1 2) (27, 812 , 2 2 2 2) (31, 816 , .2.2 0) (35, 820 , 3, 2 1, −2)

(39, 93 , 6 3) (42, 96 , 5 2 2) (45, 99 , 4 2 3) (48, 912 , 4 2 1 2) (51, 915 , 2 3 2 2) (54, 918 , 3 2 2 2) (57, 921 , 3 1 1 2 2) (60, 924 , 2 1, 3, 2+) (63, 927 , 2 1 2 1 1 2) (66, 930 , 2 1 1, 2 1, 2) (69, 933 , .2 1.2) (72, 936 , 2 2, 3, 2) (75, 939 , 2 : 2 : 2 0) (78, 942 , 2 2, 3, −2) (81, 945 , 2 1 1, 2 1, −2) (84, 948 , 2 1, 2 1, −3)

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Vertices of the following graph G1 are ordering numbers of knots from the previous table, and the edges connect knots with the Gordian distance 1 (Fig. 1.60). G1 = {{2, 1}, {3, 1}, {4, 2}, {5, 1}, {5, 2}, {5, 4}, {6, 1}, {6, 3}, {7, 1}, {7, 2}, {7, 3}, {7, 6}, {8, 1}, {8, 2}, {9, 4}, {10, 1}, {10, 5}, {11, 4}, {11, 5}, {11, 9}, {11, 10}, {12, 2}, {12, 5}, {12, 11}, {13, 2}, {13, 4}, {13, 5}, {13, 9}, {13, 10}, {14, 1}, {14, 2}, {14, 3}, {14, 5}, {14, 10}, {15, 1}, {15, 2}, {15, 3}, {16, 1}, {16, 6}, {17, 2}, {17, 4}, {17, 7}, {18, 6}, {18, 16}, {19, 3}, {19, 6}, {19, 7}, {19, 18}, {20, 7}, {21, 2}, {21, 6}, {21, 7}, {21, 16}, {21, 17}, {21, 19}, {21, 20}, {22, 1}, {22, 2}, {22, 4}, {22, 8}, {23, 2}, {23, 5}, {23, 8}, {23, 22}, {24, 1}, {24, 7}, {24, 19}, {25, 4}, {25, 5}, {25, 8}, {25, 23}, {26, 1}, {26, 5}, {26, 6}, {26, 7}, {26, 16}, {26, 17}, {27, 3}, {27, 6}, {27, 16}, {28, 1}, {28, 3}, {28, 5}, {28, 8}, {28, 22}, {29, 1}, {29, 2}, {29, 3}, {29, 5}, {29, 7}, {29, 17}, {30, 4}, {30, 5}, {31, 2}, {31, 3}, {31, 5}, {31, 8}, {32, 1}, {32, 2}, {32, 7}, {33, 2}, {34, 4}, {34, 13}, {34, 30}, {35, 1}, {35, 5}, {35, 8}, {35, 14}, {35, 25}, {36, 1}, {36, 4}, {36, 7}, {36, 20}, {37, 9}, {38, 1}, {38, 10}, {39, 9}, {39, 11}, {40, 10}, {40, 11}, {41, 5}, {41, 10}, {41, 12}, {42, 4}, {42, 9}, {42, 13}, {43, 2}, {43, 10}, {43, 13}, {44, 2}, {44, 3}, {44, 6}, {44, 14}, {45, 9}, {45, 11}, {45, 13}, {46, 11}, {46, 12}, {47, 4}, {47, 5}, {47, 11}, {47, 14}, {48, 1}, {48, 6}, {48, 10}, {48, 14}, {49, 4}, {49, 11}, {49, 12}, {49, 13}, {50, 1}, {50, 2}, {50, 5}, {50, 6}, {50, 15}, {51, 3}, {51, 5}, {51, 10}, {51, 14}, {52, 13}, {52, 34}, {53, 3}, {53, 7}, {53, 15}, {54, 5}, {54, 10}, {54, 11}, {54, 13}, {55, 1}, {55, 3}, {55, 6}, {55, 15}, {56, 2}, {56, 5}, {56, 7}, {56, 13}, {56, 14}, {57, 1}, {57, 2}, {57, 5}, {57, 12}, {57, 14}, {58, 1}, {58, 7}, {58, 8}, {58, 15}, {59, 4}, {59, 5}, {59, 13}, {60, 1}, {60, 7}, {60, 14}, {60, 35}, {61, 5}, {61, 6}, {61, 7}, {61, 14}, {62, 1}, {62, 2}, {62, 4}, {62, 5}, {62, 7}, {62, 15}, {63, 1}, {63, 2}, {63, 3}, {63, 7}, {63, 8}, {63, 14}, {64, 1}, {64, 8}, {64, 36}, {65, 8}, {65, 14}, {65, 15}, {66, 1}, {66, 6}, {66, 7}, {66, 8}, {67, 2}, {67, 4}, {67, 5}, {67, 8}, {68, 2}, {68, 3}, {68, 5}, {68, 6}, {68, 7}, {68, 15}, {68, 36}, {69, 1}, {69, 2}, {69, 3}, {69, 8}, {69, 14}, {69, 35}, {70, 1}, {70, 2}, {70, 3}, {70, 7}, {70, 15}, {71, 12}, {72, 4}, {72, 13}, {72, 14}, {73, 6}, {73, 15}, {74, 4}, {74, 12}, {74, 13}, {75, 1}, {75, 7}, {75, 12}, {75, 14}, {76, 3}, {76, 36}, {77, 5}, {77, 15}, {78, 1}, {78, 6}, {78, 7}, {78, 15}, {78, 21}, {78, 27}, {78, 32}, {79, 4}, {79, 5}, {79, 7}, {79, 14}, {79, 29}, {80, 1}, {80, 6}, {80, 8}, {80, 14}, {80, 15}, {80, 23}, {80, 31}, {81, 1}, {81, 4}, {81, 5}, {81, 7}, {81, 13}, {81, 14}, {82, 2}, {82, 6}, {82, 15}, {82, 26}, {82, 36}, {83, 2}, {83, 6}, {83, 15}, {84, 2}, {84, 12}, {84, 35}, {85, 4}, {85, 12}}.

The Mathematica function ShortestPath can compute minimal Gordian distances between knots, going only via prime knots and without distinguishing handness. In particular, the number of edges in a shortest path from any vertex v to the vertex 1 (denoting the unknot) gives the BJunknotting number of the knot corresponding to the vertex v going only via prime knots5 . In the graph G1 , most frequently occurs the unknot (35 times), then trefoil 3 (31 ) and knot 3 2 (52 ) (31 times each). The Gordian distance 1 for alternating knots can be achieved only through non-minimal diagrams. The same holds for distance 1 between 5 Compare

with Diao, Ernst, and Stasiak (2006).

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Fig. 1.60

Graph G1 .

“left” and “right” form of the same knot. Examples of alternating knots with n crossings (n ≤ 8) and the Gordian distance 1 achieved from the non-minimal diagrams with n + 1 crossings are: n=5 n=6 n=7

n=8

(5),(3 2) (4 2),(3 1 2) (7),(4 3) (5 2),(4 3) (5 2),(2 2 1 2) (2 3 1 2), (2 1, 3, 2) (3, 3, 2), (2 1, 2 1, −2) (3, 2 1, −2), (2 1, 3, 2) (6 2), (2 2 2 2) (4 1 1 2), (3 1 1 1 2) (5 1 2), (3 2 1 2) (3 3 2), (3, 3, 2) (5 1 2), (3 3 2) (4 4), (4 1 3)

(7),(3 2 2) (5 2),(3 2 2) (4 3),(3 1 3) (3, 3, −2), (2 1, 2 1, 2) (5 1 2), (2 2 1 1 2) (4 1 1 2), (2 3 1 2) (4 1 3), (3 1 1 3) (6 2), (3 2 1 2) (4 1 3), (3 3 2) (6 2), (3 3 2) (6 2), (4 4)

For example, the shortest path from 3 1 3 to 7 is 3 1 3 → 4 3 → 7 and goes via non-minimal diagrams with n = 8 crossings (Fig. 1.61).

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Fig. 1.61

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Transition 3 1 3 → 4 3 → 7 via non-minimal diagrams with n = 8 crossings.

Knots 4 2 (61 ) and 2 3 1 2 (88 ) are examples of knots with the distance 1 from their mirror images (Fig. 1.62). The LinKnot function AllStatesRational uses a Conway symbol of a rational KL projection P as an input and makes all possible combinations of crossing changes on P , i.e., choices of signs ±. The output is the unlinking number of the fixed projection P , followed by a list of ordered pairs where each second entry is a rational knot or link L, and the first entry the minimal number of crossing changes required to obtain L from P . The function AllStatesRational is very fast, since it is based on continued fractions. A similar, but much slower function, based on the function Re ductionKnotLink, can be applied to all KLs. It gives the same output, where knots or links L are given by their reduced P -data. The functions AllStatesRational and UnR can be used to illustrate the Nakanishi-Bleiler example (knot 5 1 4), or similar examples of knots from the family (2k + 1) 1 (2k), showing that the unknotting number of the fixed minimal projection of a knot (2k + 1) 1 (2k) is k + 1, and its BJ-unknotting number is k, so the BJ-unknotting gap is δBJ = 1. We can also consider links from the family (2k) 1 (2k) (k > 1). A fixed minimal projection of a link (2k) 1 (2k) can be unlinked with minimum 2k−1 crossing changes, and BJ-unlinking number, computed by the function UnR, is k, so the BJ-unlinking gap is k − 1.

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Fig. 1.62 change.

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(a) Transition from 4 2 and (b) from 2 3 1 2 to its mirror image by one crossing

In order to make a more complete and detailed research of unlinking gap, we wrote some additional LinKnot functions. LinKnot function UnR FixProj calculates unlinking number uD of a fixed diagram D of a given rational KL. The additional function fGapRat calculates the BJ-unlinking number of a given rational link L, the unlinking number of its fixed minimal projection, detects rational KLs with BJ-unlinking gap, and computes its value δBJ (L). The calculation of unlinking gaps for rational KLs is very fast because the functions UnR and fGapRat are based on continued fractions. Similar, but much slower functions UnKnotLink and fGap, based on the function ReductionKnotLink, are used for the calculation of (diagram) unlinking gap for non-rational KL diagrams. If the BJ-conjecture is not true in general, the list of KLs with unlinking gap given here will remain correct, but not exhaustive, because for every KL from that list the unlinking number obtained from a minimal projection uM (L) is greater than the BJ-unlinking number uBJ (L) which represents the upper bound for the unlinking number u(L), i.e., uM (L) > uBJ (L) ≥ u(L).

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However, in that case our list may not be complete: KLs whose unlinking number is smaller than the BJ-unlinking number will also have an unlinking gap. The following tables contain the Conway symbols of rational knots and links up to 16 crossings with a non-trivial BJ-unlinking gap, given according to the number of crossings and whether they are knots or links. Symbols given in bold denote the links with the BJ-unlinking gap 2 (the others have BJ-unlinking gap 1). The first column in each table gives the number of crossings, second the number of KLs with non-trivial BJ-unlinking gap, and the third column their list.

n 9 10 11 12

No. of KLs 1 Link 1 Knot 1 Knot 4 Links 5 Knots

List of all KLs 414 514 4142 434 614 5132 51113 714 534 4143 6132 61113 n = 13

4414 511132

6142 513112

6 1 6 6133 61123 4211113

634 7132 71113

7 Knots 41314

51322

231412

16 Links 814 34132 241312

5152 41422 411142

5332 51115 611122

n = 14 716 5414 8132 51323 81113 511142 5211113

734 6143 33152 51422 313142 711122

31 Knots 914 6152 35132 61124 314132 2141312

4163 6332 41423 61322 341312 2411132

4343 7133 51314 71123 351113 4211114

41432

41612

5 Links 413132

513212

5131112

Among the links with n = 13 crossings we find the first link 6 1 6 with the BJ-unlinking gap δBJ = 2.

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n = 15 21314112 241422 341322 414222 41712 45132 513114 51522 611232 61332 711132

43 Knots 2241312 24111312 321162 331512 4111422 4141113 41433 41514 42111132 4434 4614 5111214 513213 51324 51612 53322 6131112 61314 6144 6162 713112 8142

10 1 4 241512 341313 35133 4141212 421152 4311213 5111223 51513 54132 61224 6211113 711123 71313 811122 834

63 Links 21413112 23111412 25111212 251322 34152 34332 36132 41111232 41442 41622 4221123 4231113 43422 441312 5111322 511143 52111122 5211123 5532 611142 61323 61422 63123 6333 7 1 1 1 5 711222 7134 7 15 2 81123 8133 91113 9132

2411142 33162 414132 41613 451113 511134 51432 61112112 613212 6414

2 3 3 41 4

31412 11152 51112 21111 21131 42414 511112 51315 5352 61125 6153 6 36 71133 7332 8 16

2 2 2 13 3 4

n = 16 10 1 1 1 3 214131112 232111132 2611222 31311142 314332 3241312 33352 341512 35152 36133 412111114 4142122

138 Knots 10 1 3 2 11 1 4 215111212 2231512 241432 241612 31113142 31211152 3131422 31411132 31511122 3214132 3251113 33111412 334132 3411142 34211113 35111212 35332 3611122 371113 37132 413143 4141213 414223 414232

214111312 231622 2511232 3124132 31413112 323152 331513 341422 351322 361123 4111423 414142 414322

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414412 4183 4311214 441313 512111113 514132 5211313 52315 53323 551113 6111223 61225 61333 61522 62116 711142 71323 7211113 7414 81133 8332 916 2314132 241311112 34131112 41452 41632 43432 5111512 5131312 533212 6132112 7131112

41443 42111133 43162 51111232 5121115 51442 521152 52414 53422 55132 6111322 61315 61423 6 1 63 63124 71134 71332 73123 811123 81313 911122 934

416122 4211314 43423 5111224 5141113 51514 5221123 531115 541312 5614 611143 613213 6145 62111122 63322 71224 71422 7333 81115 8134 91123

42 Links 2314312 231712 24131212 3111532 3413212 413152 4151113 4152112 417112 41812 43612 51112132 5131114 5131132 513214 513232 611121112 61112212 613312 61432 713212

41623 4221124 4363 5111422 5141212 51523 5231113 5311213 5434 6111124 611233 61324 61513 6211123 6343 71314 7153 736 811222 81 5 2 9133

235132 3111712 4142212 41614 43414 5111332 51312112 5331112 613132 61612

First rational knots with the non-trivial unknotting gap δBJ = 2 are 6 1 6 3 and 8 1 5 2 with n = 16 crossings. First non-rational alternating knots with the BJ-unlinking gap δBJ = 1 appear for n ≥ 12 crossings: pretzel knot 5, 4, 3 (12a1242 ) and polyhedral knots 6∗ 2.4 0 : 3 0 (12a970 ), 6∗ 2.2 1 0 : 4 0 (12a76 ), and 6∗ 2.2.2.4 0 (12a1153 ). We will try to explore the effect which 2n-moves (Przytycki, 2006a) (Fig. 1.67a) have on the BJ-unlinking number uBJ , diagram unlinking number uM , and BJ-unlinking gap δBJ . Applying 2n-moves on an integer tangle decreases or increases its corresponding parameter in the Conway symbol for 2n. If we allow applying 2n-moves on an arbitrary subset of elementary

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tangles of a given link L we get its corresponding families (Definition 1.48). J.A. Bernhard (1994) and D. Garity (2001) used this approach to obtain general results for unlinking numbers of the following families of diagrams of rational knots: (2k+1) 1 (2k) (k ≥ 2) and (2k+1) (2l+1) (2k) (k ≥ 2, l ≥ 0, k > l) whose diagram unknotting gap is k + l + 1 − (k + l) = 1. Moreover, two-parameter family of rational link diagrams (2k) 1 (2l) (k ≥ 2, l ≥ 2) (Garity, 2001) has diagram unlinking number uM = k + l − 1 and unlinking number u ≤ l, so the unlinking gap of a fixed diagram is at least k − 1 and can be made arbitrarily large for a sufficiently large k. In the similar manner, we try to obtain explicit formulas for BJunlinking gap of a family obtained from a rational link with several parameters, denoted by its Conway symbol. First we consider rational links containing only 2 or 3 parameters. Proofs of the following statements are given by R. Sazdanovi´c (Jablan and Sazdanovi´c, 2005a). Lemma 1.2. For a rational link with the Conway symbol a b, the following holds: • if a, b are both odd, then for a link (2m + 1) (2n + 1) we have uBJ = uM = u = m + n + 1; • if a is odd and b is even, then for a knot (2m + 1) (2n) we have uBJ = uM = u = n; • if a, b are both even, then for a knot (2m) (2n), (m ≥ n) we have uBJ = uM = n. Lemma 1.3. For a rational link with the Conway symbol a b c, the following holds: • if a b c = (2k) (2l) (2m), then uBJ = uM = u = k + m; • if a b c = (2k + 1) (2l + 1) (2m + 1), (k ≥ m ≥ 1), then uBJ = uM = l + m + 1; • if a b c = (2k + 1) (2l) (2m + 1) (k, m ≥ 1) then uBJ = uM = u = k + l + m + 1; • if a b c = (2k + 1) (2l) (2m) (k, m ≥ 1) then uBJ = uM = u = k + m;

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• if a b c = (2k) (2l + 1) (2m), (k ≥ m ≥ 1) then l + k, if l ≤ k, m; uBJ = k + m, if k, m ≤ l. i.e. uBJ = min(k + m, k + l) and

uM

 if 1 ≤ l ≤ k, m;  k + m, = k + m − 1, if 0 = l ≤ k, m;  k + m, if k, m ≤ l.

• if a b c = (2k + 1) (2l + 1) (2m) (k, m ≥ 1) then k + min(l, m), if l, m ≤ k or k, m < l; uBJ = k + l + 1, if k, l ≤ m; uM

 if k, m < l;  k + m, = k + min(l + 1, m), if l, m ≤ k;  k + l + 1, if k, l ≤ m.

Among the KL families from Lemma 1.3, two of them have non-trivial BJ-unlinking gap: Corollary 1.3. • rational links (2k) (2l + 1) (2m) have non-trivial BJ-gap if k ≥ m ≥ 2 and m ≥ l + 1 m − 1, l = 0; δBJ = m − l, l ≥ 1. • rational knots (2k + 1) (2l + 1) (2m) have non-trivial gap δBJ = 1 if m ≥ 2 and l + 1 < m < k. Lemma 1.4. For a rational knot with the Conway symbol (2k) (2m) 1 (2n) the following holds: • diagram unlinking number is uM = n + min(k, m − 1);

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• BJ-unlinking number is n, if m ≤ n; uBJ = n + min(k, m − n), if m > n.

• BJ-unlinking gap is min(k, m − 1), if m ≤ n; δBJ = min(k, m − 1) − min(k, m − n), if m > n.

The Lemma 1.2 is used to prove the following theorem about a family of rational knots with an arbitrarily large BJ-unlinking gap6 : Theorem 1.26. For a rational knot with the Conway symbol (2k) (2m) 1 (2n) (m, k, n ≥ 0) the following holds: • diagram unlinking number is

uM = n + min(k, m − 1);

• BJ-unlinking number is n, if m ≤ n; uBJ = n + min(k, m − n), if m > n.

• BJ-unlinking gap is min(k, m − 1), if k ≤ m; δBJ = min(k, m − 1) − min(k, m − n), if m > n.

The main problem for every family is finding necessary and sufficient conditions (or, simply, conditions) for a specific family to have a non-trivial unlinking gap. Knot or link a is a torus KL of type [2, a] and therefore uBJ = uM = u, so both gap and BJ-unlinking gap are trivial. From Lemma 1.2 it follows that all rational links with 2 parameters a b have trivial BJ-unlinking gap. The same holds for all 3-parameter families a b c except two families listed in Corollary 1.3. Since computations, based on parity of parameters and symmetries of the links are long and tedious even for 3-parameter families, for multi-parameter families we give only experimental results. We choose 68 one-parameter7 families of rational links which contain all rational links up to 14 crossings with positive BJ-unlinking gap8 . For all 6 S. Bleiler (1984) asked if δ(L) = u (L) − u(L) has an upper bound. Since δ M BJ (L) ≤ δ(L), Theorem 1.26 provides more examples of links with unbounded δ(L) (compare with Stoimenow, 2003). 7 One-parameter family is obtained by applying the same 2n-move to all chosen integral tangles. 8 Compare with the tables, pages 93-95.

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families we predict values of BJ-unlinking number uBJ and BJ-unlinking gap δBJ based on computations for links with less than 48 crossings. Each family in the following table is given by its Conway symbol; the next entry is the number of components followed by experimental results for BJunlinking number uBJ and BJ-unlinking gap δBJ .

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)

Family (2k + 2) 1 (2k + 2) (2k + 3) 1 (2k + 2) (2k + 2) 1 (2k + 2) (2k) (2k + 2) (2k + 1) (2k + 2) (2k + 4) 1 (2k + 2) (2k + 3) 1 (2k + 1) (2k) (2k + 3) 1 1 1 (2k + 1) (2k + 5) 1 (2k + 2) (2k + 3) (2k + 1) (2k + 2) (2k + 2) 1 (2k + 2) (2k + 1) (2k + 4) 1 (2k + 1) (2k) (2k + 4) 1 1 1 (2k + 1) (2k + 2) (2k + 2) 1 (2k + 2) (2k + 4) 1 (2k + 2) (2k) (2k + 2) 1 (2k + 1) 1 (2k + 2) (2k + 3) 1 (2k + 1) (2k) (2k) (2k) (2k + 1) 1 (2k + 2) 1 (2k) (2k + 3) 1 1 1 (2k + 1) (2k) (2k + 4) (2k + 1) (2k + 2) (2k + 6) 1 (2k + 2) (2k + 3) 1 (2k + 3) (2k) (2k + 3) (2k + 1) (2k + 1) (2k) (2k + 4) 1 (2k + 1) (2k + 1) (2k + 5) 1 (2k + 1) (2k)

Comp. No. 2 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

(25) (26) (27) (28) (29) (30) (31) (32)

(2k + 1) (2k + 2) 1 (2k + 1) (2k) (2k + 2) 1 (2k + 2) (2k) (2k) (2k + 3) 1 1 1 (2k + 3) (2k + 4) 1 1 (2k) (2k + 1) (2k + 5) 1 1 1 (2k + 1) (2k) (2k + 2) 1 (2k + 1) 1 (2k) (2k + 2) 1 1 1 (2k + 2) (2k) (2k + 4) 1 1 1 (2k) (2k)

2 2 2 2 2 2 2 2

(33) (34) (35) (36) (37) (38)

(2k + 2) (2k) 1 1 1 1 (2k + 1) (2k + 5) (2k + 1) (2k + 2) (2k + 7) 1 (2k + 2) (2k + 2) 1 (2k + 4) (2k + 1) (2k + 2) (2k + 1) (2k + 2) (2k + 1) (2k + 3) (2k + 2) 1 (2k + 2)

2 1 1 1 1 1

uBJ k+1 k+1 k+1 2k + 1 k+2 k+1 k+1 k+2 2k + 1 k+1 k+1 k+1 k+1 k+2 k+1 k+1 k+1 k+1 2k + 2 k+3 k+3 2k + 1 k+1 3 k+1 2k + 1 2k + 1 k+2 k+1 k+1 k+1 k+1 k+1 k 2k 2k + 2 k+3 k+2 k+2 k+1

δBJ k 1 k 1 k 1 k 1 1 k 1 1 1 k k 1 1 k 1 k 1 1 1 1 if k = 1 2 if k ≥ 2 1 k k 1 k k 1 1 if k = 1, 2 2 if k ≥ 3 1 1 1 k 1 k

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(39) (40) (41) (42) (43)

(2k + 4) 1 (2k + 2) (2k + 1) (2k + 4) 1 (2k + 3) (2k) (2k + 4) (2k + 1) (2k + 1) (2k) (2k + 5) 1 (2k + 1) (2k + 1) (2k + 6) 1 (2k + 1) (2k)

1 1 1 1 1

(44)

(2k + 1) (2k + 1) 1 (2k + 3) (2k)

1

(45) (46) (47) (48) (49) (50) (51) (52)

(2k + 1) (2k + 3) 1 (2k + 1) (2k) (2k + 2) 1 (2k + 2) (2k) (2k + 1) (2k + 1) 1 (2k + 1) 1 (2k + 2) (2k + 3) 1 (2k + 1) (2k) (2k + 1) (2k + 3) 1 (2k + 2) (2k) (2k) (2k + 4) 1 1 (2k) (2k + 2) (2k + 4) 1 (2k + 1) (2k) (2k) (2k + 5) 1 1 (2k) (2k + 1)

1 1 1 1 1 1 1 1

(53) (54) (55) (56) (57) (58) (59)

(2k + 6) 1 1 1 (2k + 1) (2k + 1) 1 (2k + 1) 1 (2k + 2) (2k) (2k + 1) 1 (2k + 2) 1 (2k + 1) (2k) (2k + 1) (2k + 2) 1 (2k + 1) 1 (2k) (2k + 1) (2k + 3) 1 1 1 (2k + 1) (2k + 3) 1 1 1 (2k + 2) (2k) (2k + 5) 1 1 1 (2k) (2k)

1 1 1 1 1 1 1

(60)

(2k) 1 (2k + 2) 1 (2k + 1) 1 (2k)

1

(61) (62) (63) (64) (65) (66) (67) (68)

(2k) (2k + 2) 1 1 1 (2k + 1) (2k) (2k + 2) (2k) 1 1 1 1 1 (2k + 2) (2k + 3) (2k) 1 1 1 1 (2k + 1) (2k + 2) 1 (2k + 2) (2k + 1) (2k) (2k + 2) 1 (2k + 4) 1 (2k) (2k + 2) 1 (2k + 1) 1 (2k + 1) (2k) (2k + 3) 1 (2k + 1) (2k) 1 (2k) (2k + 3) 1 (2k + 1) 1 1 1 (2k)

1 1 1 2 2 2 2 2

k+2 k+2 2k + 1 k+1 3 k+1 2 k+2 k+1 2k + 1 k+1 k+1 2k + 1 k+1 2k 2 3 k+2 k+1 k+1 k+1 k+1 k+1 k+1 k 2k 2k 2k 2k 2k 2k + 1 2k + 1 2k + 1 2k + 1 2k + 1

k 1 1 1 1 if k = 1 2 if k ≥ 2 1 for k = 1 0 if k ≥ 2 1 k k 1 1 1 1 1 if k = 1 0 if k ≥ 2 1 k 1 k k 1 1 if k = 1, 2 2 if k ≥ 3 1 if k = 1, 2 k if k ≥ 3 1 1 1 k k 1 1 1

The following results (unless explicitly stated otherwise) are based on the properties of the generating links and experimental results for rational, pretzel and polyhedral links up to 16 crossings and present a good starting point for obtaining a better understanding of unlinking number. First, we present several multi-parameter families of rational links with an arbitrarily large BJ-unlinking gap– we either give the explicit conditions for a (nontrivial) gap, or we find a subfamily with this property: • the family (2k) (2m) 1 (2n) has an arbitrarily large BJ-unlinking gap (see Theorem 1.26);

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• the family (2k + 2) 1 (2k + 2) (2k − 1), (k ≥ 1), starting with the knot 6 1 6 3 has uBJ = k + 1 and δBJ = k; • the family (4k +4) 1 (2k +3) (2k), (k ≥ 1), starting with the knot 8 1 5 2, has uBJ = k + 2 and δBJ = k + 1; • the family (2k + 1) (2l − 1) (2m − 1) (2n − 1) (2p + 1), starting with the link 5 1 1 1 3, has a subfamily (2k + 1) 1 1 1 (2l + 1) which under the conditions k ≥ l ≥ 0 has unlinking gap  if k = l = 0;  0, δ = δBJ = l − 1, if k = l > 0;  l, if k > l > 1. • every knot family of the form

(2k1 ) . . . (2l − 1) . . . (2k2i+1 ) (k1 , . . . , k2i+1 ≥ 2, l < mini (ki )) and every link family of the form (2k1 ) . . . (2k2i−1 ) (2l − 1) (2m1 ) . . . (2m2j−1 ) (k1 , . . . , k2i−1 ≥ 2, m1 , . . . , m2j−1 ≥ 2) has members with arbitrarily large BJ-unlinking gap. In order to show this, it is sufficient to consider their subfamilies with l = 1 and k1 = k2 = ... = k, where k occurs j times in total. For every such subfamily BJ-unlinking number is uBJ = [ 2j ]k and δBJ = k − 1. Next we consider the family of pretzel knots a, b, c. For the pretzel knots 2k + 1, 2l + 1, 2m + 1 (k ≥ l ≥ m ≥ 1) we proved (Proposition 1.1b) that uBJ = l + m and δBJ = 0. For the simplicity, since pretzel KLs with three tangles remain unchanged by a permutation of tangles, we will assume that in their symbols tangles are given in non-increasing order. For the families of pretzel KLs with three tangles we have the following: Theorem 1.27. • the family 2k + 1, 2l + 1, 2m + 1 has uBJ = uM = l + m and δBJ = 0. • the family 2k, 2l, 2m has uBJ = uM = u = k + l + m, and therefore δBJ = 09 ; • for pretzel knots 2k + 1, 2l, 2m + 1 with (k ≥ m ≥ 1) we have10 : 9 Notice

that the linking number guarantees that uBJ = uM = u and δ = δBJ . that in first two cases, l = 1 and k ≥ l > 1, the signature guarantees that = uM = u and δ = δBJ .

10 Notice

uBJ

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2k + 1, 2l, 2m + 1 l=1 k≥l>1 l>k≥1

uBJ m+k m+k m+k+1

uM m+k m+k+1 m+k+1

δBJ 0 1 0

• 2k, 2l + 1, 2m (k ≥ m) has uBJ = u = k + l, and δBJ = δ = m − 111 . The following experimental results, combined with results for rational links, make calculations complete up to 12 crossings. Among alternating algebraic non-rational links, the first with positive BJ-unlinking gap is pretzel link 4, 4, 3 with 11-crossings and δBJ = 1. In the same class, we have the following seven links with 12-crossings and δBJ = 1: 2 2 1 1 2, 2, 2 3 1, 3 1, 2 1+

4 1 1, 3, 3 (2 1, 2 1 1 1) (2, 2)

2 1 1, 3 1, 3 1 (2 1, 2 2) 1 (2, 2)

5, 4, 3

The next table contains all polyhedral knots with n = 12 crossings and positive BJ-unlinking gap, together with the one-parameter families derived from them, followed by the first step of the unknotting process which reduces them to families of rational, generalized pretzel, or polyhedral knots12 :

No. 1

Knot 6∗ 2.3 1 : 3 0 uBJ = 2

Family 6∗ (2k).3 1 : 3 0 uBJ = k

2

6∗ 2.2 1 0 : 4 0

3

6∗ 2.2 2 0 : 3 0

4

6∗ 2.4 0 : 3 0 uBJ = 2 6∗ 2.2.3.3 0

6∗ (2k).2 1 0 : 4 0 uBJ = k + 1 6∗ (2k).2 2 0 : 3 0 uBJ = k + 1 6∗ 2.(2k) 0 : 3 0 (k ≥ 2) uBJ = k 6∗ (2k).2.3.3 0 uBJ = k + 1 6∗ 2.2.2.(2k) 0 (k ≥ 2) uBJ = k

5 6

11 Notice

6∗ 2.2.2.4 0 uBJ = 2

Reduction 6∗ (2k).3 1. − 1.3 0 k > 1: ≃ (2k − 1) 1 1 1 2 2 k = 1: ≃ 2 1 1 2 2 6∗ (2k).2 1 0 : 4 0 : −1 ≃ 3 1 (2k − 1) 2 2 6∗ (2k).2 2 0 : 3 0 : −1 ≃ 2 1 (2k − 1), 2 1, 2 6∗ 2.(2k) 0 : 3 0 : −1 ≃ (2k) 1 1 2 6∗ (2k).2.3.3 0. − 1 ≃ 2 1 (2k − 1), 2 1, 2 6∗ 2.2.2.(2k) 0. − 1 ≃ (2k − 1) 1 1 1 1 2

δBJ 1

1 1 1 1 1

that the signature guarantees that uBJ = u and δ = δBJ . symbol ≃ is used to denote ambient isotopy between two links; for example, in the first row symbol ≃ means that 6∗ (2k).3 1. − 1.3 0 is ambient isotopic to (2k − 1) 1 1 1 2 2 if k > 1, and 2 1 1 2 2 if k = 1. 12 The

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7

6∗ 2.2 0.3.3 0

6∗ (2k).2 0.3.3 0 uBJ = k + 1

8

6∗ 2.(3, 3)

6∗ (2k).(3, 3) uBJ = k + 1

9

6∗ 2.(3, 2).2

10

8∗ 2 : 2 : .3 0 uBJ = 2

6∗ 2.(2k + 1, 2).2 uBJ = k + 1 8∗ (2k) : 2 : .3 0

6∗ (2k).2.3.3 0. − 1 ≃ 4 1 (2k − 1) 1 2 or 6∗ (2k).2.3.3 0 : −1 ≃ 2 1 (2k − 1), 3, 2 6∗ (2k).(3, 3) :: −1 k ≥ 2: ≃ 6∗ 2.(2k − 2) : 2 0 k = 1: ≃ 2 1 1 1 2 6∗ 2.(2k + 1, 2).2. − 1 ≃ 2 2 1 (2k) 8∗ 2 : 2. − 1 : 3 0 ≃ (2k + 1) 3 2 (see Lemma 1.3.)

1

1

1 1

Moreover, the following n = 12-crossing links have BJ-unlinking gap13 δBJ = 1: 6∗ 2.2.2 : 2 1 1 0 6∗ 2.2, (2, 2) 0 6∗ 2 1 1 : .(2, 2) 0

6∗ 2.2.2 : 2 1 1 6∗ 2.(2, 2).2 1 0 6∗ 2 1 1 : .(2, 2)

6∗ (2 1, 2 2) :2 6∗ 2.2 1.2.2 0 : 2 0 6∗ (2, 2).2 1

6∗ 2.(2, 2) 1 1 : 20 8∗ 2 1 : .2 0 : 2 0

6∗ 2.(2, 2), 2 0

6∗ (2, 2) 1.2

The question of finding BJ-unlinking gap of non-alternating links is much more difficult because of the lack of classification of their minimal diagrams. For a few classes of non-alternating links partial results can be obtained using the work of W.B.R. Lickorish and M.B. Thistlethwaite (1988). Unfortunately, this is not sufficient to find all minimal diagrams corresponding to non-alternating link families and compute BJ-unlinking gap for non-alternating links. For n = 11 there are following non-alternating KL diagrams with unlinking gap δBJ = 1: 4 1 1, 3, −2

3 2, 3, −3

4, 4, −3

.(3, −2).2

.2.(3, −2)

and for n = 12 their list is following: 5, −3 1, 2 1 −5, 3 1, 2 1

−5, 3 1, 2 1 (−3, −3) (3, 2 1)

(−4, 2 1) (3, 2) (3, 3) (−3, 2 1)

(−4, −2 1) (3, 2) 3 : 2 : −4 0

5, −3 1, 2 1 −3 0.2.2 0.3 0

Non-alternating minimal diagrams 4 1 1, 3, −2 and 3 2, 3, −3 (Fig. 1.63) of the non-alternating knots 11n64 and 11n122 have the unknotting gap δM = 1. These diagrams can be extended to two-parameter families of minimal diagrams (2k + 2) 1 1, (2l + 1), (−2m) and (2k + 1) 2, (2r + 1), −3 representing Montesionos knots with the diagram unlinking gap δM = 1. As A. Stoimenow’s example shows (Fig. 1.59), different minimal projections of a non-alternating knot can have different projection BJ-unknotting 13 Links

in the first 2 rows are 2-component while ones in the last are 3-component.

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Fig. 1.63

The diagrams (a) 4 1 1, 3, −2; (b) 3 2, 3, −3.

numbers. Two different minimal projections of the knot 1436750 (Fig. 1.59) can be extended to the two-parameter families of minimal projections 124∗ − 1. − 1. − 1. − 1. − 1 : (−1, (−2l)) 0 : (2k − 1) : . − 1 and 8∗ − 2 0 : (2l + 1) 0 : −2 0.(−2k) 0. − 1.2 0 which satisfy the condition l < 2k (Fig. 1.64), where 124∗ denotes the basic polyhedron 12D from the paper by A. Caudron (1982). The experimental BJ-unknotting number of the first family of projections is k, and it is the experimental BJ-unknotting number of the knot family in question. The second family of projections has the experimental BJ-unknotting number k + 1. It is not surprising that some non-minimal diagrams can have a nontrivial unlinking gap. For example, the 11-crossing non-alternating knot 11n138 has the non-minimal diagram 3 1 1, 3, 3− with the unknotting gap δM = 1, while the (fixed) minimal diagram 3 1 1, 3, −2 1 gives the unknotting number u = 2 (Fig. 1.65). The family of non-alternating pretzel links 2k, 2k, −3 (k ≥ 2) is the candidate for non-alternating link family with an arbitrarily large unlinking gap (Fig. 1.66). This family is obtained from the family of rational links 2k 1 2k = 2k, 2k, 1 (k ≥ 2) which is a special case of the family 2k, 2l + 1, 2m (from the Corollary 1.3 for l = 0, k = m), with arbitrarily large BJ-unlinking gap

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Fig. 1.64 Two projections (a) 124∗ − 1. − 1. − 1. − 1. − 1 : (−1, (−2l)) 0 : (2k − 1) : . − 1; (b) 8∗ − 2 0 : (2l + 1) 0 : −2 0.(−2k) 0. − 1.2 0 of the same knot.

Fig. 1.65 (a) Non-minimal diagram 3 1 1, 3, 3−; (b) the minimal diagram 3 1 1, 3, −2 1 of the non-alternating knot 11n138 .

δBJ = k−1. In the similar manner as before, we may obtain that the family of standard minimal diagrams of pretzel links 2k, 2k, −3 has BJ-unlinking number k. Furthermore, the unlinking number of the standard diagram of 2k, 2k, −3 is equal to 2k − 1, hence the diagram BJ-unlinking gap is k − 1. Since the classification of all minimal diagrams of the link family 2k, 2k, −3 is, up to our knowledge, not yet achieved we are not able to show that the link family 2k, 2k, −3 has an arbitrarily large unlinking gap. One-parameter family of minimal link diagrams 2k, 2k, −3 can be extended to a three-parameter family 2k, 2l, −(2m + 1) with an arbitrarily large diagram BJ-unlinking gap.

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Fig. 1.66 The family 2k, 2k, −3 of non alternating minimal diagrams with an arbitrarily large BJ-unlinking gap.

Next we will consider two special classes of rational KLs: KLs with unknotting (unlinking) number equal to 0 or 1. For rational KLs with unknotting (unlinking) number 1, the unknotting (unlinking) number can be recognized from any minimal projection (Theorem 1.24; Nakanishi, 1996; Stoimenow, 2004). Every unknotting number one knot is prime (Scharelmann, 1985). The general form of knots with unknotting number 1 (Theorem 1.28) is described by T. Kanenobu and H. Murakami (1986), and P. Kohn formulated an analogous theorem for links (1991) (Theorem 1.29). Theorem 1.28. Every rational unknotting number 1 knot can be expressed by one of the following Conway symbols c0 c1 . . . cr−1 cr 1 1 (cr − 1) cr−1 . . . c1 c0 c1 . . . cr−1 (cr − 1) 1 1 cr cr−1 . . . c1 ,

where ci ≥ 0 for i = 0, . . . , r and cr ≥ 2 (Kanenobu and Murakami, 1986). Theorem 1.29. Every rational unlinking number 1 link can be expressed by one of the following Conway symbols c0 c1 . . . cr−1 cr 1 1 (cr − 1) cr−1 . . . c1 c0 c0 c1 . . . cr−1 (cr − 1) 1 1 cr cr−1 . . . c1 c0 ,

where ci ≥ 0 for i = 0, . . . , r and cr ≥ 2 (Kohn, 1991).

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Theorem 1.24 is the corollary of Theorem 1.28 and 1.29, because a single crossing change in minimal diagrams from Theorem 1.28 and 1.29 gives the unknot and unlink. Moreover, if L is a rational link with a minimal diagram which can not be unlinked by one crossing change and has a minimal diagram with unlinking number 2, then u(L) = 2. The LinKnot function RatKnotGenU1 gives the number and the list of Conway symbols of all rational knots with the unknotting number 1 with n crossings, and the function RatLinkU1 gives the same result for rational links with the unlinking number 1. The number of such knots is given by the formula 2[

n−2 2 ]

− 1.

The number of rational unlinking number 1 links is 0 for every even n, and for odd n it is given by the formula 2[

n−7 2 ]

.

In a similar way we can be interested in rational representations of unknots or unlinks, i.e., KLs with the unknotting (unlinking) number 0. Theorem 1.30. Every rational unknot can be expressed by one of the following Conway symbols c0 c1 . . . cr−1 cr (−1) 1 (cr − 1) cr−1 . . . c1 c0 c1 . . . cr−1 (cr − 1) 1 (−1) cr cr−1 . . . c1 , where ci ≥ 0 for i = 0, . . . , r and cr ≥ 2. Theorem 1.31. Every rational unlink can be expressed by one of the following Conway symbols c0 c1 . . . cr−1 cr (−1) 1 (cr − 1) cr−1 . . . c1 c0 c0 c1 . . . cr−1 (cr − 1) 1 (−1) cr cr−1 . . . c1 c0 , where ci ≥ 0 for i = 0, . . . , r and cr ≥ 2. Proof of these two theorems follows from the proof of Theorem 1.28 (Kanenobu and Murakami, 1986) and Theorem 1.29 (Kohn, 1991). The LinKnot function RatKnotGenU0 produce, for a given n, a list of Conway symbols of all rational unknots with n crossings, and the function

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RatLinkU0 gives the same result for rational unlinks. The number of rational unknots with n crossings is given by the formula 2[

n−2 2 ]

− 2.

The number of rational unlinks is 0 for every even n, and for odd n it is given by the formula 2[

n−7 2 ]

.

For example, knot illustrated in Figure 1.67 has the unknotting number 1. Try to find the crossing change that results in unknot. If you don’t believe that such crossing change exists, you can check its unknotting number by entering its Conway symbol 321123311232112 in the LinKnot function UnR. In a similar way you can check that the link 3211233 −112321123 from Fig. 1.68 is an unlink. Another way of checking if a KL is an unknot or unlink is exporting it to KnotPlot, using the function fCreateGraphics and relax it, so one can see how a complicated projection reduces to one or several disjoint circles.

Fig. 1.67

Knot 3 2 1 1 2 3 3 1 1 2 3 2 1 1 2.

The ∞-unknotting operation is defined by S. Jablan (1998). Every crossing of an oriented knot can be resolved by smoothing that preserves number of components, introducing a “two-sided mirror” as in Fig. 1.69. This can be repeated until we obtain the unknot. Analogously as we defined unknotting number and BJ-unknotting number (Definition 1.58), it

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Fig. 1.68

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Unlink 3 2 1 1 2 3 3 -1 1 2 3 2 1 1 2 3.

is possible to define the ∞-unknotting number u∞ (K) by the “classical” and “standard” definition, try to prove that two definitions are equivalent, define u∞ -number restricted on minimal projections (analogously to the definition of BJ-unknotting number) and make a Conjecture analogous to Bernhard-Jablan Conjecture. Every ∞-change transforms an alternating knot to an alternating knot, so the set of all alternating knots is closed with regard to ∞-changes. According to Tait’s Flyping Theorem (Theorem 1.11) and Lemma 1.1, all minimal projections of an alternating knot give the same result, so for every alternating knot it is sufficient to use only one minimal projection. In the case of a non-alternating knot, all its minimal projections must be used, since two minimal projections can give different u∞ numbers. For example, two minimal diagrams of the same pretzel knot, 3, 2 1, −2 and −2 − 1, 2 1, 2, have u∞ diagram numbers 1 and 2, respectively. The LinKnot function NoSelfCrossNo computes recursively u∞ number of a knot given by its Conway symbol, Dowker code, or P -data from minimal projections. For links, it computes the minimal number of steps necessary to obtain a link without self crossings, i.e., without crossings that belong to a single component14 . The u∞ number is a property of a family: for the knot family (2k + 1), (k ≥ 1), u∞ ((2k + 1)) = 1; for the knot family (2k) 2, (k ≥ 1), u∞ ((2k) 2) = 2; for the knot family ((2k) (2l)), (k, l > 1), u∞ ((2k) (2l)) = 3, etc. 14 A link without self-crossings can be also resolved by smoothing, i.e., reduced to the unknot.

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Fig. 1.69

∞-operation.

From experimental results we conjecture that u∞ number is a linear subfamily-dependent knot invariant (Conjecture 1.2). Conjecture 1.4. Knots belonging to subfamily (2k + 1) (31 , 51 , 71 ,...) are the only knots with u∞ = 1. An n-move on a KL is a local change illustrated in Fig. 1.70a, where the remaining part of the KL remains unchanged. We say that two KLs, L1 and L2 , are n-move equivalent if one can be transformed to the other by a finite number of n-moves and their inverses (−n)-moves (Przytycki, 2003). Nakanishi’s Conjectures (1) Every knot is 4-move equivalent to an unknot. (2) Any link is 3-move equivalent to a trivial link. The first conjecture Y. Nakanishi proposed in 1979, and the second in 1981. The counterexample to the second Nakanishi’s Conjecture is the link 2049953∗. − 1 : . − 1 : . − 1. − 1 ::: −1 :: −1. − 1 with n = 20 crossings (Fig. 1.70b) (Przytycki, 2002). M.K. Dabkowski and J.H. Przytycki (2002) proved that this link can not be reduced to a trivial link by 3-moves. The first Conjecture is proved for algebraic and 3-braid knots (Kirby, 1997; Przytycki, 2003). For a long time, the simplest potential counterexample to the first Nakanishi’s Conjecture was the (2, 1)-cable of the trefoil knot with n = 13

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Fig. 1.70 (a) n-move; (b) link which can not be reduced to a trivial link by 3-moves; (c) knot 16101∗ − 1. − 1 : . − 1. − 1 : −1. − 1 :: . − 1 : −2 0.

crossings, unknotted by N. Askitas (1999), given by the non-minimal representation with n = 18 crossings 1315∗2 1 0. − 1. − 1. − 1. − 1 : .2 0.1.3 0. − 1. − 1. − 1 By two 4-moves, it gives the knot 1315∗ − 2 1 0. − 1. − 1. − 1. − 1 : .2 0.1. − 1 0. − 1. − 1. − 1 which reduces to 12-crossing knot 2 0.− 2 − 1.− 2 0.2 1 0. From it, we obtain the unknot by the following sequence of 4-moves and ambient isotopies 2 0. − 2 − 1. − 2 0.2 1 0 → −2 0.2 − 1.2 0. − 2 1 0 ≃ 2 1 1, 2 1, −2 →

−2 1 1, −2 1, −2 ≃ 3 1 2 → −1 1 2 ≃ 1 However, the first Nakanishi’s Conjecture is still open, and now the main candidate for the smallest counterexample is the knot 16101∗ − 1. − 1 : . − 1. − 1 : −1. − 1 :: . − 1 : −2 0 (2, 1)-cable of the figure-eight knot with n = 17 crossings proposed by N. Askitas (Fig. 1.70c) (Askitas, 1999; Przytycki, 2002, 2003). An unlinking operation can be defined:

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(1) as a local operation that allows us to untie every KL in finitely many steps, or (2) as a local operation that allows us to untie certain KLs in finitely many steps. According to the first definition, crossing-changes and smoothing are unlinking operations, but n-moves (n ≥ 3), or (2, 2)-moves are not. because they can lead to solutions different than the unknot (or an unlink). For example, every rational 2-component link is 4-move equivalent to a trivial link of two components or to a Hopf link 2 (221 ) (Przytycki, 2003, Theorem 1.11). Accepting the second definition, a minimal number of n-moves required to obtain a knot or link k (k ≤ [ n2 ]) from a given KL can be defined as nmove k-unknotting (unlinking) number u(n,k) (L), taken over all projections of a KL. In the language of n-moves, unlinking number u(L) is u(2,1) (L). In the case of u(n,k) (L) we can make a conjecture analogous to BernhardJablan Conjecture, and work only with minimum projections at all levels of unknotting (unlinking) process. According to such a Conjecture, the minimal number of n-moves necessary to obtain a knot or link k (k ≤ [ n2 ]) from a rational KL given by its Conway symbol, for a given parameter n, can be computed by the LinKnot function NMoveRat. Except n-moves, there are some other operations on KLs, like (2, 2)moves, able to produce the unknot. J. Conway (1970) defined the transformation a ↔ a 0 as a reflection of a tangle a in descending diagonal line (Fig. 1.40). This reflection, preserves 1 tangle, i.e., 1 = 1 0. A (2, 2)-move ((−2, −2)-move) is the reflection of the tangle 2 (−2) followed by the crossing changes. According to this, (2, 2)move transforms 2 into −2 0 and vice versa, and −2 into 2 0 and vice versa. This approach enables us to express (2, 2)-moves in Conway notation. Together with (2, 2)-moves 2 ↔ −2 0 and −2 ↔ 2 0 (Fig. 1.71a), we will use (2, 2)-move applied to tangle 1 which transforms it into −2, −1 (Fig. 1.71b). Since 1 = 1 0, the same (2, 2)-move transforms 1 into (−2, −1) 0 and vice versa. J. Przytycki (2003, Fig. 1.26) used (2, 2)-move 1 → (−2, −1) for unknotting knot 818 = 8∗ by (2, 2)-moves. In the same way, one can develop algebra of (2, 2)-moves given in Conway notation, e.g., 1 ↔ −2 − 1, 1 ↔ −2 − 1 0, 3 ↔ (−2, 1), 4 ↔ −2 2, etc. In attempt to unknot a KL, we can apply sequences of one or several (2, 2)-moves on the same or equivalent projection.

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Fig. 1.71

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(a) (2, 2)-move and (−2, −2)-move; (b) (2, 2)-move 1 → (−2, −1).

J. Przytycki proved that every algebraic KL (according to Definition 1.46 (2)) can be reduced to a trivial link by (2, 2)-moves. Hence, in order to prove that a KL can be unlinked by (2, 2)-moves, it is sufficient to reduce it by (2, 2)-moves to an algebraic KL. For example, by one (2, 2)move knot 818 = 8∗ can be transformed into 8∗ (−2, −1) which reduces to the algebraic knot 2 2, 2 2, −2, so the knot 818 = 8∗ can be unknotted by (2, 2)-moves (Przytycki, 2003, Fig. 1.26). M.K. Dabkowski and J. Przytycki (2002) proved that knots 940 = 9∗ and 949 = −2 0 : −2 0 : −2 0 can not be unknotted by (2, 2)-moves and made a guess that they belong to the same (2, 2)-move equivalence class (Przytycki, 2003). J. Przytycki also proved that every KL with n ≤ 9 crossings is (2, 2)-move equivalent to a trivial link, knot 940 , or 949 (or their mirror images) (Przytycki, 2002, Lemma 1.8) (Fig. 1.72a). For n = 9 there are two knots, 9∗ (940 ) and −2 0 : −2 0 : −2 0 (949 ), and two links, 2 : 2 : 2 (9240 ) and 2 : −2 0 : −2 0 (9261 ) which can not be unknotted by (2, 2)-moves. Because the first link can be obtained from the knot −2 0 : −2 0 : −2 0 (949 ) by three (2, 2)-moves, and the other by one (2, 2)-move, we conclude that both of them belong to the same equivalence class of the knot 949 (Fig. 1.72b).

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Fig. 1.72 (a) Knots 9∗ (940 ) and −2 0 : −2 0 : −2 0 (949 ); (b) links 2 : 2 : 2 and 2 : −2 0 : −2 0.

For n = 10 there are two knots, 3 0 : 2 : 2 (10103 ) and −3 : 2 : 2 (10155 ) which can not be unknotted by (2, 2)-moves. From the first, by two (2, 2)-moves we obtain the knot 3 0 : −2 0 : −2 0 which further reduces to −2 0 : −2 0 : −2 0 (949 ) (Fig. 1.73a). From the second by one (2, 2)-move we obtain the knot −3 : −2 0 : 2 which also reduces to −2 0 : −2 0 : −2 0 (949 ). Hence, we conclude that both knots belong to the equivalence class of the knot 949 (Fig. 1.73b). Following tables contain all knots and all alternating links with n ≤ 12 crossings, except the basic polyhedra with n = 12 crossings, which we have not succeeded to unknot by (2, 2)-moves. All of them belong to two (2, 2)-move equivalence classes: the class of knot 9∗ (940 ), or to the class

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Notation of Knots and Links

Fig. 1.73 (a) Knots 3 0 : 2 : 2 and 3 0 : −2 0 : −2 0 that reduce to the mirror image of 949 ; (b) knots −3 : 2 : 2 and −3 : −2 0 : 2 that reduce to the mirror image of 949 .

of knot −2 0 : −2 0 : −2 0 (949 ). For each KL with n crossings is given its number of components, Conway symbol, and the sequence of (2, 2)-moves and reductions of the KLs obtained, giving 940 or 949 . Every knot is also given in Knotscape notation. n = 11

1

11a297

n = 11

1

11a317

n = 11

1

11n133

n = 11

1

11n148

n = 11

2

n = 11

2

9∗ 2 0 :::: 2 0 → 9∗ 2 0 ::: . − 2 − 1 0. − 2 ≃ 9∗ 6∗ 3 1.2.2 0 → 6∗ 3 1. − 2 0. − 2 ≃ −3 : 2 : 2 → −3 : −2 0 : 2 ≃ −2 0 : −2 0 : −2 0 8∗ 2. − 2 0.2 → ∗ 8 − 2 0. − 2 0.2 ≃ 9∗ 9∗ . − 2 − 1 → 9∗ .1 ≃ 9∗ 6∗ 3.2.3 → 6∗ (−2, 1). − 2 0.3 ≃ −3 : 2 : 2 → −3 : −2 0 : 2 ≃ −2 0 : −2 0 : −2 0 8∗ 2.2.2 → 8∗ − 2 0. − 2 0.2 ≃ 9∗

940

949 940 940

949 940

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n = 12

1

12a100

n = 12

1

12a780

n = 12

1

12a907

n = 12

1

12a921

n = 12

1

12a975

n = 12

1

12a1194

n = 12

2

n = 12

2

n = 12

2

n = 12

2

n = 12

2

n = 12

2

n = 12

2

n = 12

3

n = 12

3

n = 12

3

n = 12

3

n = 12

4

6∗ (2 1, 2).2 0 :: 2 0 → 6∗ (−1, 2).2 0 :: 2 0 ≃ −2 0 : −2 0 : −2 0 6∗ 2.2.2 1.2 0.2 0 → 6∗ − 2 0.2. − 1.2 0.2 0 ≃ −2 0 : −2 0 : −2 0 9∗ 3 :::: 2 0 → ∗ 9 3. − 2 − 1 0 ::: . − 2 ≃ 9∗ . − 2 − 1 → 9∗ .1 ≃ 9∗ 6∗ 3.2.3 1 → 6∗ 3. − 2 0.3 1 ≃ −2 1 0.3.2.2 0 → 1.3. − 2 0.2 0 ≃ −3 : 2 : 2 → −3 : −2 0 : 2 ≃ −2 0 : −2 0 : −2 0 8∗ 2 0 : 2 0 : 2 0 : 2 0 → 8∗ − 2 : −2 : 2 0 : 2 0 ≃ −3 : 2 : 2 → −3 : −2 0 : 2 ≃ −2 0 : −2 0 : −2 0 8∗ 2.3 0.2 → 8∗ − 2 0.(−2, 1) 0.2 :: . − 2 − 1 0 ≃ 9∗ 2 0 :::: 2 0 → 9∗ 2 0 ::: . − 2 − 1 0. − 2 ≃ 9∗ 6∗ 3 1 1.2.2 0 → 6∗ 3 1 1. − 2 0.2 0 ≃ 3.2.2 0. − 2 1 0 → 3. − 2 0.2 0.1 ≃ −3 : 2 : 2 → −3 : −2 0 : 2 ≃ −2 0 : −2 0 : −2 0 6∗ (2, 2+).2 0 :: 2 0 → 6∗ 2 − 1.2 0 :: 2 0 ≃ −2 0 : −2 0 : −2 0 6∗ (2, 2) 1.2 0 :: 2 0 → 6∗ 2 − 2 1.2 0 :: 2 0 ≃ 3 0 : 2 : 2 → 3 0 : −2 0 : −2 0 ≃ −2 0 : −2 0 : −2 0 6∗ 2.2.2.2 0.2 0.2 0 → 6∗ 2.2. − 2 0.2 0.2 0.2 0 ≃ 9∗ . − 4 0 → 9∗ .2 − 2 0.(−2, −1) ≃ 9∗ 8∗ 2.2 0.2 : .2 → ∗ 8 − 2 0. − 2.2 : .2. − 2 − 1 ≃ 9∗ 9∗ (2, 2) → ∗ 9 2 − 2 ≃ 8∗ 2. − 2 0.2 → 8∗ − 2 0. − 2 0.2 ≃ 9∗ 9∗ .(2, 2) → 9∗ .2 − 2 ≃ 9∗ . − 2 − 1 → 9∗ .1 ≃ 9∗ 6∗ 2.(2, 2).2 : 2 → 6∗ 2.2 − 2.2 : −2 0 ≃ −2 0 : −2 0 : −2 0 8∗ 2.2.3 0 → 8∗ − 2 0. − 2 0.3 0 ≃ 9∗ . − 2 − 1 → 9∗ .1 ≃ 9∗ 9∗ .(2, 2) 0 → 9∗ .2 − 2 0. − 2 − 1 ≃ 9∗ 9∗ (2, 2) 0 → ∗ 9 2 − 2 0 : −2 − 10 : . − 2 − 1 ≃ 8∗ − 2 − 10 : 2.2.2 → 8∗ 1 : −2 0. − 2 0.2 ≃ 9∗ 6∗ (2, 2).2 0.2.2 0 → 6∗ 2 − 2. − 2.2. − 2 ≃ −2 0 : −2 0 : −2 0

949 949

940

949

949

940

949 949

949

940 940

940

940 949

940 940

940 949

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n = 12

1

12n257

n = 12

1

12n414

n = 12

1

12n611

n = 12

1

12n745

n = 12

1

12n760

n = 12

1

12n838

n = 12

1

12n844

n = 12

1

12n847

n = 12

1

12n887

2 : (−2, −2 − 1) 0 : −2 0 → 2 : (−2, 1) 0 : −2 0 ≃ −2 0 : −2 0 : −2 0 −2 − 1 0.3.2.2 0 → 1.3.2.2 0 ≃ 3 0 : 2 : 2 → 3 0 : −2 0 : −2 0 ≃ −2 0 : −2 0 : −2 0 3.2.2 0. − 2 − 1 0 → 3.2.2 0.1 ≃ 3 0 : 2 : 2 → 3 0 : −2 0 : −2 0 ≃ −2 0 : −2 0 : −2 0 8∗ 2.3 0. − 2 0 → ∗ 8 − 2 0.(−2, 1) 0.2 : −2 − 1 : −2 − 1 ≃ 9∗ . − 2 − 1 → 9∗ .1 ≃ 9∗ ∗ 9 : −3 0.2 0 → 9∗ : (2, −1) 0.2 0. − 2 − 1 ≃ 8∗ 2.3 0. − 2 0 → 8∗ − 2 0.(−2, 1) 0.2 : −2 − 1 : −2 − 1 ≃ 9∗ . − 2 − 1 9∗ .1 ≃ 9∗ −2. − 2. − 2 0.2.2.2 0 → 2 0.2 0. − 2 0. − 2 0.2. − 2 ≃ 9∗ 8∗ 2.2 0.2 : . − 2 0 → 8∗ 2. − 2.2 : . − 2 0. − 2 − 1 ≃ 9∗ . − 4 0 → 9∗ .2 − 2 0. − 2 − 1 ≃ 9∗ 9∗ . − 4 0 → 9∗ .2 − 2 0 ::: . − 2 − 1 ≃ 9∗ 8∗ 2. − 3.2 → 8∗ − 2 0. − 3.2 ≃ 8∗ 2. − 2 0.2 → 8∗ − 2 0. − 2 0.2 ≃ 9∗

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949

949

949

940

940 940

940 940

940

According to the above results, we have the following theorem: Theorem 1.32. Among all prime knots with n ≤ 12 crossings there are two knots with n = 9, two with n = 10, four with n = 11, and 15 with n = 12 crossings which can not be unknotted by (2, 2)-moves. All of them belong to the equivalence classes of knots 9∗ (940 ) and −2 0 : −2 0 : −2 0 (949 ). Since they do not have common knots for n ≤ 20 crossings, we conjecture that these two equivalence classes are different. Among all prime non-alternating links up to 11 crossings and prime alternating links up to 12 crossings (without alternating links corresponding to basic polyhedra with n = 12 crossings) there are 2 links with n = 9, 2 links with n = 11, and 12 links with n = 12 crossings which can not be unknotted by (2, 2)-moves. All of them belong to the equivalence classes of knots 9∗ (940 ) and −2 0 : −2 0 : −2 0 (949 ). Among alternating KLs corresponding to basic polyhedra with n = 12 crossings two are algebraic: 125∗ and 1210∗ (or 12E and 12J according to Caudron). Using (2, 2)-moves we have not succeeded to unlink three of the

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remaining basic polyhedra with n = 12 crossings: 123∗ (12C), 124∗ (12D), 127∗ (12G), neither to prove that they can not be reduced to an unlink. We expect that these basic polyhedra are representatives of new classes of KLs which can not be unlinked by (2, 2)-moves.

Fig. 1.74

Basic polyhedra 12C, 12D, and 12G.

The following tables contain the list of knots with 13 ≤ n ≤ 14 crossings given in Knotscape notation, which belong to equivalence classes of knots 9∗ (940 ) and −2 0 : −2 0 : −2 0 (949 ).

n = 13

n = 14

13a3773 13n1733 13n2791 13n4674 13n4783 13n5085 14a4634 14a16965 14a18975 14n2910 14n11989 14n16197 14n18424 14n21793 14n24472 14n25143 14n25994 14n27145

13a4258 13n1836 13n3099 13n4695 13n4831

13a4387 13n1992 13n3695 13n4696 13n4870

13a4457 13n2173 13n3720 13n4735 13n4954

13n2526 13n4262 13n4742 13n5008

13n2688 13n4505 13n4771 13n5031

14a4786 14a16969

14a15312 14a17099

14a15589 14a17382

14a16042 14a17828

14a16801 14a18868

14n8313 14n13447 14n16680 14n18539 14n21966 14n24618 14n25299 14n26450 14n27152

14n8730 14n14577 14n16682 14n18590 14n22187 14n24678 14n25414 14n26505 14n27188

14n9035 14n15489 14n16965 14n18970 14n23629 14n24905 14n25686 14n26516 14n27191

14n9640 14n15505 14n17159 14n19941 14n23793 14n24908 14n25874 14n26640 14n27214

14n11899 14n16174 14n18301 14n21745 14n24016 14n24937 14n25967 14n26808

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n = 14

1.10

13a290 13n712 13n2034 13n3329 13n3127 14a4631 14n3100 14n9908 14n12705 14n16318 14n21806

13a336 13n714 13n2245 13n3386 13n3231 14a6031 14n3344 14n10168 14n13264 14n16942 14n23544

13a2980 13n828 13n2305 13n3507 13n3329 14a8654 14n3888 14n10262 14n13495 14n17138

13a3760 13n877 13n2833 13n4380 13n3386 14a8890 14n4398 14n11758 14n14690 14n18227

13a4799 13n989 13n3127 13n2305 13n3507 14a11240 14n4912 14n12403 14n14806 14n19117

13n1102 13n3231 13n2833 13n4380 14n8499 14n12463 14n15591 14n19871

Prime and composite KLs

The program LinKnot gives the opportunity to make experiments with large series of KLs given in a comprehensive way, in the Conway notation, and try to discover regularities. At first glance, KLs might seem to be an unordered, random structure, similar to that of prime numbers. Indeed, there is some analogy between KLs and numbers: in knot theory we also have prime KLs. Given any two links L1 and L2 we can define their composition (connected sum, concatenation, or direct product) denoted by L1 #L2 , as illustrated in Fig. 1.75. Suppose that a sphere in ℜ3 intersects a link L in exactly two points. This splits a link L into two arcs. The endpoints of either of those arcs can be joined by an arc lying on the sphere. This construction results in two links, L1 and L2 . The links L1 and L2 that make up the composite link L are called factor links (or simply, factors). Definition 1.63. A link is called prime if in every decomposition into a connected sum, one of the factors is unknotted. Otherwise, the link is called composite. For knots, the following properties hold: (1) (2) (3) (4)

if K1 = K2 , then for any K, K1 #K = K2 #K; for any K1 , K2 , K1 #K2 = K2 #K1 (commutativity); for any K1 , K2 , K3 , (K1 #K2 )#K3 = K1 #(K2 #K3 ) (associativity); for any K, K#1 = K, where 1 is an unknot (neutral element).

The prime decomposition theorem (Schubert, 1949; Hashizume, 1958) holds for links:

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Fig. 1.75

(a) Composition of knots; (b) commutativity.

Theorem 1.33. If L is a link, then there is a decomposition L = L1 # . . . #Lm where each Li (i = 1, . . . , m) is a prime link. Such a decomposition is unique up to order. In the set of natural numbers N with multiplication inverse elements do not exist. The same holds for KLs: inverse KLs do not exist. One KL can not cancel out another. That is, for a given KL there is no KL which, composed with the first, gives the unknot (unlink). A beautiful proof of the impossibility of knot cancellation is given by J. Conway (Gardner, 1986; Kauffman, 1995, page 16; Manturov, 2004, Theorem 2.2). Theorem 1.34. Let K1 be a non-trivial knot. Then for each knot K2 , the knot K1 #K2 is non-trivial. Proof. Let us make a direct product of knots K1 and K2 (e.g., a trefoil and a figure-eight knot) (Fig. 1.76). Put a tube around K1 #K2 so that a tube is a tubular neighborhood of K1 and engulfs K2 (i.e., construct a swallow-fallow torus for K1 and K2 ). The intersection of meridional disc c with K1 #K2 is non-trivial. On the other hand, for the unknot the only possible tubular neighborhood of the same kind is a torus, and every its intersection with a meridional disc is trivial. Hence, K1 #K2 is not an unknot. The second “speculative” proof of this theorem is given by Sossinsky (2002) and Manturov (2004), and the third proof based on genus is the corollary of Theorem 1.39.

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Fig. 1.76

Composition of KLs differs from multiplication of natural numbers. There is more then one way to form the composition of two KLs: a choice of where we remove an arc from the outside of each projection in order to get free strands and connect two knots. Since K1 #K2 = K2 #K1 , direct product is uniquely defined for prime knots, but not for prime links, where it depends from the choice of a component with removed arc. Many things about direct product of knots are still unknown. The main unanswered question is: if c(L) is the crossing number of a knot or link L, is it true that c(L1 #L2 ) = c(L1 ) + c(L2 )? This problem has been open for more then 100 years. From the Kauffman-Murasugi Theorem (Theorem 1.6) it follows that the conjecture holds when L1 #L2 is an alternating KL (Kauffman, 1988). Moreover, old conjecture on unknotting (unlinking) numbers u(L1 #L2 ) = u(L1 ) + u(L2 ) is still open. It is known that the inequality u(L1 #L2 ) ≤ u(L1 ) + u(L2 ) holds, and M. Scharlemann (1985) proved that the conjecture is true in the case u(L1 #L2 ) = 1. For u∞ numbers we also conjecture that u∞ (L1 #L2 ) = u∞ (L1 ) + u∞ (L2 ). Given a projection of a KL, let us define an overpass to be a sub-arc of the KL that goes over at least one crossing, but never goes under a crossing. A maximal overpass is an overpass that can not be made any longer, so both of its endpoints occur just before they go under a crossing. The bridge number of a projection is the number of maximal overpasses in the projection. These maximal overpasses form bridges over the rest of a link L. Definition 1.64. The bridge number of L, denoted by b(L), is the least bridge number of all of the projections of L (see, e.g., Adams, 1994).

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Theorem 1.35. A knot K has a bridge number 1 iff it is the unknot (see, e.g., Gilbert and Porter, 1994, Proposition 1.4). Proof. We accept the convention that the unknot has bridge number 1. Suppose that k is a knot with the bridge number b(K) = 1 and let D(K) be a diagram of K with one bridge and the smallest number of crossings m. Hence, m ≥ 3. No arc of D(K), except the bridge, is an overpass at any crossing. Therefore, there exist two adjacent crossings that are endpoints of an underpass. Here we can make a Reidemeister move Ω2 and reduce number of crossings by 2. This produces either a diagram of unknot, or a 1-bridge diagram with a number of crossings smaller then m, which contradicts the choice of m. Theorem 1.36. For composite knots or links L1 and L2 the following equality holds: b(L1 #L2 ) = b(L1 ) + b(L2 ) − 1 (Schubert, 1954).

Fig. 1.77

Two-bridge projections of a trefoil 3 and figure-eight knot 2 2.

There are two LinKnot functions dealing with direct product of KLs: fDToDDirect which computes Dowker code of a direct product of KLs, and fGenSignDirProd which computes signs of crossings of a direct product of KLs in the order corresponding to Dowker code or P -data. Theorem 1.37. For an oriented link L, there is an orientable, connected surface S in ℜ3 with boundary L (i.e, S spans L) (Seifert, 1934; Gilbert and Porter, 1994, Theorem 4.8; Murasugi, 1996, Theorem 5.1.1; Lickorish, 1997, Theorem 2.2; Manturov, 2004, Theorem 2.3). This theorem was first published by F. Frankl and L. Pontryagin in 1930, and then proved by H. Seifert in 1934. His proof gives an algorithm for constructing this surface, called Seifert surface.

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Proof. Take an oriented projection L′ of a link L and in each crossing, add two directed arcs (x1 , x2 ) and (y1 , y2 ) bypassing the crossing, but compatible with the orientation (please compare this operation with the derivation of mirror curves, page 389, Construction 3.1) (Fig. 1.78a). Then delete all crossings, leaving a set of oriented circles, called Seifert circles (Fig. 1.78b). Seifert circles obtained may or may not be nested. Two Seifert circles are nested if one of them is inside the other and if orientations of the two circles coincide. When Siefert circles obtained are not nested (Fig. 1.78b), the change-of-infinity operation– placing of the ∞point inside of the selected Seifert circle, nests the two circles (Fig. 1.78c). By joining Seifert circles together with half-twisted strips where crossings used to be, following the orientation of Seifert circles, an oriented surface S with boundary L is obtained. If the surface is not connected, connect components together by removing small discs and inserting tubes. Theorem 1.38. The genus of the Seifert surface S constructed from L is , where v is the number of crossings in L′ , and s is the number g = v−s+2 2 of Seifert circles obtained from L (Gilbert and Porter, 1994, Proposition 4.9; Murasugi 1996, Theorem 5.2.1). Proof. Each Seifert circle (a disk) has Euler characteristic 1. If we add a rectangle that consists of two oppositely oriented triangles (a half-twisted strip) between two disks, the number of vertices does not change, the number of edges increases by 3, and the number of faces increases by 2. We start with s disks, each contributing 1 to χ(S), and then add v rectangles, each reducing χ(S) by 1. Hence, χ(S) = s − v, and because s is . orientable, g = v−s+2 2 Different projections of L might give different surfaces with different genus. Definition 1.65. The genus of a link L is the minimal genus of all orientable surfaces which L spans (see, e.g., Gilbert and Porter, 1994; Murasugi, 1996; Lickorish, 1997; Sossinsky, 2002; Manturov, 2004). Theorem 1.39. For genus of two links L1 and L2 and their direct product L1 #L2 , the following equality holds: g(L1 #L2 ) = g(L1 ) + g(L2 ) (Schubert, 1949). For the proof see, e.g., the books by Murasugi (1996), Lickorish (1997), or Manturov (2004).

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Since the unknot has genus zero and all the other knots have genus greater then zero, Theorem 1.34 (“Knots can not cancel each other”) is a corollary of the last theorem.

Fig. 1.78

Seifert construction.

Computing knot genus is NP-complete problem, which was solved by Haken (1961), but the algorithm was too complicated for implementation. Table of knot invariants by C. Livingston and J.C. Cha (2005) contains genus for knots with n ≤ 12 crossings. From the following table based on experimental results15 Family 2k + 1 (2k) (2l) (2k) (2l + 1) (2k + 1) (2l + 1) (2m)

Genus k 1 k k+1

Family 2k + 1, 2l + 1, 2m (2k + 1, 2l + 1) (2m + 1, 2n + 1) 6∗ (2k).(2l) 8∗ (2k + 1).(2l + 1).(2m + 1)

Genus k+l+1 k+l+m+n+1 k+l+1 k+l+m+3

15 The results are obtained using the 3-genus program written by Jake Rasmussen, included in the package KnotTheory (Bar Natan, 2006).

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we can conjecture that genus of a knot is linear subfamily-dependent invariant (Conjecture 1.2). Another invariant based on Seifert surface is a signature of KL, defined using Seifert matrix (for the details, see e.g., Murasugi, 1996, Chapter 5). Definition 1.66. If M is diagonalized Seifert matrix of a link L, the difference between the number of positive and negative diagonal entries is called signature of L, and denoted by σ(L). Theorem 1.40. Signature σ(L) is the invariant of a link L (Seifert, 1934; Rolfsen, 1976; Murasugi, 1996; Kauffman and Taylor, 1976). Some of the important properties of signature are: (1) σ(L1 #L2 ) = σ(L1 ) + σ(L2 ) for every two links L1 , L2 ; (2) σ(L) = −σ(L− ), where L− is the mirror image of L; (3) |σ(D+ ) − σ(D− )| ∈ {0, 2}, where D+ and D− are diagrams of a link L that differ in only one crossing (Fig. 1.58a, Fig 2.7). The proof of these theorems can be found in Cromwell (2004). From the second property it follows that every achiral KL has zero signature. However, some chiral knots have zero signature as well (e.g., 61 , 77 , 81 , 88 ,...) and, according to experimental results, this holds for the families of chiral knots (2k) (2l) (k > l), (2k) 1 1 1 (2k), (2k) (2l + 1) 1 (2k), etc. Using LinKnot function fSignat it is possible to compute signature of large series of KLs given in Conway notation in attempt to find general formulas for signature of (sub)families of KLs. General formulas, based on these computations for KL families derived from generating KLs with n ≤ 9 crossings, are given in Appendix A (http://www.mi.sanu.ac.yu/vismath/Appendix.pdf). We conjecture that signature of KL is a linear subfamily-dependent invariant (Conjecture 1.2).

1.11

Non-invertible KLs

An orientation of a knot is a choice of direction to travel around the knot. Hence, for every non-oriented knot K, we have two different orientations and two oriented knots denoted by K ′ and K ′′ .

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Definition 1.67. A knot K is called invertible (or reversible) if the oriented knots K ′ and K ′′ are equivalent. Otherwise, it is called non-invertible. The existence of non-invertible knots was shown by H.F. Trotter (1963), who discovered non-invertible knot 7, 5, 3 and the whole family of noninvertible pretzel knots (2p + 1), (2q + 1), (2r + 1) (p 6= q, p 6= r, q 6= r) (Fig. 1.79). The proof of their non-invertibility can also be found in the book An Introduction to Knot Theory by W.B.R. Lickorish (1997, Theorem 11.11). Unlike in 1963, now we know that almost all knots are non-invertible (Murasugi, 1996). The number of non-invertible and invertible knots with 3 ≤ n ≤ 16 crossings is given in the following table (the sequences A052403 and A052402 from the The On-Line Encyclopedia of Integer Sequences by N. Sloane): 0 1

0 1

0 2

0 3

0 7

1 20

2 47

33 132

187 365

1144 1032

6919 3069

38118 8854

226581 26712

1309875 78830

The first non-invertible knot from Rolfsen’s knot tables is the knot 817 (.2.2 in the Conway notation). Composing two copies of this knot, two with matching orientations, and two with different orientations, we get two distinct composite knots which are not equivalent (Fig. 1.80). A. Kawauchi (1979) proved that there is no deformation of the knot .2.2 that reverses its orientation. Thirty six non-invertible knots with up to 10 crossings are identified by R. Hartley (1983). So far, there is no general technique for recognizing non-invertible knots.

Fig. 1.79

Non-invertible knot 7, 5, 3.

In recognizing achiral knots, we can use their antisymmetric rigid representations in ℜ3 , or on the sphere S 3 . Every KL that has an antisym-

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Fig. 1.80

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Different composite knots obtained from the knot 817 (.2.2).

metric presentation in ℜ3 , has it on S 3 as well, but not necessarily vice versa. For example, both presentations of figure-eight knot are antisymmetric: its non-minimal diagram (Fig. 1.81a) is invariant with regard to a rotational antireflection of order 4 (i.e., rotational reflection followed by crossing sign change), and its diagram obtained from a stereographic projection of S 3 is centro-antisymmetric (Fig. 1.81b). Achiral knot 817 has remarkable property: it has a centro-antisymmetric presentation coming from S 3 (Fig. 1.82), but no antisymmetric presentation in ℜ3 . Therefore, it is a topological rubber glove in ℜ3 (Flapan, 1998, 2000). The program SnapPea by J. Weeks (whose 2.0 version is also part of Knotscape) computes knot symmetry group and detects non-invertible knots. Beginning from experimental computational results (based on the list of non-invertible alternating knots with n ≤ 12 crossings) and the idea of families of knots in Conway notation, we developed a technique for deriving families of non-invertible knots and determining number of components of a KL from its Conway symbol. Our main goal was to find the general method for recognizing non-invertible knots directly from their Conway

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Fig. 1.81 S3.

Antisymmetric presentations of a figure-eight knot 2 2 (41 ) (a) in R3 ; (b) on

Fig. 1.82

Centro-antisymmetric presentation of knot .2.2 (817 ) on S 3 .

symbols. We succeed for some classes of knots, like alternating pretzel (stellar) knots, and extended this method to some other classes of algebraic and polyhedral alternating knots. L. Kauffman and S. Lambropoulou (2002a) proved that all oriented rational KLs are invertible (Theorem 1.21). Hence, the first non-invertible knots belong to stellar (pretzel) knots. In the following table, for every non-invertible knot with n ≤ 11 crossings is given its corresponding family of non-invertible knots and the conditions for their achirality16 . 16 If the fourth column is empty, this means that the corresponding family contains only chiral non-invertible knots. Single knot 1082 is the member of family 1).

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1)

817

.2.2

.(2p).(2q)

2) 3)

932 933

.2 1.2 0 .2 1.2

.(2p) 1.(2q) 0 .(2p) 1.(2q)

4) 5) 6) 7)

1067 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1090 1091 1092 1093 1094 1095 1098 10102 10106 10107 10109 10110 10115 10117 10118 10119

2 2, 3, 2 1 (3, 2) (3, 2) (3, 2) (2 1, 2) (2 1, 2) (2 1, 2) .4.2 .3 1.2 .2 2.2 .4.2 0 .3 1.2 0 .2 2.2 0 .2 1.2 1 .3.2.2 .3.2.2 0 .2 1.2.2 0 .3.2 0.2 .3 0.2.2 .2 1 0.2.2 .2.2.2.2 0 3 : 2 : 20 30 : 2 : 20 .2 1 0 : 2 : 2 0 2.2.2.2 2.2.2.2 0 8∗ 2 0.2 0 8∗ 2 : 2 0 8∗ 2 : .2 8∗ 2 : .2 0

(2p) (2q), (2r + 1), (2s) 1 ((2p + 1), (2q)) ((2r + 1), (2s)) ((2p + 1), (2q)) ((2r) 1, (2s)) ((2p) 1, (2q)) ((2r) 1, (2s))

8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29)

p=q

.(2p + 1) 1.(2q) .(2p) (2q).(2r) .(2p).(2q) 0 .(2p + 1).(2q) 0 .(2p) (2q).(2r) 0 .(2p) 1.(2q) 1 .(2p + 1).(2q).(2r) .(2p + 1).(2q).(2r) 0 .(2p) 1.(2q).(2r) 0 .(2p + 1).(2q) 0.(2r) .(2p + 1) 0.(2q).(2r) .(2p) 1 0.(2q).(2r) .(2p).(2q).(2r).(2s) 0 (2p + 1) : (2q) : (2r) 0 .(2p + 1) 0.(2q).(2r) 0 (2p) 1 0 : (2q) : (2r) 0 (2p).(2q).(2r).(2s) (2p).(2q).(2r).(2s) 0 8∗ (2p) 0.(2q) 0 8∗ (2p) : (2q) 0 8∗ (2p) : .(2q) 8∗ (2p) : .(2q) 0

p = r and q = s p = r and q = s

p 6= q

p=q

p = s and q = r p=q p=q

The list of alternating knots with n = 11 crossings, ordered according to the Knotscape tables and followed by the Conway notation and symmetry type, can be found in Table of knot invariants by C. Livingston and J.C. Cha (2005). Among 376 alternating knots (denoted in Table of knot invariants as “chiral”), there are 123 non-invertible knots. Two knots, 11a53 .2 1.4 0, and 11a262 .4 1.2 0, are the members of the family 2), and the knots 11a68 .2 1.4, and 11a265 .4 1.2, are the members of the family 3). According to experimental results, following four knots generate three-parameter families of non-invertible knots with the additional conditions for non-invertibility: 11a201 11a299 11a323 11a345

4 1, 3, 2 1 .4.2.2 .4.2 0.2 4 : 2 : 20

(2p) 1, (2q + 1), (2r) 1 .(2p).(2q).(2r) .(2p).(2q) 0.(2r) .(2p) : (2q) : (2r) 0

p 6= r p 6= r p 6= r p 6= q

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The same procedure can be applied to the remaining non-invertible knots with n = 11 crossings which generate the corresponding families of non-invertible knots without restrictions on parameters. Notice that in some families can occur repeated knots (e.g., knot 1082 given by Conway symbols .4.2 = .2.4, which belongs to the family 1) .(2p).(2q) can be obtained for p = 2 and q = 1, and for p = 1 and q = 2, respectively). Among 1288 alternating knots with n = 12 crossings, 674 are noninvertible: 37 achiral and 637 chiral. Among 37 non-invertible achiral knots (denoted in Table of Knot Invariants as “− amphicheiral”), only knot 12a1218 .4.4, is obtained from the family 1), two of them are alternating knots corresponding to the basic polyhedra (12a868 12K, and 12a1188 12B), and all the others are generators of the new families of achiral and chiral non-invertible knots. Here is their list with the conditions which parameters need to satisfy in order to obtain achiral non-invertible knots: 12a4 12a58 12a125 12a268 12a273 12a341 12a458 12a462 12a465 12a627 12a819 12a821 12a887 12a890 12a906 12a960 12a990 12a1008 12a1102 12a1123 12a1124 12a1152 12a1167 12a1209 12a1211 12a1225 12a1229 12a1249 12a1251 12a1254

(2 1 1, 2) (2 1 1, 2) (2 1, 2+) (2 1 2+) (2 2, 2) (2 2 2) 8∗ 2 1.2 1 .2 2.2 2 2.2 1.2 1.2 8∗ 2 1 0 : .2 1 0 (3, 2+) (3, 2+) .2 1.2 1.2.2 2 1 0.2.2.2 1 0 8∗ 3 : .3 ∗ 8 3 0 : .3 0 10∗∗ : 2.2 8∗ 3 0.3 0 8∗ 2 0.2 0 : .2.2 10∗∗ : 2 :: .2 8∗ 2 0.2.2.2 0 .3 1.3 1 10∗∗ : 2 0 :: .2 0 8∗ 2 0.2 0.2 0.2 0 2.2.2.2.2 0.2 0 10∗∗ 2 0 :: .2 0 10∗∗ : 2 0.2 0 8∗ 3.3 10∗ 2.2 8∗ 2.2.2.2 8∗ 2.2 : .2.2 10∗ 2 ::: .2 10∗∗ 2 :: .2 .3.3.2 0.2 0

((2p) 1 1, (2q)) ((2r) 1 1, (2s)) ((2p) 1, (2q)+) ((2r) 1, (2s)+) ((2p) 2, (2q)) ((2r) 2, (2s)) 8∗ (2p) 1.2(q) 1 (2p) (2q).(2r) (2s) (2p).(2q) 1.(2r) 1.(2s) 8∗ (2p) 1 0 : .2(q) 1 0 ((2p + 1), (2q)+) ((2r + 1), (2s)+) .(2p) 1.(2q) 1.(2r).(2s) (2p) 1 0.(2q).(2r).(2s) 1 0 8∗ (2p + 1) : .(2q + 1) ∗ 8 (2p + 1) 0 : .2(q + 1) 0 10∗∗ : (2p).(2q) 8∗ (2p + 1) 0.2(q + 1) 0 8∗ (2p) 0.(2q) 0 : .(2r).(2s) 10∗∗ : (2p) :: .(2q) ∗ 8 (2p) 0.(2q).(2r).(2s) 0 .(2p + 1) 1.(2q + 1) 1 10∗∗ : (2p) 0 :: .(2q) 0 8∗ (2p) 0.(2q) 0.(2r) 0.(2s) 0 (2p).(2q).(2r).(2s).(2t) 0.(2u) 0 10∗∗ (2p) 0 :: .(2q) 0 10∗∗ : (2p) 0.(2q) 0 8∗ (2p + 1).2(q + 1) 10∗ (2p).(2q) 8∗ (2p).(2q).(2r).(2s) 8∗ (2p).(2q) : .(2r).(2s) 10∗ (2p) ::: .(2q) 10∗∗ (2p) :: .(2q) .(2p + 1).(2q + 1).(2r) 0.(2s) 0

p = r, q = s p = r, q = s p = r, q = s p=q p = r, q = s p = s, q = r p=q p = r, q = s p = q, r = s p = s, q = r p=q p=q p=q p=q p = q, r = s p=q p = s, q = r p=q p=q p = s, q = r p = s, q = r, t = u p=q p=q p=q p=q p = s, q = r p = s, q = r p=q p=q p = q, r = s

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12a1260 12a1267 12a1280

3.2.2.3 3 0.2.2.3 0 8∗ .2 : 2 0.2 0 : 2

(2p + 1).(2q).(2r).(2s + 1) (2p + 1) 0.(2q).(2r).(2s + 1) 0 8∗ .(2p) : (2q) 0.(2r) 0 : (2s)

p = s, q = r p = s, q = r p = s, q = r

Among 637 non-invertible chiral knots, 68 are obtained from the existing families derived from knots with n = 8 and n = 10 crossings, and 554 are generators of the new families of chiral non-invertible knots without additional conditions for parameters. In the following table, the remaining 15 chiral non-invertible knots, families of chiral non-invertible knots derived from them, and non-invertibility conditions are given:

12a76 12a192 12a201 12a566 12a610 12a735 12a753 12a782 12a952 12a981 12a984 12a988 12a1191 12a1238 12a1240

.4.2 1 0.2 .4 1.2 1 0 .4.2 1.2 4 1, 2 1 1, 2 1 4 : 21 : 2 5, 3, 2 1+ 5, 2 1 1, 3 21 : 40 : 20 2.4.2 0.2 .4.2.2 0.2 0 4:3:2 4 : 2 : 30 8∗ 4 : 2 3 : 40 : 20 40 : 30 : 20

.(2p).(2q) 1 0.(2r) .(2p) 1.(2q) 1 0 .(2p).(2q) 1.(2r) (2p) 1, (2q) 1 1, (2r) 1 (2p) : (2q) 1 : (2r) (2p + 1), (2q + 1), (2r) 1+ (2p + 1), (2q) 1 1, (2r + 1) (2p) 1 : (2q) 0 : (2r) 0 (2p).(2q).(2r) 0.(2s) .(2p).(2q).(2r) 0.(2s) 0 (2p) : (2q + 1) : (2r) (2p) : (2q) : (2r + 1) 0 8∗ (2p) : (2q) (2p + 1) : (2q) 0 : (2r) 0 (2p) 0 : (2q + 1) 0 : (2r) 0

p 6= r p 6= q p 6= r p 6= r p 6= r p 6= q p 6= r q 6= r q 6= s p 6= q p 6= r p 6= q p 6= q q 6= r p 6= r

Every alternating non-invertible knot (except those corresponding to basic polyhedra) is the generator, or the member of the family of noninvertible knots. Based on the properties of its generating KL we can determine if some additional requirements are needed for the whole family of KLs to be non-invertible. For example, even though the generating knot 816 .2.2 0 is invertible, the family it generates, .(2p).(2q) 0, contains only non-invertible KLs, beginning from the knot 1085 .4.2 0, provided that p 6= q. 1.11.1

Tangle types

Disregarding the sign (+1, −1), there are two elementary tangles, [1] and [0], where for [0] we can distinguish between two different positions [0] and [∞] (Kauffman and Lambropoulou, 2002a) (Fig. 1.83).

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Fig. 1.83

The elementary tangles.

Definition 1.68. A sequence of positive integers separated by space (denoting product) and not beginning by 1 will be called R-tangle. If an R-tangle contains k numbers (k ≥ 1), we say that its length is k. Algebraic KLs are obtained from R-tangles using three operations– product, sum, and ramification. For R-tangles of an odd length the first bigon (or chain of bigons) is always drawn as horizontal, and for R-tangles of an even length as vertical. Every R tangle consists of two strands connecting SW-SE and NW-NE ends, SW-NE and SE-NW ends, or SW-NW and SE-NE ends. According to this, every tangle will be of the [0]-type, [1]-type, or [∞]-type, respectively. For every n, there are 2n−2 R-tangles with n crossings. For example, for n=2 we have an R-tangle 2 of the type [0]; for n = 3 two R-tangles: 3 of the type [1], and 2 1 of the type [∞]; for n = 4 four R-tangles: 4 and 3 1 of the type [0], 2 2 of the type [∞], and 2 1 1 of the type [1]. By taking every number modulo 2, we obtain 0-1 sequences of the length k. Numerator closures of tangles of the types [1] and [∞] are knots, and the type [0] gives 2-component links. Recall that the product of tangles a and b is defined as a b = −a + b = a 0 + b, where a 0 denotes tangle a reflected in SE-NW mirror line. The next table contains product rules for tangle types: [0] [0] = [∞] [1] [0] = [1] [∞] [0] = [0]

[0] [1] = [∞] [1] [1] = [0] [∞] [1] = [1]

[0] [∞] = [∞, ∞] =[∞2 ] [1] [∞] = [∞] [∞] [∞] = [∞]

The right multiplications by [0] form a dihedral symmetry group with the invariant point [1], and the right multiplications by [1] make the cyclic group of order 3 (Fig. 1.84).

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Fig. 1.84

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(a) The right multiplications by [0]; (b) right multiplications by [1].

The LinKnot function fTangleType calculates the type of R-tangle, giving as the result 0,1,or 2 for a tangle of the type [0],[1], or [∞], respectively. The function fMakeType produces all tangles of a given type with n crossings. The following table illustrates how to determine type of any R-tangle. For example, R-tangle 3 5 4 1 3 has 0-1 code 11011, so its type is [0]: 1 [1]

11 = [1] [1] [0]

110 = [0] [0] [∞]

1101 = [∞] [1] [1]

11011 = [1] [1] [0]

In order to simplify notation, in the case of R-tangles the brackets [ ] will be omitted, and the products of the tangles [0] and [1] will be expressed as 0-1 sequences. For example, 011 means [0] [1] [1], etc. For k = 1 there are two sequences: 0 of the type [0] and 1 of the type [1]; for k = 2 four sequences: 00 and 01 of the type [∞], 10 of the type [1], and 11 of the type [0]; for k = 3 eight sequences: 000, 010, 101 of the type [0], 001, 100, 011 of the type [1], and 010, 101 of the type [∞]; for k = 4 sixteen sequences: 0000, 0001, 0100, 0101, 1010, 1011 of the type [∞], 0010, 1000, 0110, 1101, 1111 of the type [1], and 0011, 1001, 1001, 1100, 0111, 1110 of the type [0], etc. Since every rational KL is obtained as a numerator closure of an R-tangle not beginning or ending with 1, it can be expressed as a 0-1 sequence. This means that it contains only tangles of the type [0] or [1], so as the final result can be obtained only a tangle of the type [1], [∞], or [0]. It can be only a knot (obtained as a numerator closure from [1], [∞]), or 2-component link (obtained from [0]). Hence, every rational KL is a knot, or 2-component link.

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Stellar (pretzel) tangles are obtained from at least three R-tangles using the operation of ramification. Pretzel KLs are numerator closures of pretzel tangles. For denoting the types of pretzel knots double brackets will be also omitted in the following sense: the type of pretzel knot obtained from Rtangles of the type [0], [1], [∞], will be concisely denoted as [∞, 1, 0], since [0]0 = [∞], [1]0 = [1], [∞]0 = [0]. For example, the type of the pretzel tangle 2, 3, 2 1 will be [[0], [1], [∞]] = [∞, 1, 0]. For pretzel KLs with three R-tangles we have 10 possible sets of types. [1,1,1], [1,1,∞], [0,0,1], [0,1,∞], [0,0,∞] are knots, [0,1,1], [1,∞,∞], [0,0,0], [0,∞,∞] 2-component links, and [∞,∞,∞] are 3-component links (Fig. 1.85). For pretzel KLs with four R-tangles, knots are [0,1,1,1], [1,1,1,∞], [0,1,1,∞], [0,0,0,1], [0,0,1,∞], [0,0,0,∞], 2-component links [1,1,1,1], [0,0,1,1], [1,1,∞,∞], [0,1,∞,∞], [0,0,0,0], [0,0,∞,∞], 3-component links [1,∞,∞,∞], [0,∞,∞,∞], and 4-component links [∞,∞,∞,∞]. In gensets of types. Among eral, for pretzel KLs with t R-tangles there are t+2 2 them, t + 2 will give knots, 2t − 2 will give 2-component links, and the number of i-component link types will be t − i + 1 (i = 3, 4, . . . , t).

Fig. 1.85

Types of pretzel knots consisting of three R-tangles.

For pretzel KLs consisting of more then three R-tangles, for the same type there are several different orders17 of the particular symbols 0, 1, ∞. For example, for the type [0,1,1,∞] there are two possible orders: [0,1,1,∞] and [0,1,∞,1], etc. 17 Cyclic

permutations of types, where reverse permutations are treated as equal.

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In general, the following rules hold: [1,1]=[0], [1, ∞] = [∞], [a, 0] = [a] for every a (a ∈ {0, 1, ∞}). The set of types [∞, . . . , ∞] where ∞ occurs k times will be concisely denoted as [∞k ] (k = 1, 2, . . .). Moreover, the calculation of the reduced types of pretzel KLs is commutative on the symbols 0, 1, ∞. Using these rules, every set of types can be reduced to [0], [1], or [∞k ]. The number of components of a pretzel KL of the reduced type [0] is 2, [1] are knots, and [∞k ] k-component links (k = 1, 2, . . .). For example, [1,1,1,1,1,0,0,0,∞,∞] = [1, ∞, ∞] = [∞2 ], so it is a 2component link. The symbol [∞k ] represents a tangle of the type [∞] with k − 1 already closed components. Hence, [∞k ] = [∞k−1 ], where the subscript k −1 denotes the number of already closed components. A numerator closure of [∞k ] = [∞k−1 ] is a k-component link (Fig. 1.86).

Fig. 1.86 Pretzel tangle [∞, ∞, ∞, ∞] = [∞4 ] = [∞3 ] and its numerator closure giving 4-component link.

In order to keep track of the number of already closed components we use the addition of subscripts denoting the number of closed components. For example, [0k , ∞l ] = [∞k+l ], [∞k ] [∞l ] = [∞k+l ], [∞k ] [0l ] = [0k+l ], etc.. Knowing that the numerator closure of [1] and [∞] gives one component, and the numerator closure of [0] gives two components, we conclude that the numerator closure of [1k−1 ] and [∞k−1 ] is a k-component link, and the

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numerator closure of [0k−1 ] is a (k + 1)-component link. Therefore, tangle type calculation gives the number of components of any algebraic KL. The equality of tangle types is the equivalence relation on R-tangles with three equivalence classes [0], [1], [∞], whose minimal representatives are 2, 3, 2 1. Instead of the tangle 1, the tangle 3 is taken as the minimal representative of the type [1], to avoid ambiguity coming from two different orientations (“vertical” or “horizontal”) of tangle 1. We construct families of KLs adding an even number of bigons to already existing bigons or chains of bigons. Since this addition preserves tangle type, the number of components will be preserved inside subfamilies. In general, the number of components of a KL remains preserved if we replace any tangle by a tangle of the same type. 1.11.2

Non-invertible pretzel knots

Every alternating pretzel KL is of the form r1 , r2 , . . . , ri , where r1 , r2 ,...,ri are R-tangles (i ≥ 3). Pretzel knots are obtained as the numerator closures of the pretzel tangles of the reduced type [1], or [∞]. Hence, non-reduced types of pretzel KLs can be: • [0, . . . , 0, 1, . . . 1], where 1 has an odd number, and 0 an arbitrary number of occurrences; • [0, . . . , 0, 1, . . . 1, ∞]. The main question is whether it is possible to determine non-invertibility of pretzel knots according to tangle types. Every pretzel knot can be drawn as a regular t-gon with vertices denoting R-tangles, called t-diagram. In a t-diagram vertices by themselves are treated as symmetric, and the mirror line contains at least one vertex. Conjecture 1.5. (Non-invertibility criterion for pretzel knots) A pretzel knot is non-invertible iff its type symbol consists only from 0-s and 1-s, and its t-diagram is not mirror-symmetric. We give detailed description of non-invertible pretzel knots with i ≤ 5: • for i = 3 the types giving knots are [1,1,1], [0,0,1], and all three Rtangles must be mutually different; • for i = 4 the types giving knots are [1,1,1,0], [0,0,0,1], and from the symmetry condition follows that the R-tangles at the first and third position must be different;

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• for i = 5 the types giving knots are [1,1,1,1,1], [0,0,1,1,1], [0,0,0,0,1], and the symmetry condition can be easily recognized from a t-diagram (Fig. 1.87).

Fig. 1.87 (a) Non-invertible pretzel knot 5, 3, 3, 2 1; (b) invertible pretzel knot 5, 3, 2 1, 3 with mirror-symmetric t-diagram.

For example, pretzel knots 3, 3, 3, 5, 7 and 3, 3, 5, 3, 7 are non-invertible, while 3, 5, 5, 3, 7 is invertible, and all have the same type [1,1,1,1,1]. In general, the necessary condition for invertibility of pretzel knots is that the sum of numbers in knot type must be odd. Sufficiency is determined, based on symmetry condition, from the t-diagram. As we underlined in Section 1.11, we can distinguish generating noninvertible knots and non-invertible knots that are members of already derived families. From every generating non-invertible pretzel knot its corresponding family is obtained by the following substitutions, respecting symmetry conditions: • every single bigon 2 can be replaced by the chain of bigons (2p); • chain 3 can be replaced by any odd chain (2p + 1); • tangle 1 remains unchanged. For n = 10 crossings there is only one non-invertible pretzel knot: 2 2, 3, 2 1. For n = 11 there are four of them: 4 1, 2 1, 3

2 3, 2 1, 3

2 1 1, 2 2, 2 1

3 1 1, 3, 2 1

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For the first knot 4 1, 2 1, 3, its generating knot is 2 1, 2 1, 3 with n = 8 crossings, that satisfies the necessary condition for non-invertibility of pretzel knots: its type [0,0,1] consists only from 0-s and 1-s. It contains the tangles (2p) 1 and (2q) 1, so according to the symmetry condition it generates the family of non-invertible pretzel knots (2p) 1, (2q) 1, (2r + 1) for p 6= q. Hence, the first non-invertible knot in this family is 4 1, 2 1, 3. For n = 11, the four non-invertible knots are divided into two subsets: 4 1, 2 1, 3 2 3, 2 1, 3

2 1 1, 2 2, 2 1

3 1 1, 3, 2 1

The first subset contains the knot 4 1, 2 1, 3 that belongs to the family of non-invertible knots (2p) 1, (2q) 1, (2r + 1) with the additional noninvertibility condition p 6= q, and the knots from the other subset generate families of non-invertible knots without additional requirements on parameters. For n = 12 there are 17 non-invertible pretzel knots: 4 2, 2 1, 3

2 4, 2 1, 3

2 1 1, 5, 3

2 1 1, 4 1, 2 1

3 1 2, 2 1, 3 3 1 1, 2 2, 3 3 1 1, 2 1 1, 2 1

4 1, 2 2, 3

2 1 2 1, 2 1, 3 3 2, 2 2, 2 1 2 2 1, 2 1 1, 3

2 2, 2 1, 5

2 1 1 1 1, 2 1, 3 2 2 1, 2 2, 2 1 3 2, 2 1 1, 3

2 3, 22, 3 2 1 1, 2 3, 21

divided into three subsets. The first subset contains non-invertible knots obtained from the previously derived family (2p) 1, (2q) 1, (2r + 1) (p 6= q); the second subset contains knots belonging to the new families with additional symmetry conditions (2p) 1 1, (2q + 1), (2r + 1) (q 6= r), and (2p) 1 1, (2q) 1, (2r) 1 (q 6= r); every member of the third subset generates the family of non-invertible knots with no additional conditions for parameters. For n = 13 we have 51 non-invertible knot, divided into the three subsets: 6 1, 2 1, 3 5 1 1, 2 1, 3 3 1 1, 2 1, 5 2 1 1, 4 1, 2 2

2 5, 2 1, 3 3 1 1, 4 1, 3 4 1 1, 2 2, 2 1

3 2, 5, 3 4 1 1, 2 1 1, 3

2 2 1, 5, 3 4 1, 3 2, 2 1

4 3, 2 1, 3 2 3, 2 1, 5 2 1 1, 4 2, 2 1

2 4, 2 2, 3 2 2 1, 4 1, 2 1

4 1, 2 3, 3 4 1, 2 1, 5 2 1 1, 2 4, 2 1

4 2, 2 2, 3

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3 1 3, 2 1, 3 3 2 1 1, 2 1, 3 2 1 2 1, 2 2, 3 2 1 3, 2 1 1, 3 2 1 3, 2 2, 2 1 3 2, 2 3, 2 1 2 1 2 1, 2 1 1, 2 1 3 1 1, 2 1 1, 2 2

3 3 1, 2 1, 3 2 1 1 1 2, 2 1, 3 2 1 1 1 1, 2 2, 3 2 3 1, 2 1 1, 3 2 3 1, 2 2, 2 1 2 2 1, 2 3, 2 1 2 1 1 1 1, 2 1 1, 2 1 2 1 1, 2 1, 3, 3

2 1 2 2, 2 1, 3 2 2 1 1 1, 2 1, 3 3 1 1, 2 3, 3 2 1 1 2, 2 1 1, 3 2 1 1 2, 2 2, 2 1 3 1 1, 3 2, 2 1 3 1 1, 2 2 1, 2 1 2 2, 2 1, 2 1, 3

2 2 2 1, 2 1, 3 3 1 2, 2 2, 3 2 2 1, 3 2, 3 3 1 1 1, 2 1 1, 3 3 1 1 1, 2 2, 2 1 3 1 2, 2 1 1, 2 1 2 1 1, 2 3, 2 2

The first subset contains knots from already derived families (with or without additional symmetry conditions). Knots from the second and third subset are the generators of new families, with and without additional symmetry conditions for parameters, respectively. For n = 13 we have first non-invertible knots, 2 1 1, 2 1, 3, 3 and 2 2, 2 1, 2 1, 3, with four R-tangles, where tangles at the non-adjacent positions are different, so their t-diagrams are not mirror-symmetric. The knots that consist of the same R-tangles, 2 1 1, 3, 2 1, 3 and 2 2, 2 1, 3, 2 1, with the mirror-symmetric t-diagrams, are invertible. Using LinKnot function fNinvStellar we obtained the following number of non-invertible pretzel knots with n crossings (n = 10, . . . , 17): n

10 1

11 4

12 17

13 51

12 155

15 427

16 1152

17 2983

The similar treatment can be applied to non-invertible pretzel knots with pluses, since every pretzel knot of the form r1 , r2 , . . . , ri with k pluses can be written as r1 , r2 , . . . , ri , 1, . . . , 1 where 1 occurs k times, and r1 , r2 ,...,ri (i ≥ 3) are R-tangles. The symmetry condition will be applied only to the R-part of the knot, this means, to r1 , r2 , . . . , ri . The first non-invertible pretzel knot with pluses is 2 1 1, 3, 2 1+ with n = 11 crossings. For n = 12 there are five non-invertible pretzel knots with pluses: 2 2, 2 1, 3 + + 2 2 1, 2 1, 3+

2 1, 5, 3+ 2 1 1, 2 2, 3+

3 2, 2 1, 3+

For n = 13 there are 22 non-invertible knots: 4 1 1, 2 1, 3+

2 1 1, 4 1, 3+

4 1, 2 1, 3 + +

2 2, 5, 3+

2 1 1, 2 1, 5+ 4 1, 2 2, 2 1+

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2 1 1, 2 1, 3 + ++ 2 1 3, 2 1, 3+ 3 2, 2 2, 3+ 2 3, 2 2, 2 1+

2 3, 2 1, 3 + + 2 3 1, 2 1, 3+ 2 2 1, 2 2, 3+ 3 1 1, 2 2, 2 1+

3 1 1, 2 1, 3 + + 2 1 1 2, 2 1, 3+ 2 1 1, 2 3, 3+ 2 1 1, 3 2, 2 1+

2 1 1, 2 2, 2 1 + + 3 1 1 1, 2 1, 3+ 3 1 1, 2 1 1, 3+ 2 2 1, 2 1 1, 2 1+

divided again into three subsets: knots belonging to already derived families, knots generating new families with additional symmetry conditions, and knots generating families without additional conditions for parameters. In order to obtain families of non-invertible pretzel knots with pluses, we can use the same replacements as in the case of pretzel knots, but also we can replace every sequence of k pluses with a sequence of the same parity. The number of components of a pretzel KL with pluses can be computed using the same rules as for pretzel KLs. 1.11.3

Non-invertible arborescent knots

The simplest class of arborescent KLs giving non-invertible knots is (r1 , r2 )(r3 , r4 ), where ri (i = 1, . . . , 4) are R-tangles. Using tangle type calculation, we conclude that knots will be obtained for the following (r1 , r2 )(r3 , r4 ) pretzel sets of types: [1, 1] [1, 1], [0, 1] [1, 1], [0, 0] [1, 1], [1, ∞] [1, ∞], [0, 0] [0, 1], [0, 1] [0, ∞], [0, ∞] [1, ∞], [0, 0] [0, 0], [0, ∞] [0, ∞], 2-component links will be obtained for [1, 1] [1, ∞], [0, 1] [0, 1], [0, 0] [1, ∞], [0, 1] [∞, ∞], [1, ∞] [∞, ∞], [0, 0] [0, ∞], [0, ∞] [∞, ∞], and 3-component links for [1, 1] [∞, ∞], [0, 0] [∞, ∞], [∞, ∞] [∞, ∞]. Conjecture 1.6. A knot of the form (r1 , r2 )(r3 , r4 ) is non-invertible iff r1 6= r2 , and r3 6= r4 ; it is achiral non-invertible iff r1 = r3 and r2 = r4 , and chiral non-invertible otherwise. For n = 10 there are three non-invertible knots of this form. Two of them are achiral: (3, 2) (3, 2) and (2 1, 2) (2 1, 2), and one is chiral: (3, 2) (2 1, 2). For n = 11 there are six of them: (2 1 1, 2) (2 1, 2) (3, 2 1) (2 1, 2)

(2 1 1, 2) (3, 2) (2 2, 2) (3, 2)

(2 2, 2) (2 1, 2) (3, 2 1) (3, 2)

It is clear that achiral non-invertible knots can be obtained only for (r1 , r2 )(r3 , r4 ) knots with the sets of types [1, 1] [1, 1], [1, ∞] [1, ∞], [0, 0] [0, 0], [0, ∞] [0, ∞]. The first achiral non-invertible representatives of these types are: (3, 2 1 1) (3, 2 1 1), (3, 2) (3, 2), (2 2, 2 1) (2 2, 2 1), (2 1, 2) (2 1, 2), respectively.

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Now we will consider a more general case: non-invertible knots of the form (r1 , r2 , . . . , ri )(r1′ , r2′ , . . . , rj′ ). The parts r1 , r2 , . . . , ri and r1′ , r2′ , . . . , rj′ will be called the pretzel parts of the knot. The term (r1 , r2 , . . . , ri )(r1′ , r2′ , . . . , rj′ ) is a knot iff the types of the pretzel parts are [0] [0], [0] [1], [1] [0], [1] [∞], [∞] [1], [∞] [∞]. This means that every pretzel part may have at most one R-tangle of the type [∞]. The pretzel parts are treated as ordered sequences of R-tangles, and not as cyclic structures, as in the case of pretzel KLs. From the symmetry reasons, it is sufficient to consider only knots of the type [0] [0], [0] [1], [1] [∞], [∞] [∞]. Conjecture 1.7. A knot of the form (r1 , r2 , . . . , ri )(r1′ , r2′ , . . . , rj′ ) of the type [0] [0], [1] [∞], [∞] [1], [∞] [∞] is non-invertible iff none of its pretzel parts is mirror-symmetric; it is chiral non-invertible iff i = j and rk = rk′ (k = 1, . . . , i), and chiral non-invertible otherwise. A knot of the type [0] [1], [1] [0] is chiral non-invertible iff its pretzel part of the type [0] is not mirror-symmetric. From KLs obtained as a product of three pretzel tangles, those of the types [1] [1] [1], [0] [1] [1], [1] [0] [0], [1] [∞] [1], [0] [0] [1], [1] [0] [0], [0] [1] [∞], [1] [0] [∞], [∞] [0] [1], [∞] [1] [0], [1] [∞] [∞], [∞] [1] [∞], [∞] [∞] [1], [0] [0] [∞], [∞] [0] [0], [∞] [∞] [∞], are knots. From the symmetry reasons, it is sufficient to consider knots of the type [1] [1] [1], [0] [1] [1], [1] [∞] [1], [0] [0] [1], [0] [1] [∞], [1] [0] [∞], [1] [∞] [∞], [∞] [1] [∞], [0] [0] [∞], [∞] [∞] [∞]. For example, a knot of the type [1] [1] [1] will be non-invertible iff the pretzel tangle in the middle is not mirror-symmetric, or if it is mirrorsymmetric and the first pretzel tangle is different from the last one and from its reverse. As this example shows, it is still possible to find general conditions for non-invertibility of certain classes and types of knots, but the conditions will be more complicated than before, dependent from the types of knots considered and not only from classes they belong. A KL of the form (r1 , r2 )(2k − 1)(r3 , r4 ), where ri (i = 1, . . . , 4) are Rtangles, and k = 1, 2, . . . is a knot iff its type is [1] [1] [1], [0] [1] [1], [1] [1] [0], [0] [1] [∞], [∞] [1] [0], or [∞] [1] [∞]. From the symmetry reasons, it is sufficient to consider knots of the type [1] [1] [1], [0] [1] [1], [0] [1] [∞], [∞] [1] [∞]. A knot of the type [1] [1] [1], [∞] [1] [∞] is non-invertible iff its pretzel parts (r1 , r2 ) and (r3 , r4 ) are different, and invertible otherwise. A knot of the type [0] [1] [∞] is non-invertible iff r1 6= r2 , and invertible otherwise. A knot of the type [0] [1] [1] is always non-invertible. A KL of the form (r1 , r2 )(2k)(r3 , r4 ), where ri (i = 1, . . . , 4) are R-

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tangles, and k = 1, 2, . . . is a knot iff its type is [0] [0] [1], [1] [0] [0], [1] [0] [∞], [∞] [0] [1], [0] [0] [∞], or [∞] [0] [0]. From the symmetry reasons, it is sufficient to consider knots of the type [0] [0] [1], [1] [0] [∞], [0] [0] [∞]. A knot of this form is non-invertible iff it does not contain a [∞] tangle, and invertible otherwise. Hence, all non-invertible knots are of the type [0] [0] [1]. An arborescent KL of the form (r1 , r2 ), r3 , (r4 , r5 ), where ri (i = 1, . . . , 5) are R-tangles, is a knot iff its (pretzel) type is [0, ∞, 0], [0, ∞, 1], [1, ∞, 1], [0, 1, 0], [1, 1, 1], [∞, 1, 0], [∞, 1, 1], [1, 0, ∞], [0, 0, ∞], [0, 0, 1]. A knot of the type [0, ∞, 0], [0, ∞, 1], [1, ∞, 1] (with ∞ in the middle) is noninvertible iff the tangles (r1 , r2 ), (r4 , r5 ) are different. A knot of the type [0, 1, 0], [1, 1, 1], [∞, 1, 0], [∞, 1, 1] (with 1 in the middle) is non-invertible iff the tangle (r1 , r2 ) is different from (r4 , r5 ) and from its reverse (r5 , r4 ). A knot of the type [1, 0, ∞], [0, 0, ∞], [0, 0, 1] (with 0 in the middle) is always non-invertible. The next class we consider are KLs of the form p1 , p2 , . . . , pi , where pk (k = 3, 4, . . .) are pretzel tangles of the form (r1 , r2 , . . . , rj ) (j = 2, 3, . . .). The result is a knot iff among pk tangles there is an odd number of the tangles of the pretzel type [1] and an arbitrary number of the tangles of the pretzel type [∞], or if there is exactly one tangle of the pretzel type [0]. The simplest case of such knots are those of the form: (r1 , r2 ), (r3 , r4 ), (r5 , r6 ). They are non-invertible iff they do not contain equal pretzel tangles, i.e., iff their t-diagram is not mirror-symmetric. For example, knot (5, 2), (3, 2), (2 1, 2 1) is non-invertible, and (3, 2), (3, 2), (2 1, 2 1) is invertible. Non-invertible knots (3, 4), (3, 2), (2 1, 2 1), (5, 2), (3, 2), (2 1, 2 1) belonging to the same family are obtained by breaking symmetry of their corresponding t-diagrams. In general, a knot of the form p1 , p2 , . . . , pi is non-invertible iff its t-diagram is not mirror-symmetric. For example, knots (2 1, 3), (2 1, 3), (3, 2), (3, 2 1), (2 1, 3), (2 1, 3), (2 1, 3), (3, 2 1), (2 1, 5), (2 1, 3), (2 1, 3), (2 1, 3), (2 1, 5), (4 1, 3) are non-invertible, and (2 1, 3), (3, 2), (2 1, 3), (3, 2 1) is invertible. The proposed method can be used for computing symmetry groups of non-invertible knots, where the results obtained hold for whole classes of non-invertible knots described above. For example, the symmetry group of every non-invertible pretzel knot is Z2 . 1.11.4

Non-invertible polyhedral knots

The basic polyhedron 6∗ with two vertices replaced by R-tangles gives knots of the form 6∗ r1 .r2 or 6∗ r1 .r2 0, where none of the R-tangles is of the type

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[0], and 6∗ r1 .r2 = 6∗ r2 .r1 , 6∗ r1 .r2 0 = 6∗ r2 .r1 0. Knot 6∗ r1 .r2 is achiral noninvertible if r1 = r2 , and chiral non-invertible otherwise. Knot 6∗ r1 .r2 0 is chiral non-invertible iff r1 6= r2 . The next step is to substituting three vertices of the same basic polyhedron by R-tangles. All knots of the form 6∗ r1 .r2 : r3 derived from 6∗ 2.2 : 2 are non-invertible. Knots of the form 6∗ r1 .r2 : r3 derived from 6∗ 2.2 : 2, and knots of the form 6∗ r1 .r2 0 : r3 0 derived from 6∗ 2.2 0 : 2 0 are noninvertible iff r1 6= r3 . Knots of the form 6∗ r1 .r2 .r3 0 derived from 6∗ 2.2.2 0 are non-invertible iff r2 6= r3 . Knots of the following forms: • 6∗ r1 .r2 0.r3 derived from 6∗ 2.2 0.2, • 6∗ r1 .r2 0 :: r3 0 derived from 6∗ 2.2 0 :: 2 0, and • 6∗ r1 .r2 .r3 derived from 6∗ 2.2.2 are non-invertible iff they do not contain equal R-tangles. All non-invertible knots mentioned are chiral with the trivial symmetry group. The results for knots obtained from the basic polyhedron 6∗ with four R-tangles are the following: • knots of the form 6∗ r1 .r2 .r3 .r4 derived from 6∗ 2.2.2.2 are achiral noninvertible iff r1 = r4 and r2 = r3 , and chiral non-invertible otherwise; • knots of the form 6∗ r1 .r2 : r3 .r4 0 derived from 6∗ 2.2 : 2.2 0 are chiral non-invertible iff r2 6= r4 , or the tangle types of (r1 ,r3 ) are not ([0],[∞]) or ([∞],[0]), and invertible otherwise; • knots of the form 6∗ r1 .r2 .r3 0.r4 derived from 6∗ 2.2.2 0.2 are chiral noninvertible iff r2 6= r3 , and invertible otherwise; • knots of the form 6∗ r1 .r2 .r3 .r4 0 derived from 6∗ 2.2.2.2 0 are always chiral non-invertible; • knots of the form 6∗ r1 .r2 : r3 .r4 derived from 6∗ 2.2 : 2.2 are invertible achiral iff r1 = r3 and r2 = r4 , non-invertible achiral iff r1 = r2 and r3 = r4 , invertible iff r1 = r2 or r1 = r4 and r2 = r3 , and chiral non-invertible otherwise; • knots of the form 6∗ r1 .r2 0.r3 .r4 0 derived from 6∗ 2.2 0.2.2 0 are chiral non-invertible iff r2 6= r3 , and invertible otherwise; • knots of the form 6∗ r1 .r2 .r3 0 : r4 derived from 6∗ 2.2.2 0 : 2 are achiral non-invertible iff r1 = r3 and r2 = r4 , and chiral non-invertible otherwise; • knots of the form 6∗ r1 .r2 .r3 : r4 0 derived from 6∗ 2.2.2 : 2 0 are chiral non-invertible iff r1 6= r3 , and invertible otherwise;

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• knots of the form 6∗ r1 .r2 .r3 : r4 0 derived from 6∗ 2.2.2 : 2 0 are chiral non-invertible iff r1 6= r3 , and invertible otherwise; • knots of the form 6∗ r1 .r2 0 : r3 0.r4 derived from 6∗ 2.2 0 : 2 0.2 are invertible iff r1 = r3 and tangle type of (r2 , r4 ) is not ([0],[∞]) or ([∞],[0]), r2 = r4 and tangle type of (r1 , r3 ) is not ([0],[∞]) or ([∞],[0]), and chiral non-invertible otherwise. First chiral non-invertible knots with non-trivial symmetry group are 6∗ r1 .r2 0 : r3 0.r4 with r1 = r3 or r2 = r4 , whose symmetry group is Z2 . Knots of the form 6∗ r1 .r2 : r3 0.r4 0 derived from 6∗ 2.2 : 2 0.2 0 are invertible iff r1 = r3 and tangle type of (r2 , r4 ) is not ([0],[∞]) or ([∞],[0]), r2 = r4 and tangle type of (r1 , r3 ) is not ([0],[∞]) or ([∞],[0]); they are non-invertible achiral iff r1 = r2 and r3 = r4 , and chiral otherwise. Among the chiral knots of this form, knots with r1 = r3 or r2 = r4 have the symmetry group Z2 , and a trivial symmetry group otherwise. Knots of the form 6∗ r1 .r2 .r3 : r4 derived from 6∗ 2.2.2 : 2 are chiral non-invertible iff r1 6= r3 , and invertible otherwise. There is an interesting connection between knots derived from 6∗ 2.2 : 2 0 and 6∗ 2.2 0 : 2 0. For every number of crossings n, source links 6∗ 2.2 : 2 0 and 6∗ 2.2 0 : 2 0 generate the same number of knots, and derived knots have the same symmetries and invertibility properties. The same holds for the following pairs of source links: 6∗ 2.2 0 :: 2 0 and 6∗ 2.2.2, 6∗ 2.2.2.2 and 6∗ 2.2.2 0 : 2, 6∗ 2.2 0.2.2 0 and 6∗ 2.2.2 : 2 0, 6∗ 2.2.2 0.2 and 6∗ 2.2.2 : 2 The similar connection exists between 6∗ 2.2 0 : 2 0.2 and 6∗ 2.2 : 2 0.2 0 if we consider only the number of knots derived, the number of chiral noninvertible knots among them, and their distribution according to the symmetry groups (Z2 or trivial). In the same way, it is possible to find non-invertibility criteria for all knots derived from different basic polyhedra by substituting vertices with R-tangles. The next class of non-invertible polyhedral knots will be obtained by substituting vertices of a basic polyhedron with pretzel (stellar) tangles. For example, all knots of the form 6∗ p1 .r1 , where p1 is the pretzel tangle of the pretzel type [∞], and r1 is R-tangle of the type [0], are chiral non-invertible. We already mentioned that a pretzel tangle p1 will be of the type [∞] if it consists of an arbitrary number of R-tangles of the types [1], [∞], and from exactly one R-tangle of the type [0]. First examples are 11-crossing chiral non-invertible knots 6∗ (3, 2).2, 6∗ (2, 3).2, 6∗ (2 1, 2).2, and 6∗ (2, 2 1).2. We conclude that all knots derived from the basic polyhedron 6∗ with one

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pretzel tangle p1 and one R-tangle r1 will be chiral non-invertible. The first class of polyhedral knots with pretzel tangles that requires symmetry discussion are knots of the form 6∗ p1 .p2 , where p1 , p2 are pretzel tangles of the types ([∞], [∞]), or ([0],[0]), respectively. If p1 , p2 are pretzel tangles of different types, [0] and [∞], the corresponding knots are chiral non-invertible. Knots of the form 6∗ p1 .p2 where p1 , p2 are pretzel tangles of the types ([∞], [∞]) are achiral non-invertible iff p1 , p2 are mutually palindromic pretzel tangles, and chiral non-invertible otherwise. For example, the knot 6∗ (3, 2).(2, 3) is achiral non-invertible, and 6∗ (3, 2).(3, 2) is chiral non-invertible. The same criterion holds for the knots of the same form 6∗ p1 .p2 , where p1 , p2 are pretzel tangles of the types ([0],[0]). In the same way we can conclude that all knots of the form 6∗ p1 .r1 0, where p1 is a pretzel tangle, and r1 is R-tangle, are chiral non-invertible. Knots of the form 6∗ p1 .p2 0, where p1 , p2 are pretzel tangles of different types [0], [∞], are chiral non-invertible. If the pretzel tangles p1 , p2 are of the same type ([0],[0]), or ([∞], [∞]), knots of the form 6∗ p1 .p2 0 are chiral non-invertible iff p1 6= p2 . For example, knot 6∗ (3, 2).(3, 2) 0 is invertible, and 6∗ (5, 2).(3, 2) 0 is chiral non-invertible. In general, for every basic polyhedron infinite classes of non-invertible knots derived from it can be recognized. Replacement of the tangle of particular reduced type ([0k ], [1k ], or [∞k ]) by a tangle of the same type preserves the number of components. Hence we have the following main conjecture: Conjecture 1.8. In every non-invertible knot K, substituting a tangle by a tangle of the same type, respecting symmetry conditions, gives a noninvertible knot of the same chirality. In other words: non-invertibility is a type-dependent and symmetrydependent property.

1.12

Reduction of R-tangles

In standard Conway symbols of KLs (Conway, 1970; Rolfsen, 1966) there are no rational parts of a KLs with mixed signs, since Conway symbols represent minimal KL diagrams. For rational tangles and rational KLs holds the Theorem 1.18: two rational tangles are equivalent iff their continued fractions yield the same rational number. For every rational KL with mixed signs it is possible to

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calculate its corresponding continued fraction and express it as a rational KL with all entries of the same sign. For example, 4 − 3 − 1 1 2 2 − 2 reduces to 3 1 5 1 2, so all rational KLs can be expressed by their rational reductions. After rational reductions applied to some non-rational KL, where every tangle with mixed signs is replaced by its rational reduction, in general, a KL different from the original KL will be obtained. For example, the knots 8∗ 4 − 3 − 1 1 2 2 − 2 and 8∗ 3 1 5 1 2 are different. Every R-tangle with mixed signs has its reduced equivalent, but in general this equivalent is not a tangle obtained by rational KL reduction based on continued fractions. The reason is that in the case of rational KLs we are working with numerator closures of rational tangles, and here we reduce open tangles. For example, as the open R-tangle 3 − 2 − 2 1 reduces to 2 1 1 1 1 0, that gives as the numerator closure 2 1 2. For R-tangles with n = 3 crossings there is one tangle with mixed signs −2 1 and its equivalent tangle 2 0, where 0 has the standard meaning according to Conway notation of KLs. In the following tables are given ordered pairs of R-tangles with mixed signs and their equivalents for n ≤ 6: (−3 1, 2 1 0) (−2 1 − 1, 1)

(−2 2, 2 1) (−2 − 1 1, 3 0)

(−4 1, 3 1 0) (−3 − 1 1, 4 0) (−2 − 2 1, 2 1 1 0) (2 − 1 1 1, 0) (−2 − 1 1 1, 4) (−5 1, 4 1 0) (−4 1 − 1, 3 0) (3 − 2 1, 2 2 0) (3 − 1 2, 2 0) (3 − 1 − 1 1, 2 1 1 0) (−3 1 − 1 − 1, 1) (−3 − 1 − 1 1, 3 2 0) (−2 3 1, 2 2 1) (−2 2 2, 2 1 2) (2 − 2 1 − 1, 2) (−2 − 2 1 1, 2 1 2) (2 − 1 3, 1) (2 1 − 2 1, 4 0) (2 − 1 − 2 1, 3 1 0) (−2 − 1 2 1, 3 1 1)

(−2 1 1, 3)

(−3 2, 2 1 1) (−2 3, 2 2) (2 − 1 2, 0) (2 − 1 − 1 1, 2 1 0) (−2 − 1 1 − 1, 2)

(−3 1 1, 2 2) (2 − 2 1, 3 0) (−2 1 2, 4) (−2 1 1 1, 3 1) (−2 − 1 − 1 1, 2 2 0)

(−4 2, 3 1 1) (−4 − 1 1, 5 0) (−3 2 1, 2 1 1 1) (−3 1 2, 2 3) (−3 1 1 1, 2 2 1) (−3 − 1 1 1, 5) (−2 4, 2 3) (−2 − 3 1, 2 2 1 0) (−2 − 2 2, 2 1 1 1) (2 − 2 − 1 1, 2 2 0) (−2 − 2 1 − 1, 2 1 0) (−2 1 3, 5) (2 − 1 2 1, 0) (−2 1 2 1, 4 1) (−2 − 1 − 2 1, 2 1 1 1 0)

(−3 1 − 1, 2 0) (−2 2 1, 2 1 1) (−2 − 12, 3 1) (−2 1 − 1 1, 2)

(−4 1 1, 3 2) (−3 3, 2 1 2) (−3 − 2 1, 3 1 1 0) (−3 − 1 2, 4 1) (−3 1 − 1 1, 3) (−3 − 1 1 − 1, 3) (2 − 3 1, 2 1 1 0) (2 − 2 2, 3 1) (2 − 2 1 1, 4) (−2 2 1 1, 2 1 1 1) (−2 − 2 − 1 1, 2 3 0) (−2 − 1 3, 3 2) (2 − 1 2 − 1, 0) (−2 1 − 2 1, 0) (2 − 1 1 2, 1)

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(2 − 1 − 1 2, 2 1 1) (−2 − 1 1 2, 5) (2 1 − 1 1 1, 2 0) (2 − 1 1 1 − 1, 0) (2 − 1 − 1 1 − 1, 2 0) (−2 1 − 1 1 1, 2 1) (−2 − 1 − 1 1 1, 2 3)

(−2 1 1 2, 3 2) (−2 − 1 1 − 2, 1) (2 1 − 1 − 1 1, 3 1 0) (2 − 1 1 − 1 1, 2 0) (2 − 1 − 1 − 1 1, 4 0) (−2 − 1 1 1 1, 4 1) (−2 − 1 − 1 1 − 1, 2 1)

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(−2 1 − 1 2, 3) (−2 − 1 − 1 2, 2 2 1) (2 − 1 1 1 1, 0) (2 − 1 − 1 1 1, 2 2) (−2 1 1 1 1, 3 1 1) (−2 − 1 1 − 1 1, 2 1) (−2 − 1 − 1 − 1 1, 2 1 2 0)

The general reduction procedure for R-tangles, analogous to the rational KL reduction is the following: take the R tangle, reverse it, express it as continued fraction, and compute its corresponding rational tangle. In this way, it is very easy to reduce any R-tangle with mixed signs. Hence, there are two possible procedures: (1) the first is the KL reduction, meaning that every R-tangle (with mixed signs) can be replaced by its reduced equivalent; (2) the other is KL extension: each reduced tangle can be replaced by its unreduced equivalent. For example, because the reduced equivalent of the tangle −3 1 − 1 − 1 is 1, −2 1 − 2 − 2 1, 2 − 1 1 1 2 1, 2 1 − 1 4 is the stellar (pretzel) representation of trefoil knot, and 2 − 1 2 2 − 1, 2 − 1 1 1 2 1, 2 1 − 1 4 is the stellar unknot. Every KL has an infinite number of non-minimal representations, but is it possible to find a k-crossing representation of a given KL with n crossings (k > n) for a given k? For example, it is clear that a trefoil can not be represented as a 4-crossing knot, so its minimal non-alternating representation has five crossings. There are additional questions of this type: for which values of k we can represent a trefoil as a k-crossing knot? Some answers to this and similar questions provides R-tangle equivalence. Working in Conway notation, it is possible to obtain (sub)families of R-tangles with mixed signs and their reduced equivalents. For example, the R-tangle 3 − 1 4 − 1 2 p reduces to p + 3. Hence, instead with particular KLs, we can work again with KL families. Conjecture 1.9. All link diagrams belonging to a subfamily of a minimal non-alternating KL diagram are minimal.

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If this conjecture is true, the minimality problem, i.e., crossing number problem can be reduced to computation of crossing numbers for generating KL diagrams, where every addition of two bigons will change crossing number by 2. 1.12.1

KLs with unlinking number one

Unlinking number is a kind of problem which can be described to a child, but even the forefront techniques from different areas of mathematics can not give an answer. In this section we will address only unknotting number 1 knots and links, new conjectures and results obtained with the help of computers. Only rational unlinking number 1 knots and links have been completely described: they have unlinking number 1 minimal diagrams (Theorem 1.24), and their general form is given in the Theorem 1.28, and Theorem 1.29, respectively. C.McA. Gordon and J. Luecke (2006) have recently obtained important results for unknotting number 1 knots Definition 1.69. An incompressible or essential sphere is a 2-sphere in a 3-manifold that does not bound a 3-ball. An algebraic knot that has an essential Conway sphere is called large algebraic knot. Theorem 1.41. Let K be a large alternating algebraic knot with unknotting number 1. Then K can be unknotted by a crossing change in any alternating diagram of K (Gordon and Luecke, 2006, Theorem 12.5). Theorem 1.42. Having unknotting number 1 is invariant under mutation (Gordon and Luecke, 2006, Theorem 12.5). M. Eudave-Mu˜ noz (1997), C.McA. Gordon and J. Luecke (2006), and P. Ozsvath and Z. Szabo (2005), using Heegaard Floer homology, completed tables of unknotting number 1 knots with n ≤ 11 crossings. In this section we describe infinite classes of pretzel and arborescent KLs with unlinking number 1, and conjecture that the only algebraic KLs with unlinking number 1 are rational KLs, or members of the classes described in Theorems 1.45-1.51. t−1 -tangles play the main role in the derivation of KLs with unlinking number one: Definition 1.70. R-tangle which can be transformed into a tangle re-

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ducible to −1 by one crossing change is called t−1 -tangle. For example, by a single crossing change the tangle 2 1 1, gives 2 − 1 1 = −1, 3 1 1 1 gives 3 − 1 1 1 = −1, 2 1 1 2 gives 2 1 − 1 2 = −1, etc. The following table contains all t−1 -tangles with n ≤ 12 crossings: n=4 n=6 n=8 n = 10 n = 12

211 3111 4 112 5 113 23 1121 6 114 33 1122 24 1131 213 11211

2112 31 13 41 14 221 131 51 15 321 132 231 141 2121 1311

22 32 212 42 312 222 2112

1111 1112 11111 1113 11112 11121 111111

211 311 2111 411 3111 2211 21111

121 122 1211 123 1212 1221 12111

where bold 1’s represent the crossings that need to be changed in order to obtain tangle that reduces to −1. n−4 The number of t−1 -tangles with n crossings is 2 2 for even n (n ≥ 4), and 0 otherwise. LinKnot function fTanUn1 gives the list of t−1 -tangles with n crossings (n ≥ 4). The number of R-tangles l (l ≥ 3) in a pretzel KL is called length of a pretzel KL. Every alternating pretzel (Montesinos) KL of the length 3 remains the same after any permutation of R tangles, i.e., r1 , r2 , r3 + k = r1 , r3 , r2 +k = ... = r3 , r2 , r1 +k. The same holds for non-alternating pretzel KLs of the length 3. Theorem 1.43. There are no pretzel (Montesinos) KLs of length l > 3 with unlinking number 1. K. Motegi (1996) gave a proof of this theorem for knots, and I. Torisu (1998) for links. We propose the following conjecture: Conjecture 1.10. For every pretzel (Montesinos) knot or link L of length l the following inequality holds: u(L) ≥ l − 2. For pretzel KLs given by their standard diagrams, denoted by M (k; (p1 , q1 ), ..., (pl , ql )) (pi ≥ 0, GCD(pi , qi ) = 1, i = 1, ..., l), where (pi , qi ) denotes R-tangle ri of slope pqii , and k is the number of pluses (see, e.g., Burde and and Zieschang, 1985), I. Torisu (1996) proved the following theorem:

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Theorem 1.44. Let K be a Montesinos knot (link L) with l ≥ 3. Then u(K) = 1 (u(L) = 1) and the unknotting (unlinking) operation is realized in a standard diagram iff the following condition holds: • K = M (0; (p, −r), (q, s), (2mn ± 1, 2n2 )), where p, q, r, s, m, n, are some non zero integers, m and n are coprime, and ps − rq = 1; • L = M (0; (p, −r), (p, q), (2mn ± 1, 2n2 )), where p, q, m, n, are some non zero integers, and m and n are coprime. I. Torisu conjectured that all Montesinos KLs with l ≥ 3 and with unlinking number 1 are described by this theorem. A reverted R-tangle r given in Conway notation we denote by r (e.g., if r = 3 2 1, r = 1 2 3). Theorem 1.45. All alternating pretzel KLs of the form r1 , r2 , r3 + k, with the following properties: • r1 − 1 r2 is a rational unlink; • r3 k is a t−1 -tangle, have unlinking number 1. Proof. If r3 k is a t−1 -tangle, by one crossing change in r3 the pretzel link r1 , r2 , r3 + k reduces to the rational unlink r1 , r2 , −1 = r1 − 1 r2 . For example, the unlinking number of the pretzel link 2 1 1, 2, 2 is one, because 2 1 1 is a t−1 tangle, and 2 − 1 2 is a rational unlink, so it holds the relation: 2 − 1 1, 2, 2 = −1, 2, 2 = 2 − 1 2 = 1. In the same way, 2 1, 2, 3 1 1 1 is a pretzel knot with unknotting number one, since 3 1 1 1 is a t−1 -tangle, 2 1 −1 2 is a rational unknot, and 2 1, 2, 3 −1 1 1 = 2 1, 2, −1 = 2 1 −1 2 = 1. For pretzel KLs with pluses, we have the following example: 3 1 1, 2, 2 + 3 is a 2-component link with unlinking number one, since 3 1 1 3 is a t−1 -tangle (3 − 1 1 3 = −1), 2 − 1 2 = 1 is unlink, and 3 − 1 1, 2, 2 + 3 = −1, 2, 2 = 2 − 1 2 = 1. The LinKnot functions fStUnNo1 and fStPlusNo1 compute the number of pretzel unlinking number 1 KLs with n crossings without and with pluses. For 8 ≤ n ≤ 20 crossings there are 1, 2, 5, 10, 18, 33, 54, 97, 146, 258, 400, 707, and 1058 KLs with unlinking number 1, respectively, for each class. By computing all pretzel KLs and selecting those with unlinking number 1, we checked that the derivation according to Theorem 1.45 is exhaustive for n ≤ 16.

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Theorem 1.46. All alternating KLs of the form (r1 , r2 + k) r3 (r4 , r5 + l), with the following properties: • r4 − 1 r5 is a rational unlink; • the tangles r2 k and r1 1 r3 l are t−1 -tangles, and all KLs obtained from them by mutation have unlinking number 1. Proof. If r2 k is a t−1 -tangle, after one crossing change in r2 the arborescent link (r1 , r2 + k) r3 (r4 , r5 + l) reduces to (r1 , −1) r3 (r4 , r5 + l) = r1 − 1 r3 l, r4 , r5 . Since r1 1 r3 l is a t−1 -tangle, further reduction gives r1 − 1 r3 l, r4 , r5 = −1, r4 , r5 = r4 − 1 r5 = 1. For example, (4 1 1 1 2, 2 2 2 2 2 11 1 2 2 2 + 1) 1 1 1 1 1 (2 1 3 2, 4 1 1 + 3) is an arborescent alternating knot with unknotting number 1, because 2 1 3 2 1 1 1 4 is a rational knot with unknotting number 1 (2 1 3 2 − 1 1 1 4 = 1), and 2 2 2 2 2 1 1 1 2 2 2 1 (2 2 2 2 2 − 1 1 1 2 2 2 1 = −1), and 4 1 1 1 2 1 1 1 1 1 1 3 (4 1 1 1 2 − 1 1 1 1 1 1 3 = −1) are t−1 -tangles. In the same way, 2-component link (3 1 1 2 1, 4 2 1 1 12 2 + 3) 3 1 1 (5 2, 5 1 1 + 2) has the unlinking number 1, where the crossing that need to be changed in order to unlink it is denoted by bold number 1. Conjecture 1.11. All algebraic alternating KLs with unlinking number 1 are: • rational KLs described in Theorems 1.28 and 1.29, • pretzel KLs described in Theorem 1.45, • arborescent KLs described in Theorem 1.46. According to the last conjecture, every alternating algebraic KL with unlinking number 1 is a knot or a 2-component link. This extremely restrictive conjecture is based on extensive computations for different classes of arborescent KLs, made according to the BernhardJablan Conjecture (page 83) and no counterexample has been found so far. Another argument supporting Conjecture 1.11 is based on the structure of the arborescent world, containing all other worlds (rational, stellar,...) as

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subworlds. There is natural hierarchy where KLs from a “higher” worlds are obtained by composing tangles from the lower ones. A nice way to see the structure of arborescent world is by looking at graphs of KLs (Caudron, 1982). For example, Montesinos KLs are obtained by substituting bigons in pretzel source links (2, 2, . . .) by R-tangles, arborescent source links of the form (2, 2, . . .) (2, 2, . . .) are obtained by substituting bigons in the source link 2 2 by pretzel source links (2, 2, . . .), etc. Hence, the process of unlinking of a pretzel or arborescent link L with unlinking number 1 always ends with a rational unlink r1 − 1 r2 = r1 , r2 , −1, where its parts r1 and r2 are nested in L in the form r1 , r2 . A link of the form t1 , t2 , ..., tn , where ti (i = 1, 2, ..., n) are arbitrary algebraic tangles is called generalized pretzel link of the length n. In the case of pretzel (Montesinos) KLs with unlinking number 1 the maximum length is 3 (Theorem 1.43). For generalized pretzel parts of arborescent source links which generate KLs with unlinking number 1, we expect that maximum length is 2. This implies that all arborescent links with unlinking number 1 are of the form (r1 , r2 + k) (r3 , r4 + l), or (r1 , r2 + k) r3 (r4 , r5 + l). The first is a special case of the second form, that is the subject of Theorem 1.46. Next we consider non-alternating algebraic KLs with unlinking number 1 of the form: r1 , r2 , r3 −, (r1 , r2 + k) (r3 , r4 −), (r1 , r2 + k) r3 (r4 , r5 −), (r1 , r2 + k) − (r3 , r4 ), and (r1 , r2 + k) r3 − (r4 , r5 ). Theorem 1.47. All non-alternating pretzel KLs of the form: (1) r1 , r2 , 2 k− (k ≥ 1), or (2) r1 , r2 , r3 k 1 l− (k ≥ 1, l ≥ 1), or (3) r1 , r2 , r3 (k + 1) l− where r1 − 1 r2 is a rational unlink, and r3 k is a t−1 -tangle, have the unlinking number 1. Proof. (1) After a crossing change in tangle 2, KL of the form r1 , r2 , 2 k− gives: r1 , r2 , (1, −1) k− = r1 , r2 , 0 k− = r1 , r2 , −1 = r1 − 1 r2 = 1; (2) after a crossing change in t−1 -tangle r3 k which results in −1, KL of the form r1 , r2 , r3 k 1 l− reduces to r1 , r2 , −11 l− = r1 , r2 , 0 l− = r1 , r2 , −1 = r1 − 1 r2 = 1; (3) after a crossing change in t−1 -tangle r3 k which results in −1, KL of the form r1 , r2 , r3 (k + 1) l− reduces to r1 , r2 , −1 1 l− = r1 , r2 , 0 l− =

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r1 , r2 , −1 = r1 − 1 r2 = 1.

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For example, (1) every knot of the form 4 1 2, 2 1, 2 k −1 (k ≥ 1, r1 −1 r2 = 4 1 2 −1 1 2 = 1) has unknotting number 1 (case 1); (2) every knot of the form 2 1 1 2, 2 1 1, 7 2 1 1 1 6 1 l − 1 (case 2) or 2 1 1 2, 2 1 1, 7 2 1 1 1 7 l − 1 (case 3), (k = 6, r1 − 1 r2 = 2 1 1 2 − 1 1 1 2 = 1, and r3 k = 7 2 − 1 1 1 6 = −1) has unknotting number 1. Proofs of the following three theorems are omitted, since they are similar to the last one. Theorem 1.48. All non-alternating arborescent KLs of the form: (1) (r1 , r2 + k) (2 l, k 1−) (k ≥ 1, l ≥ 1), or (2) (r1 , r2 + k) (r3 m 1 l, k 1−) (k ≥ 1, l ≥ 1, m ≥ 1), or (3) (r1 , r2 + k) (r3 (m + 1) l, k 1−) (k ≥ 1, l ≥ 1, m ≥ 1), and their mutants, such that r1 − 1 r2 is a rational unlink, and r3 m is a t−1 -tangle, have unlinking number 1. Theorem 1.49. All non-alternating arborescent KLs of the form: (r1 (m + 1), r2 + k) 1 m r1 l (r3 , r4 −) (k ≥ 1, l ≥ 1) and their mutants, satisfying the following properties: • r2 k (k ≥ 1) is a t−1 -tangle, • r3 − 1 r4 is unlink, have unlinking number 1. Theorem 1.50. All non-alternating arborescent KLs of the form: (1) (r1 , r2 + k) − (2 l, k + 1) (k ≥ 1, l ≥ 1), or (2) (r1 , r2 + k) − (r3 m 1 l, k + 1) (k ≥ 1, l ≥ 1, m ≥ 1), or (3) (r1 , r2 + k) − (r3 (m + 1) l, k + 1) (k ≥ 1, l ≥ 1, m ≥ 1), and their mutants, such that r1 − 1 r2 is a rational unlink, and r3 m is a t−1 -tangle, have unlinking number 1. Definition 1.71. Let r1 , r2 be R-tangles. A tangle of the form −r1 0 r2 is called t1 -tangle if reduces to the tangle 1.

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For example, the tangles −2 − 1 − 2 0 2 1 3, −3 − 2 0 2 2 2, −3 − 2 0 2 3 1 with n = 11 crossings are t1 -tangles. Theorem 1.51. All non-alternating arborescent KLs of the form: (1) (r1 , r2 + k) r3 − ((k + 1) r3 , 2 l) (k ≥ 1, l ≥ 1), where r1 − 1 r2 is a rational unlink, and r2 k is a t−1 -tangle, or (2) (l 1, r2 + k) r3 − (r4 , r5 ) (k ≥ 1, l ≥ 1), where r4 − 1 r5 is a rational unlink, r2 k is a t−1 -tangle, and −(l + 1) 0 r3 is a t1 -tangle, or (3) (r1 l 1, r2 + k) r3 − (r4 , r5 ) (k ≥ 1, l ≥ 1), where r4 − 1 r5 is a rational unlink, r2 k is a t−1 -tangle, and −r1 − (l + 1) 0 r3 is a t1 -tangle and all KLs obtained from them by mutation, have unlinking number 1. The following knots with unknotting number 1 illustrate preceding four theorems (2 1 1, 2+) (2 1 1 1 1, 2−) (5 3 11 4, 2 + 4) 2 2 (2 2 1, 4−) (2 1 1, 2+) − (3 1 1 2 5, 2) (4 1 1, 4 3 5 1 + 2) 6 3 4 9 1 − (2 2 1 1, 3 2) Conjecture 1.12. All algebraic non-alternating KLs with unlinking number 1 are those constructed according to Theorems 1.47-1.51. Non-alternating KLs from the preceding four theorems can be used for the derivation of alternating KLs with unlinking number 2. Theorem 1.52. Let r1 , r2 , r3 − = r1 , r2 , r3 , −1 be a non-alternating pretzel knot or link from Theorem 1.47. If r4 is a t−1 -tangle, every KL of the form r1 , r2 , r3 , r4 obtained by substituting −1 with r4 , and all its mutants have unlinking number 2. Analogous substitution of −1 by a t−1 -tangle in pretzel tangles of the form r1 , r2 − = r1 , r2 , −1 also gives KLs with unlinking number 2. The LinKnot function fMakePretNaltNo1 computes the number of non-alternating pretzel unlinking number 1 KLs with n crossings. For 7 ≤ n ≤ 20 crossings we obtained 1, 3, 5, 14, 26, 49, 86, 152, 259, 426, 702, 1121, 1790, and 2852 KLs with unlinking number 1, respectively.

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Alternating polyhedral KLs with unlinking number 1 are harder to derive, because there is no classification of polyhedral KLs. The only possibility to make some order in the recognition and derivation of alternating polyhedral KLs with unlinking number 1 is their exhaustive detection for each particular basic polyhedron and source links derived from it. For example, the basic polyhedron 8∗ gives, respectively, 2, 12, 34, 87, 124, 136, 72, and 30 source links with n = 1, ..., 8 bigons. For n = 1 there are two source links 8∗ 2 0 and 8∗ 2. The family corresponding to 8∗ 2 0 contains single unknotting number 1 knots: 8∗ 2 0 with n = 9 crossings, 8∗ 3 0 with n = 10 crossings, 8∗ 2 2 0 with n = 11 crossings, 8∗ 2 1 2 0, and 8∗ 3 2 0 with n = 12 crossings. For n ≥ 13 we obtain two families of knots with unknotting number one: 8∗ (n − 11) 1 1 2 0 and 8∗ (n − 11) 2 2 0. It is interesting that all knots from these families can be unknotted by the same crossing change: 8∗ (n − 11) 1 1 2 0 : −1 and 8∗ (n − 11) 2 2 0 : −1 are unknots. The other unknotting 8∗ 2 0 ::: −1 is equivalent to the first from symmetry reasons. For the other source link 8∗ 2 and its family, we have single unknotting number 1 knots: 8∗ 2 1 with n = 10 crossings, 8∗ 2 1 1 with n = 11 crossings, 8∗ 2 1 1 1 with n = 12 crossings, and 8∗ 2 1 1 1 1, 8∗ 3 1 1 1 with n = 13 crossings. For n ≥ 14 we obtain two families of knots with unknotting number one: 8∗ (n − 12) 1 1 1 1 1 and 8∗ (n − 12) 2 1 1 1. The both can be unknotted by crossing changes in the same vertex: 8∗ (n − 12) 1 1 1 1 1. − 1, and 8∗ (n − 12) 2 1 1 1. − 1. Consider a polyhedral alternating link L in Conway notation, which can be unlinked by a crossing change in a vertex V that contains a single tangle 1. Vertex V is called V −1 -vertex, and L− is called V −1 -unlink. Substitution of a V −1 vertex by a t−1 -tangle t or t 0 is called t−1 -substitution. Theorem 1.53. Let L be an alternating polyhedral V −1 -unlink L. All KLs obtained by from L− by t−1 -substitution are KLs with unlinking number one. For example, 8∗ (n−11) 1 1 2 0 : t, 8∗ (n−11) 1 1 2 0 : t 0, 8∗ (n−11) 2 2 0 : t, and 8∗ (n − 11) 2 2 0 : t 0 are knots unknotting number 1 knots for an arbitrary t−1 -tangle t. The smallest polyhedral alternating unknotting number 1 knot is .2.2 with n = 8 crossings. By one V −1 crossing change it gives the unknot .2.2 : . − 1 = 1. After substituting −1 in .2.2 : . − 1 with any t−1 -tangle, we obtain two alternating unknotting number 1 knots. For example, .2.2 :

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.2 1 1 and .2.2 : .2 1 10 are the smallest polyhedral alternating unknotting number 1 knots with n = 11 crossings that can be derived from .2.2, and here are the unknotting sequences: .2.2 : .2 − 1 1 = .2.2 : . − 1 = 1 and .2.2 : .2 − 1 1 0 = .2.2 : . − 1 = 1. Knots .2.2 : .2 1 1 and .2.2 : .2 1 10 are obtained by inserting t−1 -tangle into two different positions. However, due to symmetry of a generating KL, different substitutions can give the same KL. Since 8∗ 2 0 : 2 1 1 = 8∗ 2 0 ::: 2 1 1, and 8∗ 2 0 : 2 1 1 0 = 8∗ 2 0 ::: 2 1 1 0, substituting t−1 tangle t = 2 1 1 into −1-vertices of unlinks 8∗ 2 0 : −1 and 8∗ 2 0 ::: −1 (in two different positions each), yields four knots, but only two distinct ones. Continuing derivation from .2.2 we obtain: • .2.2 : .3 1 1 1, .2.2 : .3 1 1 1 0, .2.2 : .2 1 1 2, .2.2 : .2 1 1 2 0 with n = 13 crossings, • .2.2 : .4 1 1 2, .2.2 : .4 1 1 2 0, .2.2 : .3 1 1 3, .2.2 : .3 1 1 3 0, .2.2 : .2 2 1 1 1 1, .2.2 : .2 2 1 1 1 1 0, .2.2 : .2 1 1 1 2 1, and .2.2 : .2 1 1 1 2 1 0 with n = 15 crossings, etc. Conjecture 1.13. Every polyhedral KL with unlinking number one is: • V −1 -unlink; • a link derived from some V −1 -unlink by a t−1 -tangle substitution. Usually, detection and derivation of KLs with unlinking number 1 from source links with more then one bigon is extremely difficult. For example, from the following table of unknotting number 1 knots derived from the source link 8∗ 2 : .2 0 with n = 10 crossings n = 11 n = 12 n = 13

8∗ 3 : .2 0 8∗ 2 1 : .2 1 0 8∗ 2 : .3 2 0

8∗ 2 : .2 2 0 8∗ 2 1 : .2 2 0

8∗ 3 1 : .2 0 8∗ 2 2 1 : .2 0

8∗ 4 : .2 0 8∗ 5 : .2 0

it is hard to recognize any pattern. In order to find a general criterion for KLs of the form 8∗ r1 : .r2 0 with unlinking number 1 we need to discuss many different cases. However, for the class 8∗ (r1 , r2 ) : .2 2 0 the pattern is easily recognizable: members of this class are KLs with unlinking number 1 if r1 − 1 r2 is a rational unlink, and all of them are V −1 -unlinks which unlink in the same vertex V , namely 8∗ (r1 , r2 ) : .2 2 0 : −1. In the same way, all non-alternating KLs of the form 8∗ (r1 , r2 ) : .3 0 have unlinking number 1 if r1 − 1 r2 is a rational unlink, etc.

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We can even obtain some multi-parameter classes of polyhedral KLs with unlinking number 1. For example, all alternating KLs of the form 6∗ (r1 , r2 + k).l : .l − 1.k − 1 (k ≥ 2, l ≥ 1) have unlinking number 1 if r1 − 1 r2 is a rational unlink, and the same holds for non-alternating KLs of the form 6∗ − (r1 , r2 ).k 1, or 6∗ − (r1 , r2 ).k 1 0 (k ≥ 1). Hence, for recognizing patterns in polyhedral KLs with unlinking number 1, we need a more profound understanding of the polyhedral world. In the same way as for arborescent KLs, we expect that the maximum length of any pretzel part of a polyhedral KL with unlinking number one is 2. We finish this part with the following theorem: Theorem 1.54. For any number of components c there exist c-component (non-alternating) link with unlinking number 1. Proof. We give a simple constructive proof: for c = 1, 2 there is an infinite number of KLs with unlinking number 1 (e.g., rational ones). For c = 3 the non-alternating link . − (2, 2) (9321 ) has the unlinking number one. From it, by recursively adding in each step one component (a circlecomponent concentric with a preceding one), we obtain a series of nonalternating links with unlinking number 1. All of them are V −1 links and can be unlinked by a single crossing change in the same vertex. Corollary 1.4. For any c there is an infinite number of non-alternating c-component links with unlinking number 1. An infinite collection of such links can be obtained from already constructed V −1 -unlinks from Theorem 1.54 by t−1 -substitutions. For alternating KLs we propose the following conjecture: Conjecture 1.14. Every alternating link L with c components has unlinking number u(L) ≥ c − 1 (c ≥ 2).

1.13

Braids

K.F. Gauss was the first to notice that braids can be used to describe knotting phenomena18 . J.W.H. Alexander in 1923 discovered a remarkable connection between KLs and closed braids. 18 A drawing of a braid from Gauss notebooks, with strand permutation at each height, dates between 1814 and 1830 (Przytycki, 2004).

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A braid is, as a mathematical object, a formal description of what is meant by braid in everyday language– several strings tangled in a certain way. Definition 1.72. An n-strand braid consists of n disjoint arcs running vertically in ℜ3 space, where the starting points lie on the same horizontal line (Fig. 1.88). The set of starting points for the arcs must lie exactly above the set of end-points.

Fig. 1.88 (a) Braid giving as the closure trefoil knot 3; (b) braid giving as the closure Hopf link 2.

In a similar way as with KLs, the strands of a braid can be rearranged (without detaching the top and the bottom, and of course without tearing or reattaching them) to get a braid that looks different, but is isotopic (or equivalent) to the original braid. Definition 1.73. Two braids are isotopic iff there is a smooth deformation with fixed points from the first one to the second one. As in the case of KLs, the piecewise-linear category and smooth category give the same result, but further in the sequel we will use piecewiselinear drawings. As with KLs, we do not distinguish between isotopic braids, thinking of them as representations of the same object. Definition 1.74. A braid is an equivalence class of braids with regard to braid isotopy. Braid theory was introduced by E. Artin in 1920s. This theory connects different fields of mathematics: topology, geometry, algebra (group theory), and algorithmic methods. We obtain the closure of a braid by joining the upper ends of the strands by arcs to the lower ends (Fig. 1.88). J.W.H. Alexander proved the theorem:

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Theorem 1.55. Every KL can be represented as a closed braid (Alexander, 1923). Definition 1.75. A product of two braids consists of placing two n-strand braids end to end, and joining the upper part of the second braid to the lower part of the first (Fig. 1.89a).

Fig. 1.89

(a) Product of two braids; (b) braid b1 and its inverse b−1 1 .

The product b1 × b2 of any two n-strand braids b1 and b2 is a new n-strand braid (closedness), any three n-strand braids satisfy the relation (b1 × b2 ) × b3 = b1 × (b2 × b3 ) (associativity), for every n there is an n-

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strand unit braid e such that for any braid b with the same number of strands eb = be = b (the existence of a neutral element), and for each nstrand braid b there exists an inverse n-strand braid b−1 whose product with b gives the trivial (unit) n-strand braid e. Let us notice that the braid diagrams of a braid b and its inverse braid b−1 are mirror-symmetric in a mirror line containing the joint end of their product b × b−1 (Fig. 1.89b). Hence, the following theorem holds: Theorem 1.56. All n-strand braids make a group called a braid group and denoted by (Bn ,×). The group (Bn , ×) is not commutative: the product of two braids generally depends on the order of the factors. A braid with a single crossing is called an elementary braid. Pictures representing braids can be algebraically encoded. Moving along a braid from top to bottom in successive levels, we see that the braid can be represented as the successive product of elementary braids. If in an n-strand braid we denote the crossings of the strands si and si+1 , when si overcrosses si+1 by bi , and by b−1 when si+1 overcrosses si (i = 1, . . . , n − 1), i we obtain algebraic codes for braids– braid words. Expressed in terms of braid words, the equivalence relation– isotopy of braids is described by the following relations: (1) commutativity for distant braids bi bj = bj bi

for

|i − j| ≥ 2,

i, j = 1, . . . , n − 1,

(1.1),

and (2) Artin’s relation (or the braid relation) bi bi+1 bi = bi+1 bi bi+1 ,

i = 1, . . . , n − 2

(1.2).

Together with trivial relations bi b−1 = b−1 i i bi = e, the relations (1.1) and (1.2) are sufficient for replacing geometric manipulations related to isotopy by algebraic calculations. Thus, two braids are isotopic iff the word representing one of them can be transformed into the word representing the other by a finite series of replacements satisfying the relations (1.1), (1.2). In order to simplify the notation, we will use A, B, C, . . . instead of b1 , b2 , −1 −1 p b3 ,. . ., and a, b, c, . . . instead of b−1 1 , b2 , b3 ,. . ., and a stands for a . . . a, −1 where a letter a occurs p times. For example, ABCd2 C = b1 b2 b3 b−1 4 b4 b3 . In this way, we use words to denote braids and describe isotopies as word equivalences.

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KLs and braids

In order to use braids in knot theory we need to know more than which words represent equivalent braids: we need to determine when the closures of two braids represent the same oriented KL, and how to express KL isotopies via relations between closed braid words. Two braids are called Markov equivalent if their closures yield the same oriented KL. Analogously to Reidemeister moves, we need to consider Markov moves– a set of moves on braids that give all equivalents of any given closed braid. In a paper published in 1935, A.A. Markov formulated the theorem, now known as the Markov’s Theorem (Theorem 1.57). This theorem describes Markov moves and proves their necessity and sufficiency. In addition to relations between open braids, we need two more operations for closed braids. The first of them is called conjugation (Fig 1.84d). It is a multiplication of a braid word w by b and b−1 , on one and the other side, resulting in the word bwb−1 or b−1 wb. This operation corresponds to the second Reidemeister move. Because none of the operations introduced till now change the number of strings in a braid, we need the operation called stabilization (Fig. 1.90e), which enables adding and deleting loops in closed braids. This operation takes a word w describing an n-strand braid and replaces it with the word wbn or wb−1 n , each of them corresponding to an (n+1)-strand braid. In this case, the resulting word wbn or wb−1 n corresponds to a braid with one more strand. We also use the inverse operation, where a word of the form wbn or wb−1 n is replaced with the single word w, assuming that w does not contain the letters bn or b−1 n . In this case, the number of strands is decreased by 1. Two additional Markov moves are visible on a closure of a braid. A conjugation is a trivial move on a closure of a braid, since closing the braid bwb−1 or b−1 wb allows b−1 to cancel the effect of b or vice versa. Loops that appear as the result of closing a braid can be removed using stabilization. Theorem 1.57. (Markov’s Theorem) The five operations described are sufficient to obtain from one closed braid representation of an oriented KL any other closed braid representation of the same oriented KL (Birman, 1976).

It is interesting that Markov’s Theorem was not proved by A.A. Markov (1935), but by J. Birman in 1976. With Markov moves we have the same problem as with Reidemeister

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Fig. 1.90 (a) Commutativity of distant braids; (b) Artin’s relation; (c) trivial identity; (d) conjugation; (e) stabilization; (f) a sequence of Markov moves to get from one closed braid representation of figure-eight knot the other.

moves. Although they are sufficient for transforming a closed braid to any of its equivalents, there is no algorithm for finding the corresponding sequences of Markov moves (Fig. 1.90f). In the language of Markov moves and braid words, the transformations shown in Fig. 1.90f are BaBac = BaBa =aBaBaA =aBaB.

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A knot or link L can be formed from an infinite number of braids. Definition 1.76. Within the set of braids from which L is formed there exist braids that have the least number of strands. Any such braid is called the minimal braid presentation of L (or simply minimal braid), and its number of strings is called the braid index of L. The minimal braid is not unique since many minimal braids representing a given link L have the same number of strands. A braid representation corresponding to a given KL can be found using Vogel’s algorithm (1990). Probably the best explanation of Vogel’s algorithm, using the geopolitical language from the 1990s that may remain current till our own days, is given by A. Sossinsky (2002), so we paraphrase his description. Let us consider a planar map determined by a projection of a knot or link L. A country in this map is said to be in turmoil if it has two edges that belong to two different circles labelled with arrows going in the same direction (Fig. 1.91). In the case of the knot 3 2, only the region T is in turmoil. An operation called perestroika can be applied to any country in turmoil. It consists in replacing two faulty edges by two “tongues”, one of which passes over the other, forming two new crossings. The aim is to create a new bigonal country (not in turmoil) and several new countries, some of which (in our example two of them) may be swallowed up by neighbouring countries. If some non-nested Seifert circles remain, we apply the change-of-infinity operation. Vogel’s algorithm is repeated as long as there are regions in turmoil or some non-nested Seifert circles. Vogel’s algorithm applied to knot 3 2 is illustrated in Fig. 1.91. It can be used to unroll a KL and obtain its braid representation (Fig. 1.92). The Knot 2000 (K2K) function GetBraidRep is the implementation of Vogel’s algorithm. From an input, P -data of a KL, using Reidemeister’s moves Ω2 it transforms the diagram until obtaining braid representation. As an external program this function also uses the program Braid-9.0 written by A. Bartholomew. In most cases, you can get a shorter braid word if you first reduce input KL, given by P -data, using the function Reducti onKnotLink, and then apply the function GetBraidRep. To get the graphical output– braid diagram for the braid word, one can use the function ShowBraid. The inverse function KnotFromBraid produces P -data from an arbitrary braid word. In order to work with braids using Conway symbols of KLs as an input, first you need to convert the input using the LinKnot function fCre atePData. After that you can make experiments with KLs and look for

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Fig. 1.91

Fig. 1.92

Vogel’s algorithm applied on the knot 3 2.

Braid diagram obtained using Vogel’s algorithm.

dependency relations between Conway symbols and braid words. For example, every KL of the family p (p = 2, 3, . . .) has a braid word Ap , every KL of the family p 1 2 (p = 2, 3, . . .) has a braid word Ap bAb, to every KL of the family p 1 1 q corresponds a braid word Ap bAbq , etc. General formulas of this kind can be obtained not just for rational KLs, but for all kinds of KL families. Braid words can be used for calculating different KL invariants. The LinKnot function fSeifert calculates the Seifert matrix of a KL given by its Conway symbol, Dowker code, or P -data. The function fSignat calculates

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the signature of a KL given by its Conway symbol, Dowker code, or P -data. These functions are based on the functions ssmW and SeifertMatrix, written by S. Orevkov. 1.14

Braid family representatives

T. Gittings (2004) defined, described and generated minimum braids for knots up to ten crossings and oriented links up to nine crossings. Later he used them for studying graph trees, amphicheirality, unknotting numbers and periodic tables of KLs. According to Alexander’s Theorem (Theorem 1.55), any oriented KL can be represented as a closed braid. Braid representation is not unique: every KL has an infinite number of braid representations. Accepting as the first criterion the minimal number of strands, i.e., the smallest braid index we obtain minimal braids. For a chosen (minimal) number of strands, the minimal braid will still be not unique. T. Gittings (2004, Definition 1) gave the following four restrictions which insure the uniqueness of minimal braid words: Definition 1.77. Among the set of braids for any KL, the minimum braid is the one that has the following properties: (1) (2) (3) (4)

minimum minimum minimum minimum

number of braid crossings; number of braid strands; braid universe; binary code for alternating braid crossings.

These criteria are listed in descending order of importance for determining minimum braids. A braid universe is an ordered sequence of integers, where the element i represents an unsigned crossing of the ith and (i + 1)th braid strands. A braid universe becomes a braid word when to each crossing is assigned a sign +1 or −1 in the same manner as was done before with KLs. In a braid word a positive (negative) crossing of the ith and (i + 1)th strands is represented by ith capital (lower case) letter. A braid is alternating if even numbered generators have the opposite sign of odd numbered generators. Therefore, crossings in alternating braids have capitals for the odd letters, and lower cases for the even letters. The same convention can be applied to non-alternating braids: a crossing is called alternating if it is capital for

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an odd letter and lower case for an even letter. A binary code for any braid can be generated by assigning 0 for an alternating crossing and 1 for a non-alternating crossing. With the four criteria that define a minimum braid, there is always a unique minimum for any set of braids (Kawauchi and Tayama, 2004, 2006; Gittings, 2004). We will present another approach: after defining braid family representatives (BF Rs) we will establish a correspondence between BF Rs and families of KLs given in the Conway notation. For better understanding of this correspondence, together with the standard Conway notation, a braidmodified Conway notation will be introduced and used. First we define a reduced braid word, describe a general form for all reduced braid words with s = 2 strands, generate all families of two-strand braid words, and establish a correspondence between them and families of KLs given in the Conway notation. Then we consider the same problem for s ≥ 3. Some applications of minimum braids (Gittings, 2004) and braid family representatives will be discussed in Subsection 1.14.1. All computations are made using the program LinKnot. We use the standard definition of a braid and description of minimum braids given by T. Gittings (2004). Instead of a . . . a, where a capital or lower case letter a appears p times, we write ap ; p is the degree of a (p ∈ N ). It is also possible to work with negative powers, satisfying the relations: A−p = ap , a−p = Ap . A number of strands is denoted by s, and a length of a braid word by l. Definition 1.78. The operation a2 → a applied on any capital or lower case letter a is called idempotency. Idempotency can be repeatedly applied to any braid word until a reduced braid word is obtained. Definition 1.79. A reduced braid word is a braid word with degree of every capital or lower case letter equal to 1. By an inverse procedure, braid word extension, from every reduced braid word we obtain all braid words which can be derived from it by assigning a degree greater or equal to 1 to each letter. In this case, a reduced braid word plays the role of a generating braid word. Definition 1.80. A braid word with one or more parameters denoting degrees greater then one represents a family of braid words. If values of all parameters are equal to 2, it is called a source braid.

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For the minimality of reduced braids we are using the following criteria: (1) minimum number of braid crossings; (2) minimum number of braid strands; (3) minimum binary code for alternating braid crossings. According to the first and second criterion minimal reduced braids are the shortest reduced braids with the smallest possible number of different letters among all equivalent reduced braids representing certain KL. A binary code for braid crossings can be generated by assigning a zero for an alternating, and one for a non-alternating crossing. Hence, priority will be given to alternating braids, and then to braids which are closest to alternating. Analogous minimality criteria can be applied to source braids.

Definition 1.81. Among the set of all braid families representing the same KL family, the braid family representative (BF R) is the one that has the following properties: (1) minimum number of braid crossings; (2) minimum reduced braid; (3) minimum source braid. These criteria are listed in descending order of importance for determining BF Rs. Notice that some members of BF Rs will not be minimum braids, defined by T. Gittings (2004). For example, the minimum braid of the link .2 1 : 2 (9311 in Rolfsen (1976)) is 9:03-05a AAbACbACb (Gittings, 2004, Table 2). According to the second BF R criterion it will be derived from the generating minimum braid AbAbACbC corresponding to the link .2 1 (8213 ), but not from the non-minimum generating braid AbACbACb corresponding to the same link. Hence, to the three-component link .2 1 : 2 (9311 ) obtained as the first member of BF R AbAp bACbC for p = 2 corresponds the braid AbAAbACbC, which is not minimum braid according to the minimum braid criteria (Gittings, 2004). The third criterion: minimum source braid, enables us to obtain KLs of a certain family from a single BF R, and not from several different BF Rs. For example, applying this criterion, KLs .3.2.2 0, .2.3.2 0 and .2.2.3 0 belonging to the same KL family .r.p.q 0 will be obtained from the

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single BF R Ap bAq bAbr . Otherwise, using the minimum braid criteria (Gittings, 2004), the knot .3.2.2 0 will be obtained from the family Ap bAbq Abr , three-component link .2.3.2 0 will be obtained from Ap bAq bAbr , and the knot .2.2.3 0 will be obtained from Ap bAq br Ab for p = 3, q = 2, r = 2. Source braids corresponding to the families Ap bAbq Abr , Ap bAq bAbr and Ap bAq br Ab are A2 bAb2 Ab2 , A2 bA2 bAb2 and A2 bA2 b2 Ab, respectively, and the second source braid is minimal. Hence, the representative of the KL family .r.p.q 0 is BF R Ap bAq bAbr . According to this, single family of KLs given in the Conway notation can be associated to every BF R and vice versa. Notice that families of KLs obtained from BF Rs can overlap on a finite number of KLs at their beginnings. For example, distinct BF Rs AbAp bACbC and Ap bCbAbCb, giving KL families .2 1 : p and .p 1 : 2, respectively, for p = 2 will have in common three-component link .2 1 : 2 (9311 ) mentioned above. According to the second BF R criterion, it will be derived from the minimum generating braid AbAbACbC, and not from AbACbACb. Hence, BF R AbAp bACbC giving KLs of the form .2 1 : p starts for p = 2, and Ap bCbAbCb giving KLs of the form for .p 1 : 2 starts for p = 3. In this way, all ambiguities can be avoided. Every KL is algebraic if its basic polyhedron is 1∗ or polyhedral otherwise. According to this criterion, all KLs are divided into two main categories: algebraic and polyhedral. Since the correspondence between members of a BF R and KLs is one-to-one, we can introduce the following definition: Definition 1.82. An alternating BF R is polyhedral iff its corresponding KLs are polyhedral. Otherwise, it is algebraic. A non-alternating BF R is polyhedral iff its corresponding alternating BF R is polyhedral. Otherwise, it will be called algebraic. Division of non-alternating BF Rs into algebraic and polyhedral does not coincide with the division of the corresponding KLs (Conway, 1970; Caudron, 1982; Adams, 1994), because minimum number of braid crossings is used as the first criterion for the BF Rs. Accepting minimum reduced braid universe (Gittings, 2004) as the first criterion, all KLs derived from the basic polyhedron .1 will be algebraic, because they can be represented by non-alternating minimal (but not minimum) algebraic braids. For example, alternating knot .2.2 0 (816 ) with the polyhedral braid A2 bA2 bAb can be represented as the algebraic knot (−3, 2) (3, −2) with the correspond-

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ing algebraic braid A3 b2 a2 B 3 . In this case, to the knot 816 corresponds algebraic braid A3 b2 a2 B 3 that reduces to AbaB, and not A2 bA2 bAb that reduces to AbAbAb. Another way to solve the discrepancy is changing the definition of algebraic KLs: Definition 1.83. KL is algebraic if it has an algebraic minimum crossing number representation. In this case, all KLs derived from the basic polyhedron .1 (with the Conway symbols beginning with a dot) will be polyhedral KLs, because their minimum crossing number representations are polyhedral. We will consider only BF Rs corresponding to prime KLs. Every 1-strand BF R is of the form Ap , with the corresponding KL family p in the Conway notation. Theorem 1.58. Every reduced BF R with s = 2 is of the form (Ab)n , n ≥ 2. This BF R corresponds to the knot 2 2 and the family of basic polyhedra .1 = 6∗ , 8∗ , 10∗ , 12∗ (or 12A according to A. Caudron (1982)), etc. For n ≥ 3 all of them are n-antiprisms. Let us notice that the first member of this family, the knot 2 2, is not an exception: it is an antiprism with two bigonal bases.

Theorem 1.59. All algebraic alternating KLs with s = 2 are the members of the following families: • • • • •

p 1 2 with the BF R Ap bAb (p ≥ 1); p 1 1 q with the BF R Ap bAbq (p ≥ q ≥ 2); p, q, 2 with the BF R Ap bAq b (p ≥ q ≥ 2); p, q, r 1 with the BF R Ap bAq br (r ≥ 2, p ≥ q ≥ 2); (p, r) (q, s) with the BF R Ap bq Ar bs (p, q, r, s ≥ 2, p ≥ r, p ≥ s, s ≥ q and if p = s, then r ≥ q).

Minimum braids include one additional braid (Ap bq Abr ) in the case of algebraic alternating KLs with s = 2.

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Alternating polyhedral KLs corresponding to BF Rs with s = 2 are given in the following table. KLs in this table are given in the Conway notation according to Rolfsen’s book (1976) (for knots with n ≤ 10 and links with n ≤ 9 crossings). This table can be extended to an infinite list of antiprismatic basic polyhedra (2n)∗ described by the BF Rs (Ab)n , n ≥ 3 and BF Rs with s = 2 obtained as their extensions. Table 1 Basic polyhedron .1 = 6∗ Ap bAbAb Ap bAbAbq Ap bAq bAb Ap bAbq Ab Ap bAq bAbr Ap bAq bAr b

.p .p.q .p.q 0 .p : q 0 .r.p.q 0 p:q:r

(1) (2) (3) (4) (5) (6)

Ap bAbAq br Ap bAbq Ar bs Ap bAq bAr bs Ap bAq br Abs Ap bAq br As bt Ap bq Ar bs At bu

r : p0 : q0 p.s.r.q q 0.p.r 0.s 0 .p.s.r 0.q 0 p.t.s.r.q p.q.r.s.t.u

(7) (8) (9) (10) (11) (12)

If we apply minimum braid criteria (Gittings, 2004), there are ten additional braids corresponding to the basic polyhedron .1 = 6∗ : (1’) Ap bAbq Abr , (2’) Ap bAq br Ab, (3’) Ap bq AbAbr , (4’) Ap bAq br As b, (5’) Ap bq AbAr bs , (6’) Ap bq Abr Abs , (7’) Ap bq Ar bAbs , (8’) Ap bq Abr As bt , (9’) Ap bq Ar bAs bt , (10’) Ap bq Ar bs Abt . Applying BF R criteria, according to the minimum source braid criterion all KLs obtained from the braids (1’) and (2’) will be obtained from BF R (5), KLs obtained from (3’) will be obtained from (7), KLs obtained from (4’) and (6’) will be obtained from (9), KLs obtained from (5’) and (7’) will be obtained from (8), and KLs obtained from (8’), (9’) and (10’) will be obtained from (11). Using minimum braid criteria (Gittings, 2004), we need to make analogous additions to all classes of BF Rs considered. For the basic polyhedron 8∗ we have: Basic polyhedron 8∗ Ap bAbAbAb Ap bAbAbAbq Ap bAq bAbAb Ap bAbAbq Ab Ap bAbAq bAb Ap bAq bAbAbr

8∗ p 8∗ p.q 8∗ p : q 8∗ p : .q 8∗ p :: q 8∗ p.r :: .q

Ap bAq bAbr Abs Ap bAbq Ar bAbs Ap bAq bAr bAs b Ap bAbAq br As bt Ap bAq bAbr As bt Ap bAq br As bAbt

8∗ p : q : .r : s 8∗ p.s : .r.q 8∗ p : s : r : q 8∗ p.t.s.r.q 8∗ p.t.s.r : .q 8∗ p : q.r.s : .t

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Ap bAbAq bAbr Ap bAq bAr bAb Ap bAq bAbr Ab Ap bAbAbAq br Ap bAbAbq Ar bs Ap bAq bAbAr bs Ap bAbq Ar bs Ab Ap bAq br AbAbs Ap bAq bAr bAbs

8∗ p.r : .q 8∗ p : q : r 8∗ p : .r : .q 8∗ p.r.q 8∗ p.s.r.q 8∗ p.s.r :: q 8∗ p : .s.r.q 8∗ p.s :: r.q 8∗ p.s : .r : q

Ap bAq bAr bAs bt Ap bAq bAr bs Abt Ap bAbq Ar bs At bu Ap bAq bAr bs At bu Ap bAq br As bAt bu Ap bAq br As bt Abu Ap bAq br As bt Au bv Ap bq Ar bs At bu Av bw

8∗ p.t.s : r : q 8∗ p.t : s.r : q 8∗ p.u.t.s.r.q 8∗ p.u.t.s.r : q 8∗ p.u.t : s.r.q 8∗ p : q.r.s.t : u 8∗ p.v.u.t.s.r.q 8∗ p.q.r.s.t.u.v.w

Recall that the correspondence between Conway symbols and KLs is not one-to-one. For example, the same polyhedral knot .p can be given by : p, : .p, . . ., or even as 6∗ p, 6∗ .p, 6∗ : .p, etc. Therefore, we usually order them according to the notation introduced in the original Conway’s paper (1970) and in the papers or books following it (Adams, 1994; Caudron, 1982; Rolfsen, 1976). In order to obtain better understanding of the correspondence between BF Rs and Conway symbols of KLs, we introduce the modified Conway notation, more suitable for denoting KLs obtained from BF Rs. In this notation the same degree p occurs at the first position of a braid, and as the first element (chain if bigons) in the Conway symbol of a KL corresponding to it. Whenever possible, the order of other degrees will be preserved in the corresponding Conway symbol as well. In this notation we can recognize a simple pattern for BF Rs derived from the generating minimum braids of the form (Ab)n : substituting in a given braid every sequence of length k, containing single letters, by k + 1 dots, we obtain its Conway symbol. In order to recognize this pattern for KLs derived from basic polyhedra, we use only one basic polyhedron 6∗ with n = 6 crossings, and not two of them (.1 and 6∗ ). In this case, Table 1 will look as follows: Basic polyhedron 6∗ Ap bAbAb Ap bAbAbq Ap bAq bAb Ap bAbq Ab Ap bAq bAbr Ap bAq bAr b

6∗ p 6∗ p :: .q 6∗ p : q 6∗ p : .q 6∗ p : q : .r 6∗ p : q : r

Ap bAbAq br Ap bAbq Ar bs Ap bAq bAr bs Ap bAq br Abs Ap bAq br As bt Ap bq Ar bs At bu

and for the basic polyhedron 8∗ we have:

6∗ p :: q.r 6∗ p : .q.r.s 6∗ p : q : r.s 6∗ p : q.r : s 6∗ p : q.r.s.t 6∗ p.q.r.s.t.u

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Basic polyhedron 8∗ Ap bAbAbAb Ap bAbAbAbq Ap bAq bAbAb Ap bAbAbq Ab Ap bAbAq bAb Ap bAq bAbAbr Ap bAbAq bAbr Ap bAq bAr bAb Ap bAq bAbr Ab Ap bAbAbAq br Ap bAbAbq Ar bs Ap bAq bAbAr bs Ap bAbq Ar bs Ab Ap bAq br AbAbs Ap bAq bAr bAbs

8∗ p 8∗ p ::: .q 8∗ p : q 8∗ p :: .q 8∗ p :: q 8∗ p : q :: .r 8∗ p :: q : .r 8∗ p : q : r 8∗ p : q : .r 8∗ p ::: q.r 8∗ p :: .q.r.s 8∗ p : q :: r.s 8∗ p : .q.r.s 8∗ p : q.r :: s 8∗ p : q : r : .s

Ap bAq bAbr Abs Ap bAbq Ar bAbs Ap bAq bAr bAs b Ap bAbAq br As bt Ap bAq bAbr As bt Ap bAq br As bAbt Ap bAq bAr bAs bt Ap bAq bAr bs Abt Ap bAbq Ar bs At bu Ap bAq bAr bs At bu Ap bAq br As bAt bu Ap bAq br As bt Abu Ap bAq br As bt Au bv Ap bq Ar bs At bu Av bw

8∗ p : q : .r : s 8∗ p : .q.r : .s 8∗ p : q : r : s 8∗ p :: q.r.s.t 8∗ p : q : .r.s.t 8∗ p : q.r.s : .t 8∗ p : q : r : s.t 8∗ p : q : r.s : t 8∗ p : .q.r.s.t.u 8∗ p : q : r.s.t.u 8∗ p : q.r.s : t.u 8∗ p : q.r.s.t : u 8∗ p : q.r.s.t.u.v 8∗ p.q.r.s.t.u.v.w

Unfortunately, it is not possible to express every family of KLs in the braid-modified Conway notation. In the same way, it is possible to derive BF Rs from basic polyhedra with a higher number of crossings. Corollary 1.5. All alternating KLs with s = 2 are described by Theorem 1.59 and by an infinite extension of tables for antiprismatic basic polyhedra (2n)∗ described by the BF Rs (Ab)n , n ≥ 3. From alternating BF Rs we obtain non-alternating BF Rs by crossing changes. In this way, from BF Rs derived from the generating minimum braid (Ab)2 we obtain the following families of non-alternating BF Rs and corresponding new KL families: Ap BaB Ap baq b Ap BAq B Ap BaB q Ap B q aB r

(p − 1) 3 p, (q − 1) 1, −2 p, q, −2 (p − 1) 2 q p − 1, q, r+

Ap baq br Ap BAq B r Ap B q ar B s Ap B q Ar B s

p, (q − 1) 1, −(r + 1) p, q, −r 1 (−p, r) (q, s) (p, r) − (q, s)

In the same way, we can derive non-alternating BF Rs with s = 2 from the generating BF R (Ab)n , n ≥ 3.

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The first step towards derivation of BF Rs and corresponding KLs for s ≥ 3 is the derivation of all different reduced minimum braid words. Here are the general construction rules for generating minimum braid words: Definition 1.84. For a given generating minimum braid word W = wL which ends with a capital or lower case letter L, a replacement of L by a word w1 in W is called extending by replacement. An addition of the word w1 to W is extending by addition. The both operations are called extending operations. Definition 1.85. Let W = wLs and w1 = Ls+1 Ls Ls+1 be generating minimum braids with s and s + 1 strings, where Ls denotes sth letter and Ls+1 denotes (s + 1)th letter. The word extending operations obtained in this way are, respectively, (s + 1)-extending by replacement, and (s + 1)extending by addition. For example, the first operation applied on AbAb gives AbACbC, and the other AbAbCbC. The (s + 1)-extending by replacement is sufficient for constructing generating minimum braids for a given s, with l = 2s, corresponding to KLs of the form 2 . . . 2 = 2s , where 2 occurs s times. For 2 ≤ s ≤ 6 as the result we obtain: AbAb, AbACbC, AbACbdCd, AbACbdCEdE, AbACbdCEdf Ef , ... The generating minimum braids for given s, with l = 3s − 2, corresponding to KLs of the form 2 1 . . . 1 2 = 2 13s−6 2, where 1 occurs 3s − 6 times, can be obtained using only (s + 1)-extension by addition. For 3 ≤ s ≤ 6 we obtain: AbAbCbC, AbAbCbCdCd, AbAbCbCdCdEdE, AbAbCbCdCdEdEf Ef , . . . In a similar way, from A3 we obtain the series A3 BaB, A3 BaBCbC, 3 A BaBCbCDcD, A3 BaBCbCDcDEdE . . ., corresponding to the knots 3 2, 5 2, 7 2, 9 2, . . . Starting with w1 = AbAbCbdCd and using the (s + 1)-extension by replacement, the generating minimum braids with l = 2s+1, corresponding to KLs of the form 2 2 1 . . . 1 2 = 22 12s−5 2 are obtained for given s. However, for exhaustive derivation of reduced minimum braids we use all combinations of (s + 1)-extending operations. Theorem 1.60. Every generating algebraic minimum braid can be derived from AbAb by a recursive application of (s + 1)-extending operations.

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The minimal generating braid words for s ≤ 5 with their corresponding KLs are given in the following table: s=1 l=1

A

1

s=2 l=4

AbAb

22

s=3 l=6 s=3 l=7

AbACbC AbAbCbC

222 21112

s = 4 l = 8 AbACbdCd s = 4 l = 9 AbAbCbdCd s = 4 l = 10 AbAbCbCdCd

2222 221112 21111112

s=5 s=5 s=5 s=5 s=5 s=5

22222 2221112 2211122 221111112 211121112 21111111112

l l l l l l

= 10 = 11 = 11 = 12 = 12 = 13

AbACbdCEdE AbAbCbdCdEdE AbACbCdCEdE AbAbCbCdCEdE AbAbCbdCdEdE AbAbCbCdCdEdE

Generalizations can be made in the case of polyhedral generating minimum braid words as well. We have already considered the infinite class of generating polyhedral minimum braid words (Ab)n with s = 2. The first infinite class with s = 3 will be (Ab)n−1 ACbC, with the corresponding KLs of the form (2n)∗ 2 1 0. For s = 3 there are two generating alternating algebraic minimum braid words: • AbACbC, l = 6, with the corresponding link 2 2 2; • AbAbCbC, l = 7, with the corresponding knot 2 1 1 1 2, which generate prime KLs. From AbACbC we derived 17 alternating BF Rs and their corresponding families of KLs, given in the following table: Ap bACbC AbACbp C Ap bACbq C Ap bAq CbC Ap bACbC q

p122 p, 2, 2+ p 1, q, 2+ p, q, 2 2 p121q

Ap bAq Cbr C Ap bq Ar Cbs C Ap bq ACbr C s Ap bAq Cbr C s Ap bAq C r bC s

(p, q) (r, 2+) (p, r) (q, 2, s) p 1, q, s 1, r (p, q) (r, s 1+) (p, q) 2 (r, s)

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Abp ACbq C Ap bq ACbr C Ap bACbq C r Ap bAq CbC r

p, 2, q, 2 p 1, q, r, 2 p 1, q, r 1+ p, q, r 1 2

Ap bq Ar Cbs C t (p, r) (q, t 1, s) Ap bAq C r bs C t (p, q), s, (t, r)+ Ap bq Ar C s bt C u (p, r), q, (u, s), t

The next generating alternating algebraic minimum braid AbAbCbC of the length 7, with s = 3, gives the following results: Ap bAbCbC AbAbCbp C AbAbp CbC Ap bq AbCbC Ap bAq bCbC Ap bAbq CbC Ap bAbCbC q AbAbp Cbq C Abp AbCbq C Ap bq Ar bCbC Ap bq Abr CbC Ap bq AbCbr C Ap bAq bCbr C Ap bAq br CbC Ap bAq bCbC r Ap bAbq Cbr C Ap bAbq CbC r Ap bAbCbq C r Abp Abq Cbr C Ap bq Ar bs CbC

Ap bq AbCbr C s Ap bAq br Cbs C Ap bAq br CbC s Ap bAq bC r bC s Ap bAq bCbr C s Ap bAbq Cbr C s Ap bAbC q br C s Ap bq Ar bCbs C Ap bq Abr Cbs C Ap bq Ar bCbs C t Ap bq Abr Cbs C t Ap bAq br C s bC t Ap bAq br Cbs C t Ap bAq bC r bs C t Ap bAbq C r bs C t Ap bq Ar bs Cbt C Ap bq Ar bs Cbt C u Ap bq Ar bC s bt C u Ap bAq br C s bt C u Ap bq Ar bs C t bu C v

p11112 p, 2 1 1, 2 21p12 p 1, q, 2 1 1 p, q, 2 1 1 1 p11q12 p11111q 2 1 p, q, 2 (p, 2) 1 (q, 2) (p, r) (q, 2 1 1) p 1, q, 2 1 r (p 1, q) 1 (r, 2) (p, q) 1 1 (2, r) p, q, 2 1 r 1 p, q, r 1 1 1 1 p 1 1 q, r, 2 p11q11r p 1 1 1, q, r 1 (p, 2) q (r, 2) (p, r) (q, 2 1 s)

(p 1, q) 1 (s 1, r) (p, q) 1 r (2, s) p, q, s 1 1 r 1 (p, q) 1 1 1 (r, s) (p, q) 1 1 (r, s 1) p 1 1 q, r, s 1 (p 1 1 1, r) (q, s) (p, r), q, (2, s) 1 (p 1, q) r (2, s) (p, r), q, (t 1, s) 1 (p 1, q) r (t 1, s) (p, q) 1 r 1 (s, t) (p, q) 1 r (s, t 1) (p, q) 1 1, (t, r), s (p 1 1 q, s) (r, t) (p, r), q, (t, 2) s ((p, r), q) s (u 1, t) ((p, r), q) 1 ((u, s), t) (p, q) 1 r ((u, s), t) ((p, r), q) s ((u, t), v)

Except AbACbC and AbAbCbC, all generating minimum braids with s = 3 are polyhedral. For s = 3 and l ≤ 12, the polyhedral generating braids and their corresponding KLs are given in the following table, in the notation for basic polyhedra with 12 crossings according to A. Caudron (1982): l=8 l=8

AbAbACbC AbCbAbCb

.2 1 .2 : 2

l=9 l=9 l=9

AbAbCbAbC AbAbAbCbC AbACbACbC

8∗ 2 0 .2 1 1 9∗

l l l l l

= 11 = 11 = 11 = 11 = 11

AbAbAbAbCbC AbAbAbCbCbC AbAbACbAbCb AbAbACbACbC AbAbCbACbCb

8∗ 2 1 1 11∗∗∗ 10∗∗ .2 0 11∗∗ 11∗

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l = 10 l = 10 l = 10 l = 10 l = 10 l = 10

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AbAbAbCbCb AbAbAbACbC AbAbACbAbC AbAbCbAbCb AbAbACbCbC AbAbCbACbC

.2 1 2 8∗ 2 1 0 9∗ .2 9∗ 2 10∗∗∗ 10∗∗

l l l l l l l

= 12 = 12 = 12 = 12 = 12 = 12 = 12

AbAbAbAbACbC AbAbAbACbCbC AbAbAbCbACbC AbAbACbAbCbC AbAbCbAbACbC AbAbCbAbCbCb AbCbAbCbAbCb

10∗ 2 1 0 12I 12F 12H 12G 12D 12C

Generating braid words listed above are used for deriving BF Rs with no repetitions. For example, for l = 8, the generating minimum braid AbAbACbC (.2 1) gives 70 BF Rs, and AbCbAbCb (.2 : 2) gives 19 BF Rs. Overlapping of those families occurs only if all parameters are equal to 2, i.e., for source braids and source KLs corresponding to them. According to the minimality criteria, all these source braids belong to the first BF R. The generating minimum braid AbAbACbC (.2 1) gives the following BF Rs: Ap bAbACbC AbAp bACbC AbAbACbp C AbAbACbC p AbAbp ACbC Ap bAq bACbC Ap bAbAq CbC Ap bAbACbq C Ap bAbACbC q Ap bAbq ACbC Ap bq AbACbC AbAp bACbq C AbAp bACbC q AbAp bq ACbC AbAbACbp C q AbAbAC p bC q AbAbp ACbq C AbAbp ACbC q Abp Abq ACbC Ap bAq bAr CbC Ap bAq bACbr C Ap bAq bACbC r Ap bAq br ACbC Ap bAbAq Cbr C Ap bAbAq CbC r Ap bAbACbq C r Ap bAbAC q bC r Ap bAbq Ar CbC Ap bAbq ACbr C Ap bAbq ACbC r

.2 1.p 0 .2 1 : p .(p, 2) .p 1 1 .2 1.p .2 1.p 0.q 210 : p0 : q0 .(2, q).p 0 .q 1 1.p 0 .q.2 1.p 0 21 : p : q0 .(2, q) : p .q 1 1 : p .2 1.q.p .(q 1, p) .(q, p) 1 .(q, 2).p .q 1 1.p p : q : 210 .2 1.p 0.q : r .q.p 0.(r, 2) .r 1 1.p 0.q .2 1.r.q.p 0 p 0 : q 0 : (r, 2) 0 r110 : q0 : p0 .p.(r 1, q) 0 .p.(r, q) 1 0 q.2 1.r 0.p .q.(2, r).p 0 .q.r 1 1.p 0

AbAp bAC q bC r AbAp bq ACbr C AbAp bq ACbC r AbAbp ACbq C r AbAbp AC q bC r Abp Aq br ACbC Abp Abq ACbr C Abp Abq ACbC r Ap bAq bAr Cbs C Ap bAq bAr CbC s Ap bAq bACbr C s Ap bAq bAC r bC s Ap bAq br As CbC Ap bAq br ACbs C Ap bAq br ACbC s Ap bAbAq Cbr C s Ap bAbAq C r bC s Ap bAbq Ar Cbs C Ap bAbq Ar CbC s Ap bAbq ACbr C s Ap bAbq AC r bC s Ap bq Ar bACbs C Ap bq Ar bACbC s Ap bq Ar bs ACbC Ap bq AbACbr C s Ap bq AbAC r bC s Ap bq Abr As CbC Ap bq Abr ACbs C Ap bq Abr ACbC s AbAp bq ACbr C s

.p : (q, r)1 .p.q.(2, r) .r 1 1.q.p .p.(r 1, q) .p.(r, q) 1 2 1.p.r 0.q 0 p : q : (2, r) 0 q : p : r110 (2, s).p 0.r.q 0 s 1 1 0 : r 0.q.p 0 .(s 1, r).p 0.q (s, r) 1 : p.q 0 q.r.s.2 1 0.p .(s, 2).r.q.p 0 .s 1 1.r.q.p 0 (r, s 1) 0 : q 0 : p 0 (s, r) 1 0 : q 0 : p 0 q.(2, s).r 0.p q.s 1 1.r 0.p .q.(s 1, r).p 0 .q.(s, r) 1.p 0 r.q.p.(2, s) 0 r.q.p.s 1 1 0 s.r.q.p.210 (s 1, r) : p : q 0 (s, r) 1 : p : q 0 s.r.q.p.2 1 0 p 0.(s, 2).q.r 0 p 0.s 1 1.q.r 0 .(r, s 1).q.p

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Ap bq Ar bACbC Ap bq AbACbr C Ap bq AbACbC r Ap bq Abr ACbC AbAp bACbq C r

r.q.p.2 1 0 p : (r, 2) : q 0 r11 : p : q0 p 0.2 1.q.r0 .p : (r 1, q)

AbAp bq AC r bC s Abp Aq br ACbs C Abp Aq br ACbC s Abp Abq ACbr C s Abp Abq AC r bC s

.(s, r) 1.q.p (s, 2).p.r 0.q 0 s 1 1.r.p 0.q 0 q : p : (s 1, r) 0 p : q : (r, s) 1 0

The generating minimum braid .2 : 2 gives the following BF Rs: p

A bCbAbCb Abp CbAbCb Ap bq CbAbCb Ap bCbAq bCb Ap bCbAbC q b Abp CbAbq Cb Abp CbAbCbq Ap bq CbAr bCb Ap bq CbAbr Cb Ap bq CbAbCbr

.p 1 : 2 .2.p 0.2 .p 1.q 0.2 .(p, q) : 2 .p 1 : q 1 .p.2 0.q.2 0 2.p 0.q.2 0 .2.q 0.(p, r) .p 1.r 0.2.q 0 p 1.q 0.r.2 0

Ap bq CbAbC r b Ap bq Cbr AbCb Ap bCbAq bC r b Abp CbAbq Cbr Ap bq CbAr bs Cb Ap bq CbAr bC s b Ap bq CbAr bCbs Ap bq CbAbr C s b Ap bq CbAbC r bs

.r 1.q 0.p 1 2.q 0.r.p 1 0 .r 1 : (p, q) 2.p 0.r.2 0.q .(p, r).s 0.2.q 0 .s 1.q 0.(p, r) (r, p).q 0.s.2 0 .p 1.r 0.s 1.q 0 p 1.s 0.q.r 1 0

In the similar manner, all alternating and non-alternating BF Rs and their corresponding families of KLs can be derived from generating minimum braid words with s = 3. For example, the following table contains non-alternating BF Rs with at most two parameters, derived from the minimum reduced braid AbACbC: Ap BacBc Ap BacBcq Ap bACB q C Ap BacB q c Ap bACB q C Ap bACbcq Ap BacBC q

(p − 1) 3 2 (p − 1) 3 1 q p 1, (q − 1) 1, 2 p − 1, q, 2 + + p 1, (q − 1) 1, 2 p 1 3 (q − 1) (p − 1) 4 (q − 1)

Ap baq CbC Ap BAq cBc Abp ACB q C Abp AcB q c Abp Acbq c AB p ACB q C AB p ACbq C

p, 2 2, −q p, 2 1 1, −(q − 1) 1 p, 2, 2, −q p, 2, −q, −2 p, q, 2, −2 p, q, −2, −2 p, 2, q, −2

From the generating minimum braid word W = (Ab)n (n ≥ 2), that defines the family of basic polyhedra (2n)∗ , by word extension w1 = CbACbC, we obtain the second family of basic polyhedra 9∗ (AbACbACbC), 10∗∗ (AbAbCbACbC), 11∗∗ (AbAbACbACbC), 12F (AbAbAbCbACbC), etc. From W = (Ab)n (n ≥ 3), for w1 = CbCbC, we derive the third family of basic polyhedra 10∗∗∗ (AbAbACbCbC), 11∗∗∗ (AbAbAbCbCbC), 12I (AbAbAbACbCbC), etc. In the same way, • for W = (Ab)n (n ≥ 1), w1 = CbAbCbAbCb, the family of basic polyhedra beginning with 12C (AbCbAbCbAbCb) is obtained;

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• for W = (Ab)n (n ≥ 2), w1 = CbAbCbCb the family of basic polyhedra beginning with 12D (AbAbCbAbCbCb) is obtained; • for W = (Ab)n (n ≥ 2), w1 = CbAbACbC the family of basic polyhedra beginning with 12G (AbAbCbAbACbC) is obtained; • for W = (Ab)n (n ≥ 2), w1 = CbAbCbC the family of basic polyhedra beginning with 12H (AbAbACbAbCbC) is obtained, etc. Families derived above can be divided into subfamilies according to the type of extension (replacement or addition) used for their derivation. Theorem 1.61. For s = 4 generating algebraic minimum braids are: • AbACbdCd, l = 8, with the corresponding link 2 2 2 2, • AbAbCbdCd, l = 9, with the corresponding link 2 2 1 1 1 2, • AbAbCbCdCd, l = 10, with the corresponding knot 2 1 1 1 1 1 1 2. All other generating minimum braid words with s = 4 are polyhedral. For s = 4 and l ≤ 12, the polyhedral generating braids and their corresponding KLs are given in the following table, with the notation for basic polyhedra with 12 crossings according to A. Caudron (1982): l l l l

= = = =

10 10 10 10

AbAbACbdCd AbACbCbdCd AbACbdCbdC AbACdCbCdC

.2 2 1 .2 1.2 1 .2 1 : 2 1 0 .2 2 : 2

l l l l l l l l l

= = = = = = = = =

11 11 11 11 11 11 11 11 11

AbAbACbCdCd AbAbCbCbdCd AbAbCbdCbdC AbAbCdCbCdC AbACbACbdCd AbACbCdCbCd AbACbCdCdCd AbACbdCbCdC AbACdCbCdCd

.2 1 1 1 1 .2 1 1.2 1 0 .2 1 1 : 2 1 .2 1 1 1 : 2 9∗ 2 1 0 8∗ 2 1 0 :: 2 0 .2 2 1 1 8∗ 2 1 : .2 0 8∗ 2 2 0

AbAbAbACbdCd AbAbACbAbdCd

8∗ 2 2 1 0 9∗ .2 2

l = 12 l = 12

l l l l l l l l l l l l l l l l l

= 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12 = 12

AbAbACbdCdCd AbAbACdCbCdC AbAbCbAbdCbd AbAbCbCdCbCd AbAbCbdCbCdC AbAbCbdCbdCd AbAbCdCbCdCd AbACbAdCbdCd AbACbCbCbdCd AbACbCbdCbCd AbACbCbdCbdC AbACbCdCbCdC AbACbCdCbdCd AbACbdCbCdCd AbACbdCbdCdC AbCbAbCdCbCd AbCbACbdCbCd

12J 11∗∗∗ : .2 0 9∗ 2 2 8∗ 2 1 1 :: 2 0 8∗ 2 1 1 0 : .2 0 9∗ 2 1 1 8∗ 2 1 1 1 0 12L 8∗ 2 1 0.2 1 0 9∗ .2 1 : .2 8∗ 2 1 0 : .2 1 0 9∗ 2 1 : 2 10∗∗ : 2 1 0 10∗∗ .2 1 10∗∗ : 2 1 10∗∗ : 2 0 :: .2 0 10∗∗ 2 0 :: .2 0

The family of basic polyhedra starting from 12J (AbAbACbdCdCd) is obtained for W = (Ab)n (n ≥ 2), w1 = ACbdCdCd, and the family of basic polyhedra starting from 12L (AbACbAdCbdCd) is obtained for W = (Ab)n (n ≥ 1), w1 = ACbAdCbdCd.

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1.14.1

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Applications of minimum braids and braid family representatives

A. Caudron (1982) was the first to use graph-theoretical approach to knot theory. T. Gittings (2004, Conjecture 1) established a mapping between minimum braids with s strands and trees with s+1 vertices and conjectured that the number of graph trees of n vertices with alternating minimum braids is equal to the number of rational KLs with n crossings (see page 70, Theorem 1.16). KL is achiral (or amphicheiral) if its “left” and “right” forms are equivalent, meaning that one can be transformed to the other by an ambient isotopy. If an oriented knot or link L can be represented by an antisymmetric vertex-bicolored graph on a sphere, whose vertices with the sign +1 are white, and vertices with the sign −1 are black, it is achiral. In this case, for an oriented knot or link L there exists an antisymmetry (sign-changing symmetry) switching orientations of vertices, i.e., mutually exchanging vertices with the signs +1 and −1. In the language of braid words, this means that its corresponding braid word is antisymmetric (or palindromic): there exists a mirror antisymmetry transforming one letter to another and vice versa and changing their case (i.e., transforming capital to lower case letters and vice versa). For example, the reduced braid words Ab |Ab or ABac |BDcd are palindromic, where the anti-mirror is denoted by |. Conjecture 1.15. An oriented KL is achiral iff it can be obtained from a palindromic reduced braid by symmetric assigning of degrees. For s = 2 all alternating BF Rs are of the form (Ab)n (n ≥ 2), defining a series of the basic polyhedra (2n)∗ , beginning with 2 2, .1 = 6∗ , 8∗ , 10∗ , 12∗ , etc. All of them are achiral KLs, representing a source of other achiral KLs. From 4:1-01 AbAb (2 2 or 41 ) by symmetric assigning of degrees we can derive achiral alternating knots with n ≤ 10 crossings: 6:1-02 A2 bAb2 (2 1 1 2 or 63 ), 8:1-05 A3 bAb3 (3 1 1 3 or 89 ), 10:1-017 A3 b2 A2 b3 ((3, 2) (3, 2) or 1079 ), and one achiral alternating link with n ≤ 9 crossings: 8:3-05a A2 b2 A2 b2 ((2, 2) (2, 2) or 834 ), etc. In general, from AbAb the following families of achiral alternating KLs are derived: Ap bAbp

p11p

Ap bq Aq bp

(p, q) (p, q)

Borromean rings 6:3-02 AbAbAb (.1 = 6∗ or 632 ) are the origin of achiral alternating knots 8:1-07 A2 bAbAb2 (.2.2 or 817 ), 10:1-020 A2 bA2 b2 Ab2 (.2.2.2 0.2 0 or 1099 ), 10:1-022 A2 b2 AbA2 b2 (2.2.2.2 or 10109 ), and of the link 8:3-04a Ab2 AbA2 b (.2 : 2 0 or 836 ), etc. The following families of achiral

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alternating KLs are derived from AbAbAb: Ap bAbAbp Abp AbAp b Ap bq AbAq bp

.p.p .p : p 0 p.q.q.p

Ap bAq bq Abp Ap bq Ar br Aq bp

.p.p.q 0.q 0 p.q.r.r.q.p

Achiral basic polyhedron AbAbAbAb (8∗ ) is the origin of the following families of alternating achiral KLs: Ap bAbAbAbp AbAp bAbp Ab Ap bq AbAbAq bp Ap bAq bAbq Abp

8∗ p.p 8∗ p : .p 8∗ p.q.q.p 8∗ .p : q.q : p

Ap bAbq Aq bAbp Ap bq Ar bAbr Aq bp Ap bAq br Ar bq Abp

8∗ p.q : .q.p 8∗ p.q.r.r.q.p 8∗ p.q.q.p : r.r

In the same way, it is possible to derive achiral alternating KLs from achiral basic polyhedra (Ab)n for n ≥ 5. From the antisymmetry condition it follows that every palindromic braid has an even number of strands. For s = 4 and l ≤ 12 palindromic algebraic generating braids are: • AbACbdCd, l = 8 with the corresponding achiral link 2 2 2 2, • AbAbCbCdCd, l = 10, with the corresponding achiral knot 2 1 1 1 1 1 1 2. The palindromic polyhedral generating braids are: • • • •

AbACbCbdCd, l = 10, with the corresponding achiral knot .2 1.2 1, AbAbACbdCdCd, l = 12, with the corresponding achiral link 12J, AbACbAdCbdCd, l = 12, with the corresponding achiral knot 12L, AbACbCbCbdCd, l = 12, with the corresponding achiral link 8∗ 2 1 0.2 1 0, • AbCbAbCdCbCd, l = 12, with the corresponding achiral knot 10∗∗ : 2 0 :: .2 0, • AbCbACbdCbCd, l = 12, with the corresponding achiral knot 10∗∗ 2 0 :: .2 0. From the generating braid AbACbdCd the following families of alternating achiral KLs are derived: Ap bACbdCdp AbAC p bp dCd Abp AC q bq dC p d

p1221p (p, 2+) (p, 2+) (p, q, 2) (p, q, 2)

Ap bAq C r br dq Cdp Ap bq AC r br dC q dp Ap bq Ar C s bs dr C q dp

(((p, q), r)+) (((p, q), r)+) (q, p 1, r) (q, p 1, r) (q, (p, r), s) (q, (p, r), s)

From the same palindromic non-alternating generating braid the following families of achiral KLs are obtained:

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Ap BacBDcdp AbAcp B p dCd

pp (p, 2) (q, 2)

Ap BacBDcdp Ap bAcq B q dCdp

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2pp2 (p 1, q) (p 1, q)

The family of achiral odd crossing number knots discovered by J. Hoste, M. Thistlethwaite and J. Weeks (1998) can be extended to the twoparameter BF R defined by the palindromic braid ABaB q C p BAdcbp cq Dcd corresponding to the family of non-alternating achiral odd-crossing knots with n = 7 + 4p + 4q crossings 10∗∗ (−2p) 0. − 1. − 2 0.(2q) : (−2p) 0. − 1. − 2 0.(2q). T. Gittings (2004) emphasized that in some cases it is possible to calculate unlinking numbers from minimum braids. Unfortunately, this is true only for KLs with n ≤ 10 crossings, including the link 4 1 4 (924 ) and the Nakanishi-Bleiler example 5 1 4 (108 ) with an unlinking gap. Definition 1.86. The minimum braid unlinking gap is the positive difference between the unlinking number obtained from a minimum braid uB (L) and BJ-unlinking number uBJ (L) of a link L, i.e., δB = uB (L) − uBJ (L) > 0. The unlinking gap for minimum braids appears for n = 11 crossings. The following alternating links given in the Conway notation, followed by their minimum braids have the minimum braid unlinking gap: .5.2 .3.4 8∗ 4 .2.3.3 0

A5 bAbAb2 A4 bAbAb3 A4 bAbAbAb A3 bA3 bAb2

8∗ 3.2 8∗ 3 : 2 8∗ 2.2 : .2 10∗ 2

A3 bAbAbAb2 A3 bA2 bAbAb A2 bAbA2 bAb2 A2 bAbAbAbAb

For links .5.2, .3.4 the value of the minimum braid unlinking gap is δB = 2, and for other links from this list δB = 1. Hence, the minimum braid unlinking number is different from the unlinking number and represents a new KL invariant. Periodic tables of KLs can be organized in three ways: according to families of KLs given in the Conway notation, minimum braids (Gittings, 2004), or BF Rs. Since we have established one-to-one correspondence between BF Rs and families of KLs in Conway notation, it follows that the same patterns (with regard to all KL polynomial invariants and KL properties) will appear in all cases. For example, for every family of KLs is

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possible to obtain a general formula for Alexander polynomials, with coefficients expressed by numbers denoting integer tangles in Conway symbols, from their corresponding parameters from minimum braids, or from BF Rs. Based on experimental results we believe that the same holds not only for KL polynomials, but for many other properties of KLs: writhe, amphicheirality, number of projections, unlinking number, signature, periods, etc.

1.15

More KL invariants

Rational KLs are easy to distinguish and compare: very fast and simple function RatReduce (based on two Mathematica functions: Continued Fraction and FromContinued Fraction) reduces KLs given in the Conway notation. To recognize and compare KLs from other KL-worlds, we need more refined invariants and reduction methods. For alternating KLs, one can use minimization of Dowker codes. In the language of the Conway notation, product p q can be expressed as the ramification (p,1,. . .,1), where 1 occurs q times. The flyping sequence for a product p q is then: (p,1,. . .,1), (1,p,. . .,1), . . ., (1,1,. . .,p). Using the same flyping algorithm for Conway symbols, we can obtain all projections of an alternating KL given by its Conway symbol. The LinKnot function fProjections, calculates all projections of alternating KLs given in the Conway notation, their Conway symbols, and the overall number. Furthermore, all non-isomorphic projections of an alternating KL are output of the function fDiffProjectionsAltKL. Obtained projections can be minimized using the function MinDowProjAltKL. This whole process is contained in the function MinDowAltKL which computes the minimal Dowker code for an alternating KL. Please notice that the function MinDowAltKL can not compute unique minimal Dowker code of an alternating polyhedral KL that can be obtained from different basic polyhedra. In this case one can compute all particular minimal Dowker codes, and take the minimal one. Two alternating prime KLs are equal (up to their mirror images) iff their minimal Dowker codes without signs are equal. This means that minimal Dowker codes (without signs) are sufficient for comparing two arbitrary alternating knots or links L1 and L2 given by their Conway symbols. The function SameAltProjKL compares two alternating KL projections given by their Conway symbols. The result is 1 for isomorphic, and 0 for

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non-isomorphic projections. The function SameAltConKL compares two alternating KLs given by their Conway symbols. The result is 1 for ambient isotopic, and 0 for non-equivalent KLs. The function fDiffProjectionsAltKL enables calculating some invariants (or properties) of alternating KLs that depend on all non-isomorphic projections and to extend these properties from individual projections to the corresponding KLs. For example, if we find at least one achiral projection of a KL, we know that KL in question is achiral. This mostly holds for the functions dealing with symmetry of KLs. In the same way as with graph automorphisms, we can view KL projections as weighted graphs, where a weight is a sign of the crossing, and instead of automorphisms we consider sign-preserving automorphisms. These automorphisms form a group: the automorphism group of a KL. The order of the automorphism group Aut(L) of a link L is the number of (sign-preserving) automorphisms it contains. The LinKnot function AmphiProjAltKL tests chirality of a given projection of an oriented alternating KL given by its Conway symbol, Dowker code, or P -data, and the function AmphiAltKL tests the chirality of an alternating oriented KL given by its Conway symbol. Both functions are based on antisymmetric representations of achiral projections. For example, the projection (2 2, 2) (2 2, 2) of the achiral knot (2 2, 2) (2 2, 2) is not achiral (Fig. 1.93a), but achirality can be seen on the projection ((1, 1, 2), 2) ((2, 1, 1), 2) (Fig. 1.93b). Figure 1.93c shows its centroantisymmetric representation. Unfortunately, these functions are restricted to minimal projections of alternating KLs, so they fail to detect achirality of oriented KLs without antisymmetric minimal projections (e.g., achiral 3-component links 3.2.3 0 : 2, 8∗ 2.2 : 2 0 : .2 0, 8∗ 2 0.2 0 : .2 0.2 0, and 10∗∗ .2 :: .2 with n = 12 crossings, the knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0 with n = 18 crossings, etc.). If a projection of KL is preserved under rotation for 2π p angle, we say that it has a period p. Notice that a single projection of KL can have several different periods. For example, trefoil has one minimal projection. For rotation axis we have two possible choices, the first corresponding to three-fold, and the second to two-fold rotation (half-turn). Hence, the periods of the trefoil knot minimal projection are 3 and 2 (Fig. 1.94). The list of periods of a KL contains periods of all its projections. Since every KL has infinitely many projections, we are working only with alternating KLs and their minimal projections. The LinKnot function PeriodProjAltKL calculates periods of a given

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Fig. 1.93 (a) Chiral projection of the knot (2 2, 2) (2 2, 2); (b) its achiral projection; (c) centro-antisymmetric representation of the projection (b).

Fig. 1.94 (a) Projection of a trefoil with the period 3; (b) the same projection showing the period 2.

projection of an alternating KL given by its Conway symbol, Dowker code, or P -data, and the function PeriodAltKL calculates the period of a given alternating KL given by its Conway symbol. The function PeriodAltKL works only with minimal projections.

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The function Symm calculates all automorphisms of an alternating KL projection given by its Conway symbol, Dowker code, or P -data. The output contains the list of automorphisms given by permutations, and the list of the corresponding cycles. The function MaxSymmProjAltKL finds the most symmetric projection of an alternating KL given in Conway notation. The output, number of automorphisms and the most symmetric projection, are computed over all minimal non-isomorphic projections. For example, among six nonisomorphic projections of the knot 2 1 1 1 1 1 2, the most symmetric projection is 2 1 1 1 1 1 2, with the automorphism group of order 2 (Fig. 1.95).

Fig. 1.95 (a) The most symmetric projection of a knot 2 1 1 1 1 1 2; (b) one of its lesssymmetric projections.

The next invariant we consider, defined only for links, is the splitting number: Definition 1.87. A splitting number s(L) is the minimum number of crossing changes over all projections of a link L required to obtain a split link, i.e., a link with split components, not necessarily unknotted. Comparing splitting number and unlinking number, C. Adams (1996) gave the example of 2-component link 112∗ .2 0 :: −1. − 1. − 1. − 1. − 1 with splitting number 1, and unlinking number 2. A single crossing change turns it into a split link, changing at the same time one of its unknotted components into a trefoil, so its unlinking number is greater then splitting number. We propose a simpler example of the link with the splitting number 1, and unlinking number 2: the link .2 (or 726 in the classical notation) (Fig. 1.96). It splits by one crossing change, which turns one of its un-

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knotted components into a trefoil knot. The next link with this property is .2.2.2 0, and we propose the family of links .p 1, (p ≥ 2) with the same property. After a crossing change, circled on the drawing (Fig. 1.97), link .p 1 splits into an unknot and knot p 2 (obtained from the unknotted component which contains the crossing where the change was made). We can also define a BJ-splitting gap– difference between BJ-unlinking number and splitting number of a link L. Instead of analyzing splitting number and splitting gap in great detail, we rather illustrate it with several interesting experimental results obtained using the LinKnot function SplittNo that calculates splitting number of a minimal projection of link given by its Conway symbol, Dowker code, or P -data. A family of polyhedral links .(2k), (k ≥ 1) has the BJ-unlinking number k + 1 and splitting number 1 (see Fig. 1.96 for k = 1), so BJ-splitting gap is at least k and can be made arbitrarily large. The same results we obtain for the family .(2k) (2l), (k, l ≥ 1). Moreover, every link of the form .(2a1 ) (2a2 ) . . . (2an ), (a1 , . . . , an ≥ 1) has the splitting number 1, and P[ n+1 ] BJ-unlinking number i=12 a2i−1 . All our examples with splitting number 1 were 2-component links, so the question is: is there exists a 3-component link with the splitting number 1? The answer is: probably not!

Fig. 1.96

The link .2 (or 726 ) before (a) and after crossing change (b).

Another similar, but probably weaker invariant is the cutting number: we cut one component of a link L, but without gluing it again and repeat the same procedure until we obtain a split link.

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Fig. 1.97

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(a) The link .8 1 before (a) and after crossing change (b).

Definition 1.88. The minimum number of component breaks necessary to obtain a split link is the cutting number of a link. Cutting number is uninteresting for rational links– they are 2component links, so their cutting number is always 1. However, it becomes more interesting for links with more than two components. 1.16

Borromean links

No two elements interlock, but all three do interlock. A three-component link with this property was named Borromean rings after Borromeos, an Italian family from the Renaissance, that used them as their family crest symbolizing the value of collaboration and unity. B. Lindstr¨om and H.O. Zetterstr¨om (1991) proved that Borromean circles are impossible: Borromean rings can not be constructed from three flat circles, but can be constructed from three triangles. The Australian sculptor J. Robinson assembled three flat hollow triangles to form a structure (called Intuition), topologically equivalent to Borromean rings. A cardboard model of Intuition collapses under its own weight, to form a planar pattern. P. Cromwell recognized the same construction in a picture-stone from Gotland (1995), and H.S.M. Coxeter (1994) considered these and other symmetric combinations of three and four hollow linked triangles. In geometry, Borromean rings appear in a regular octahedron, in Venn diagrams, in DNA, and in other various areas (Fig. 1.98).

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Fig. 1.98

Borromean rings.

In knot theory Borromean rings are the foremost examples of links having two remarkable properties: three mutually disjoint simple closed curves form a link, yet no two curves are linked. Hence, if any one curve is cut, the other two are free to separate. In the case of 3-component links these two properties are inseparable: one follows from the other. In the case of ncomponent links (n ≥ 3), n-Borromean links can be defined as n-component non-trivial links such that any two components form a trivial link. Among them, those with at least one non-trivial sublink, for which we will keep the name Borromean links, will be distinguished from Brunnian links in which every sublink is trivial (Liang and Mislow, 1994c). It seems surprising that besides the Borromean rings, represented by the link 632 in Rolfsen notation, no other link with the properties mentioned above can be found in link tables (Rolfsen, 1976; Adams, 1994). The reason for this is very simple: all existing knot tables contain only links with at most 9 crossings. In fact, an infinite number of n-Borromean or n-Brunnian links exist, and they can be derived as infinite series. The first infinite series of 3-component links, beginning with the Borromean rings, was discovered by P.G. Tait (1876/77b). Their geometrical equivalent is a regular octahedron for k = 1, and (3k)-gonal antiprisms for k ≥ 2. Their corresponding alternating links are achiral 3-Borromean links (Fig. 1.99). If we relax the condition that every two components do intersect, an infinite number of “fractal” Borromean links can be derived from each nBorromean link. The construction is simple: it is enough to surround an even number of the appropriately chosen crossing points of any two components by circles (Fig. 1.100). However, our consideration will be restricted to n-Borromean links such that each two components intersect.

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Fig. 1.99

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Tait’s series of Borromean rings.

Fig. 1.100

“Fractal” Borromean rings.

Another infinite series of 3-Borromean links, starting again from Borromean rings is derived from the family of 2-component links (4n − 2)21 (221 , 621 , 1021 . . .) by introducing the third component: a circle intersecting opposite bigons (Fig. 1.101a). In a similar way, from the family of 2-component links (2n)21 (221 , 421 , 621 . . .) we derive an infinite series of 3-component Borromean links without bigons (Fig. 1.101b), that can be used for creating new series of Borromean links. In a self-crossing point of the oriented component, we introduce an even chain of bigons in the appropriate position (Fig. 1.101c). Notice that the first series of Borromean links with bigons (Fig. 1.101a) can also be derived from Borromean rings by introducing identical even chains of bigons in crossing-points of two different components. Therefore, we can first get different infinite series of n-Borromean links without bigons, and then introduce bigons in a way which preserves the Borromean property.

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Fig. 1.101 Infinite series of Borromean rings: (a) with bigons; (b) without bigons; (c) after introducing new chains of bigons.

Tessellations of an (2n + 1)-gonal prism, such that to every ring of the corresponding graph we add “left” or “right” diagonals (Fig. 1.102), yield new infinite series. Tessellations and their corresponding links are denoted by symbols (2n + 1, k), where all compositions of the number k = (2n + 1)l − 2, are denoted by k, so that every composition is identified with its inverse. From a tessellation with k rings satisfying the properties stated above we obtain 2k − 2 + 2[ 2k ]− 1 different (2n+ 1)-Borromean links without the Brunnian property. All their components are equivalent, and bigons can be introduced in the same way as before. The same method applied on “centered” rectangular tessellations gives series of (2n + 1)-Borromean links (Fig. 1.103). We can construct Borromean links with an even number of components and without the Brunnian property. C. Liang and K. Mislow (1994c) proposed two methods for the construction of n-Borromean links with at least one non-trivial sublink, but they both result in n-Borromean links with

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Fig. 1.102 prism.

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The first infinite series of Borromean links derived from a (2n + 1)-gonal

some non-intersecting components (n ≥ 4). In the first method, involving duplication of one or more rings, the duplicate rings are interchangeable by a continuous deformation. For example, by duplicating one ring in Borromean rings, we obtain a 4-Borromean link, and continuing in the same manner, n-Borromean links (n = 5, 6, 7, . . .) (Fig. 1.104a). Series depend on the choice of duplicate rings. The following example illustrates the method, similar to the one producing “fractal” Borromean rings, which can be used for constructing Borromean links with an even number of components: in the link 421 (4) two crossing points are surrounded by non-intersecting circles (Fig. 1.104b). Adding new non-intersecting circles, Borromean links with an even number of components are obtained. One open question remains: are they exist (2n)-Borromean links with no non-intersecting components, and with all components that are equivalent. n-component links (n ≥ 4) without non-trivial sublinks were described by H. Brunn (1892) (Fig. 1.105).

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Fig. 1.103 prism.

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The other infinite series of Borromean links derived from a (2n + 1)-gonal

For a link with ordered components, given by Conway symbol, Dowker code, or P -data, the LinKnot function fBreakComp calculates P -data of the link with k th component cut. The function BreakCoAll gives all KLs obtained by cutting components of a KL given by its Conway symbol, Dowker code, or P -data. The output is the list of KLs obtained by cutting each of components, where for a split link the output is {0}. The function CuttNo computes the cutting number of a link given by its Conway symbol, Dowker code, or P -data. The functions GetPdatabyTracking and fFindCon can be used for recognition of KLs coming from different sciences or ornamental knotwork. From drawings of KLs we can find their Conway symbols. Using the function GetPdatabyTracking we obtain as output P -data of an input KL drawing. For a KL given by its Dowker code, or P -data, the function fBasicPoly recognizes its corresponding basic polyhedron, and for every alternating KL with at most 12 crossings given by its Dowker code with

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Fig. 1.104

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(a) The first and (b) second Liang-Mislow construction.

signs, or by P -data, the function fFindCon gives its Conway symbol. For example, Tait’s series (Fig. 1.101a) is the family .(2k + 1) : (2k + 1) 0, (k ≥ 1). The first link in the series of Borromean links without bigons from Figure 1.100b is the basic polyhedron 1312∗, from which originates the family of Borromean links with bigons 1312∗ .(2k + 1) 0, (k ≥ 0) (Fig. 1.101c). Their cutting number 1 can be checked using the function CuttNo. Definition 1.89. A torus knot or link [m, n] is a simple closed curve on the torus which wraps around m times meridianally and n times longitudinally. Detailed description of torus knots is given by Murasugi (1996), in the Chapter 7. If the integers m, n are relatively prime, the result is a torus knot; otherwise, it is a torus link. For a given torus knot or link [m, n], the function fTorusKL calculates its P -data, braid word, minimal number of crossings, unknotting number, number of components, bridge number, Alexander polynomial and (Murasugi) signature. An infinite series of torus links will be obtained for GCD(m, n) = 3, where GCD(m, n) is the greatest common divisor for m and n. All of them are non-alternating links derived from basic polyhedra which are m-gonal antiprisms (m ≥ 3). Making them alternating, we obtain an infinite series of Borromean links. The main difference between “real” and “mathematical” KLs is that the first are open-ended. We can take closed mathematical KLs and turn them into real KLs by cutting them, fix endpoints, and compute the number of different (non-isomorphic) classes of obtained real KLs. For a given projection of a KL given by its Conway symbol, Dowker

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Fig. 1.105

Brunnean links.

code, or P -data, the function fCuttRealKL computes the number of “real” cuttings, i.e., the number of cuttings with a different cutting point in the projection. An output is the number of different “real” cutting classes with preserved signs, or with preserved or reversed signs.

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Chapter 2

Recognition and Generation of Knots and Links

2.1

Recognition of KLs

According to D.J.A. Welsh (1993), two fundamental algebraic problems about KLs are: (1) the unknotting problem– how to decide whether a KL is really knotted (linked); (2) the recognition problem– decide if two KLs are ambient isotopic. These two problems are almost inseparable and closely related with the problem of minimizing the number of crossings of KLs. In the case of rational KLs these problems were solved in a very simple, elegant and fast way (in the sense of algorithm complexity), using continued fractions. Outside the rational world various problems arise. For the unknotting problem (and the unknotting number problem) we propose a finite algorithm based on Bernhard-Jablan Conjecture. Trying to solve the recognition problem, we implemented an algorithm for recognizing alternating KLs given in Conway notation. The LinKnot function SameAltConKL for two given alternating KLs computes their minimal Dowker codes and compares them. In case they are equal, the result is 1, otherwise 0. Before we begin derivation of KLs from other worlds: stellar (or prismatic), arborescent stellar, arborescent generalized, and polyhedral, we will consider some invariants, mostly polynomial, which will enable us to distinguish and recognize KLs. Those invariants can be used for eliminating duplicates that can occur in KL derivation. R. Fox defined an elementary invariant of KLs, called Fox’s 3-coloring (Crowel and Fox, 1965). Given an oriented KL, denote each oriented arc 195

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Fig. 2.1

Relations in a crossing point.

connecting two successive undercrossings by xi (i = 1, 2, . . . , m). Oriented arcs xi are called generators1 (of the KL group). To each generator we assign one color from a set of 3 colors. Definition 2.1. KL diagram is 3-colorable if at each crossing all generators have the same color, or they all have different colors. Theorem 2.1. If a link L has at least one 3-colorable diagram, then each of its diagrams is 3-colorable. The number of 3-colorings is an invariant of link isotopy (Livingston, 1993, page 33; Manturov, 2004, Theorem 3.5). Proof. Consider diagrams D and D′ related by a Reidemeister move. Denote colors by 0,1, and 2. For each coloring, all edges outside the shown region are colored identically in D and D′ . In the case of Ω1 only one color can be used, so desired one-to-one correspondence between colorings is evident (Fig. 2.2a). The same holds for “one color” cases of Ω2 and Ω3 . The invariance under Ω2 and one-to-one correspondence between colorings is illustrated in Fig. 2.2b. In the case of Ω3 , the invariance is illustrated in Fig. 2.2c. In this figure are given four 3-colorings, and all the other can be obtained from them by permutations of colors. To each 3-coloring of D corresponds exactly one 3-coloring of D′ obtained from D by applying a Reidemeister move, so if a link L has at least 1 Justification for the name generators will be given later, when introducing the group of a KL (page 201).

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one 3-colorable diagram, all its diagrams are 3-colorable and the number of 3-colorings is invariant under Reidemeister moves. Three-colorability is one of the simplest methods used in knot theory to distinguish knots. For example, a trefoil can be distinguished from the unknot or figure-eight knot, because the first is three-colorable, and two others are not. Definition 2.2. KL is perfectly 3-colorable if generators at each crossing all have different colors.

Fig. 2.2

Invariance of 3-colorings under (a) Ω1 ; (b) Ω2 ; (c) Ω3 .

The 3-coloring can be generalized to k-coloring (k > 3). Definition 2.3. A link diagram is k-colored if every generator is labelled by one of the numbers 0,1,2,. . .,k − 1 in such a way that the sum of the labels of the undercrossings (incoming and outgoing generator) is equal to twice the label of the overcrossing (passing generator) modulo k in every crossing. In other words, if iV , oV , and pV are the labels of the incoming, outgoing, and passing generator in the crossing V, then in every crossing we

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have the relation iV + oV = 2pV (mod k). Theorem 2.2. (Labelling theorem) If a diagram of a KL can be labelled mod k, then every diagram of that KL can be labelled mod k. The number of k-colorings is an invariant of link isotopy. (Livingston, 1993; Przytycki, 2004, 2006). If colk (L) denotes the number of k colorings of a link L, then colk (L1 )colk (L2 ) = k colk (L1 #L2 ) (Przytycki, 2006, Lemma 2.2). Definition 2.4. The set of different numbers of colors which can be used for coloring L is called coloring number set of L. Fox’s k-colorings are special cases of quandles introduced by Joyce and Matveev (Joyce, 1982; Matveev, 1982). A detailed discussion of quandles is given in the book Knot Theory (Chapter 5) by V. Manturov (2004). Definition 2.5. A set Q with a binary operation ◦ satisfying the rules: (1) a ◦ a = a for all elements of Q; (2) for every a,b ∈ Q there is a unique element x ∈ Q such that x ◦ a = b; (3) (a ◦ b) ◦ c = (a ◦ c) ◦ (b ◦ c) for all a, b, c ∈ Q. is called a quandle. In any quandle Q, the inverse operation for ◦ is denoted by /, meaning that element b/a is defined as the unique solution of the equation x ◦ a = b. In this case, by replacing the second rule from the definition of quandle by the rule (a ◦ b)/b = a that holds for all elements of Q and treating Q as a set with two binary operations ◦ and /, we obtain the equivalent definition of the quandle. For all a, b, c ∈ Q the following identities hold: (1) (a ◦ b)/c = (a/c) ◦ (b/c); (2) (a/b) ◦ c = (a ◦ c)/(b ◦ c). Theorem 2.3. To any group G it is possible to associate a quandle with Q = G and the quandle operations defined using conjugation in group: a ◦ b = b−1 ab and a/b = bab−1 . Proof. In order to prove this theorem it is sufficient to show that the rules given in the definition of quandle are satisfied after replacements a ◦ b = b−1 ab and a/b = bab−1 . We will use the equivalent definition of quandle, i.e., (a ◦ b)/b = a as the second rule. The following identities:

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a ◦ a = a−1 aa = a; (a ◦ b)/b = (b−1 ab)/b = bb−1 abb−1 = a; (a ◦ b) ◦ c = (b−1 ab) ◦ c = c−1 b−1 abc = c−1 b−1 cc−1 acc−1 bc = (c−1 ac) ◦ −1 (c bc) = (a ◦ c) ◦ (b ◦ c), hold for any a, b, c ∈ G, where G is arbitrary group. To relate a quandle with KLs we define the rule of coloring (Fig. 2.3a). Theorem 2.4. If some diagram of a KL can be colored by elements of a quandle Q, then this holds for every diagram of that KL. The number of colorings by elements of any quandle is a link invariant (Manturov, 2004, Proposition 5.1). We need to show the invariance of quandle colorings under Reidemeister moves and one-to-one correspondence between colorings. The sketch of the proof is given in Fig. 2.3b,c,d.

Fig. 2.3

(a) Rule of coloring and invariance of quandle under (b) Ω1 ; (c) Ω2 ; (d) Ω3 .

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Definition 2.6. Two non-isotopic knots are equivalent if one can be obtained from the other by changing both the orientation of the ambient space and that of the knot. According to this definition, “left” and “right” trefoil are equivalent and have isomorphic quandles. Theorem 2.5. The knot quandle is a complete knot invariant (in the sense of Definition 2.6) (Manturov, 2004). One of the most important examples is the Alexander quandle, i.e., the way of obtaining Alexander polynomial (page 211) as a quandle. Let A be a free module over Laurent polynomial ring (with respect to a variable t) (Definition 2.10). Then A is a quandle defined by the operations a ◦ b = ta + (1 − t)b, 1 1 a + (1 − b). t t A knot or link given by its Conway symbol, Dowker code, or P -data is an input for the LinKnot function fGenerators, which computes generators with the list of corresponding signs of crossings. The result is a list of ordered triples containing incoming, outgoing, and passing generator (I O P = Incoming-Outgoing-Passing, Fig. 2.1) for each crossing, divided according to the components of the KL. a/b =

Fig. 2.4 (a) The link 2, 2, 2 with denoted generators and its colorings with (b) k = 3(2p − 1) colors (p = 1, 2, 3, . . .); (c) k = 2p (p = 1, 2, 3, . . .) colors.

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The function fColTest has the same input as fGenerators with the additional number k (k ≥ 3) denoting the number of colors used for KL coloring. The result is the list of generators, list of their labelling, and the list of generator colors. For example, the link 2, 2, 2 (or 631 ) is k-colorable if k = 0 (mod 2) or k = 0 (mod 3), so its color set is {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, . . .} (Fig. 2.4), where the perfect colorings are obtained for k = 0 (mod 3). 2.1.1

Group of KL

Every finitely-generated group can be given by a presentation: set of generators x1 , . . . , xm and their relations r1 , . . . , rn . For example, a cyclic group Cn is given by the set of generators {x1 } and a single relation xn1 = e, where n is the order of the group Cn , and e is the identity. A dihedral group is given by the set of generators {x1 , x2 } satisfying the relations xn1 = x22 = (x1 x2 )2 = e, or by the set of generators {y1 , y2 } satisfying the relations y12 = y22 = (y1 y2 )n = e. Two presentations are called isomorphic if one is algebraically equivalent to the other (see, e.g., Coxeter and Moser, 1980). From the first presentation of the dihedral group, by the substitution y1 = x1 x2 we obtain the second. Presentations can be reduced in order to obtain minimal presentations, i.e., presentations with a minimum number of generators. For example, the presentation (x1 , x2 , x3 :

x31 = x22 = x23 = x1 x2 x−1 3 = e)

can be reduced to (x1 , x2 :

x31 = x22 = (x1 x2 )2 = e)

which is a minimal presentation of the group D3 . A Wirtinger presentation of a link group G(L) (or link complement fundamental group) can be defined for every knot or link diagram L. The group G(L) has a presentation G(L) = (x1 , x2 , . . . , xm : r1 , . . . rn ), where x1 , . . . , xm are generators, r1 , . . . , rn are relations satisfied by generators in crossing points, and n is a number of crossings. The rule for making relations is: for each crossing write down the corresponding generator with the exponent 1 if the arc is entering the crossing and −1 if it is leaving it. For all crossings do that in the same cyclic order (“left” or “right”). If xi is an incoming generator, xo is an outgoing, and xp is a passing generator, −1 we have the relation xi xp x−1 o xp = e (in a “right” cycle order) (Fig. 2.1). Beginning from a generator other then xi we obtain conjugate relation. An m-generator presentation obtained in this way can always be reduced to a minimal presentation. One group can have several isomorphic minimal

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Fig. 2.5

Generators of the cinquefoil knot 5 (or 51 ).

presentations, as we noticed in the case of dihedral group. For example, the group of the cinquefoil knot 5 in Conway notation (or 51 in the classical notation) (Fig. 2.5) has the following presentation G(5) = (x1 , x2 , x3 , x4 , x5 :

−1 −1 −1 x3 x1 x−1 4 x1 = e, x1 x4 x2 x4 = e,

−1 −1 −1 −1 −1 x4 x2 x−1 5 x2 = e, x2 x5 x3 x5 = e, x5 x3 x1 x3 = e)

that after a series of reductions results in the minimal presentation G(5) ≃ (a, b :

a5 = b2 ).

Theorem 2.6. Group of a link G(L) is an invariant of ambient isotopy. This means that if L1 and L2 are ambient isotopic KLs, their corresponding groups G(L1 ) and G(L2 ) will be isomorphic. The inverse statement does not hold: two different KLs can have isomorphic KL-groups (see, e.g., Crowell and Fox, 1965; Rolfsen 1976). For example, a granny knot 3#3 and a square knot 3#(−3) have isomorphic groups, with the same presentation (x, y, z : z −1 xz = xzx−1 , z −1 yz = yzy −1 ). For some KL families one can explicitly describe their groups (or presentations). For example, for every knot from the family (2k + 1) the knot group is G((2k + 1)) = (a, b : a2 b2k+1 ); for every link from the family (2k) it is G((2k)) = (a, b : abk a−1 b−k ),(k ≥ 1), etc. The program SnapPea by J. Weeks can be used for computing fundamental groups and many other topological properties of KLs (such

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as hyperbolic volume, symmetry group, Dirichlet domain, etc.). A conversion from P -data to data in SnapPea format or vice versa is provided by Knot2000 (K2K) functions SnapPeaDataFromPLData and PLDataFromSnapPeaData. Arf invariant, introduced independently by C. Arf and R.A. Robertello (1965), is defined for knots (not links) and takes values in the group Z2 . It is defined using Seifert surface of a knot (Theorem 1.37). For every oriented knot K we can obtain an orientable connected surface that spans K. Let l be a band that is a part of the Seifert surface of the knot K. Definition 2.7. Knots K and K ′ are Arf equivalent if K is obtained from K ′ by twisting a band l by two full turns. This means that we add to diagram of l (or delete it from l) a chain of three bigons that has four half-twists. Theorem 2.7. Each knot is either Arf equivalent to the unknot or to the trefoil. Arf invariant takes value 0 for unknot, and 1 for a trefoil (where “left” and “right” trefoils are Arf equivalent). Proof. The sketch of the proof (Kauffman, 1983; Adams, 1994, page 223; Manturov, 2004) is the following: each Seifert surface can be thought of as a disc with several bands attached to its boundary. Each band can be twisted and bands can be knotted. The number of half-turn twists for each band can be taken to be zero or one according to Arf equivalence. If we put an orientation on the boundary of Seifert surface, then the two edges of each band are always oppositely oriented. The Arf equivalence allows to unknot bands: passing of one band through the other is also Arf equivalence. Definition 2.8. The passing of one band with oriented edges through one another is called a pass-move (Fig. 2.6a). After unknotting bands, we can continue with the reduction of the obtained surface. Since the original Seifert surface was orientable, each band has an even number of half-twists, so we can lower the number of half twists in each band until two or zero half-twists remain. Bands with two half twists can be replaced by a curl. Every band must have another end of a band between its two ends on the boundary of a disk, so if one of the ends of each of two distinct bands l1 and l2 lie between the ends of a third band l3 on the edge of the disk, we can slide the end of l1 along one edge

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(a) A pass-move; (b) four types of pieces obtained by cutting Seifert surface.

of l2 to move it outside the two ends of l3 . Repeating these steps, we can make sure that there is at most one end of a band between two ends of any single band on the edge of the disk. In particular, this means that the bands match up in pairs and there is an even number of them. The Seifert surface obtained can be cut into four types of pieces, each with two bands attached to it. The first two of them (Fig. 2.6b) are the spannings of an unknot, and the other two of a “left” and “right” trefoil, so they are pass equivalent. Since the unknot has a role of neutral element in knot composition, every knot K is pass equivalent to either unknot or a composition of trefoil knots. Moreover, a composition of two oppositely oriented trefoils is pass equivalent to an unknot, and each trefoil is pass equivalent to its mirror image, so we can eliminate pairs of trefoils and reduce number of trefoils modulo 2. Hence, the final result of reduction is a single trefoil or unknot. The proof that trefoil and the unknot are not pass equivalent is given by Kauffman (1983). Hence, each knot is Arf equivalent (or pass equivalent) to the unknot, or trefoil. As in the Seifert construction, let us apply to a crossing of K the operation of smoothing that transforms the knot K into a 2-component link L0 . Theorem 2.8. If K+ , K− , and L0 are projections that are identical outside the shown region (Fig. 2.7), the Arf invariants of two knots K+ and K− are related through the equation: a(K+ ) ≡ a(K− ) + lk(L0 )(mod Z2 ) where lk(L0 ) is the linking number of the link L0 (see Definition 1.32) (Adams, 1994; Manturov, 2004).

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Arf invariant can be computed from the Alexander polynomial ∆K (t) of a knot K. Arf (K) = 1 if ∆K (−1) = ±3 (mod 8), and 0 otherwise (Jones, 1985; Lickorish, 1997, Theorem 10.7). We conjecture that Arf invariant is a subfamily-dependent invariant, and that for every knot family a general formula for Arf invariant can be obtained. For example, for the family (2k − 1), Arf ((2p − 1)) = 1 if 2p − 1 = ±3 (mod 8), and 0 otherwise; for the family (2p + 1) (2q), Arf ((2p + 1) (2q)) = 1 if 4pq + 2q + 1 = ±3 (mod 8), and 0 otherwise; for the family (2p) (2q), Arf ((2p) (2q)) = 1 if 4pq + 1 = ±3 (mod 8), and 0 otherwise, etc.

Fig. 2.7

Diagrams of regions D+ , D− and D0 .

In addition to 3- and k-colorings and quandle colorings (pages 195-200), KLs can also be labelled with elements of a group G, which is usually a symmetric group Sn with the standard presentation {g1 , . . . , gn }

(gi gj )ni,j = E,

i, j = 1, . . . , n − 1

where ni,i = 1,

ni,i+1 = 3,

ni,j = 2 for i < j + 1,

and gi = (i i + 1). Definition 2.9. The labelling an oriented KL diagram with elements of a group G consists of assigning an element of G to each generator xi of the KL group (page 201), subject to the following two conditions: (1) Consistency Three generators (I, O, P ) = Incoming-Outgoing-Passing appear at each crossing V of the diagram. If their labels are group elements i, o, p, and σ is the sign of the crossing V , then the labels satisfy the relation pσ op−σ = i; (2) Generation The labels generate the group G.

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Once two out of three knot generators I, O, P are labelled, the label on the third is forced by the consistency condition. The following theorem holds for KL labeling: Theorem 2.9. If a diagram of a KL can be labelled with elements from a group G, then every diagram of the KL can be labelled with the elements from G, regardless of the choice of orientation (Livingston, 1993). To prove this theorem it is sufficient to check what happens to the labelling after performing each Reidemeister move. The use of labelling is a powerful tool for distinguishing KLs. For example, it was very efficiently used by M. Thistlethwaite for the computer derivation of knots. Among 12965 knots with n ≤ 13 crossings there were only 5639 different Alexander polynomials. Using labelling from all subgroups of S5 , enabled him to reduce the number of unresolved cases to about a thousand (Thistlethwaite, 1985). Elements g and g ′ in a group G are called conjugates if there is an element h ∈ G such that h−1 gh = g ′ . The conjugacy is an equivalence relation, that preserves the cyclic structure of a permutation group G: every element g and all its conjugates are represented as the products of permutation cycles in the same way. The relation of conjugacy induces partitions of G into equivalence classes. For example, in S5 there are seven conjugacy classes, that can be represented by (1), (1 2), (1 2 3), (1 2 3 4), (1 2 3 4 5), (1 2)(3 4), (1 2)(3 4 5). If a diagram of an oriented knot can be labelled with elements of a group G with the labels coming from a conjugacy class C of G, then every diagram of the same knot can be labelled with the elements from C. In the case of links, labels on each component of a labelled link belong to the same conjugacy class. Using consistency relationships, once a few labels are chosen, the rest are forced. In practice, we take a knot K and fix a group G that will be used for knot labelling. After labelling two knot generators in some crossing by two group generators x and y, the third generator in that crossing will be labelled by the consistency condition. Hence, each crossing determines a label on the next arc, forced by the labels that preceded it. Equations satisfied in G can be read from the labelling of arcs. In this way, the knot labelling problem is reduced to solving the equations obtained from the group G. C. Livingston (1993) proposed an interesting example of knots 4 2 (61 ) and 3, 3, −3 (946 ), that can not be distinguished using colorings, or by

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their Alexander polynomials, but are distinguishable by a suitably chosen labelling.

Fig. 2.8

The labelling of knot 3, 3, −3 using transpositions from S4 .

Let us consider knot 4 2 and its labelling using transpositions from S4 . By the choice of two labels x and y in any crossing of the knot 4 2, all the other labels are forced by the consistency relation. Since only two transpositions from S4 are not enough to generate S4 , we conclude that it is impossible to construct a labelling of knot 4 2 using transpositions from S4 . The labelling of the knot 3, 3, −3 using transpositions from S4 (Figure 2.8) proves that knots in question are different. 2.2

Polynomial invariants

Non-empirical recognition of KLs became possible after the introduction of polynomial invariants. The first of them, Alexander polynomial, was used by Alexander and Briggs (1926-27) to prove that knots with at most nine crossings claimed to be distinct in empirically obtained knot tables were actually distinct. K. Reidemeister completed the rigorous classification of knots with up to nine crossings in his book Knotentheorie published in 1932. For more than 40 years, Alexander polynomial remained the only polynomial invariant. The tangle approach was introduced by J. Conway in 1967, together with a new polynomial invariant, the Conway polynomial, based on a skein relation. J.W. Alexander knew about the skein relation, but J. Conway was the first to prove (in 1967) that it can be used for an axiomatic definition of the polynomial (Conway, 1970). A modification of the skein relation produced the HOMFLYPT polynomial (1985). Probably

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the most famous polynomials are Jones and Kauffman polynomial, since they established connections between knot theory and other branches of mathematics (algebra of operators, braid theory) and physics (statistical models and quantum groups). Despite of all these important achievements, there is one disappointing fact: every polynomial invariant sometimes fails, meaning that two (or more) different KLs may have equal polynomials. Even worse: some KLs that are really knotted are impossible to distinguish from the unknot by some polynomial invariants. For example, there is an infinite number of non-alternating knots with Alexander polynomial equal to one, and an infinite number of non-alternating links with a trivial Jones polynomial. The infinite series of non-trivial non-alternating 2-component links: 9∗ 3 : −1. − 1.2. − 1. − 1 : −3, 9∗ 5 1 2 : −1. − 1.2. − 1. − 1 : −5 − 1 − 2, 9∗ 5 1 4 1 2 : −1. − 1.2. − 1. − 1 : −5 − 1 − 4 − 1 − 2, 9∗ 5 1 4 1 4 1 2 : −1. − 1.2. − 1. − 1 : −5 − 1 − 4 − 1 − 4 − 1 − 2, etc.,

and 4-component links: 16370∗ . − 2 : −3. − 1 : . − 1. − 1 : . − 1.3 : −2, 16370∗ . − 2 : −5 1 2. − 1 : . − 1. − 1 : . − 1.5 1 2 : −2, 16370∗ . − 2 : −5 1 4 1 2. − 1 : . − 1. − 1 : . − 1.5 1 4 1 2 : −2, 16370∗ . − 2 : −5 1 4 1 4 1 2. − 1 : . − 1. − 1 : . − 1.5 1 4 1 4 1 2 : −2, etc.,

with a trivial Jones polynomial were recently discovered by S. Eliahou, L. Kauffman and M. Thistlethwaite (2003) (Fig. 2.9). For every polynomial we can define its coefficient of efficiency in distinguishing KLs: the number of different KLs it can recognize as different, divided by the number of all different KLs with a given number of crossings n. However, for non-alternating KLs, the exactness of the second number is impossible to prove using polynomials: it is possible that all existing polynomial invariants for two different KLs (e.g., mutants) coincide. This is more likely to happen than one might expect, because some polynomial invariants are only special cases of more general invariants. For example, Alexander polynomial can be obtained from Conway polynomial or Jones polynomial.

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Fig. 2.9 Links 9∗ 3 : −1. − 1.2. − 1. − 1 : −3 and 9∗ 5 1 2 : −1. − 1.2. − 1. − 1 : −5 − 1 − 2 with trivial Jones polynomial.

It is possible to construct infinite classes of different KLs that particular invariant can not distinguish. For example, all KLs obtained from a KL of the form p1 , p2 , . . . , pn by any permutation of tangles p1 , p2 , . . ., pn can not be distinguished by any polynomial. An interesting class of KLs are those with trivial Alexander and Conway polynomial. We assume that every link with trivial Alexander and Conway polynomial is a member of a family of links, with trivial Alexander and Conway polynomial. For n = 8: • the 4-component link 2, 2, −2, −2 is a member of the family 2k, 2k, −2k, −2k with trivial Alexander and Conway polynomial. For n = 9: • the 3-component link . − (2, 2) that has the non-minimal algebraic 10crossing representation (2, −2), (−2, 2), 2 is a member of the family (2k, −2k), (−2m, 2m), 2n with trivial Alexander and Conway polynomial. For n = 10:

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• the 3-component link (3, 2 1) − (2, 2) is a member of the family (2k + 1, 2k 1) − (2, 2); • the 3-component link 3, 2 1, −2, −2 is a member of the family 2k + 1, 2k 1, −2, −2; • the 3-component link 3, −2, 2 1, −2 is a member of the family 2k + 1, −2, 2k 1, −2; • the 4-component link (2, 2) − 2 (2, 2) that has the non-minimal 12crossing representation (2, 2−) 2 (2, 2−) is a member of the family (2, 2−) 2k (2, 2−); • the 4-component link . − (2, 2) : 2 that has the non-minimal 12-crossing algebraic representation (2, 2) − 1, 3, (2, −2) is a member of the family (2, 2) − 1, 2k + 1, (2, −2); • the 4-component link 103∗ − 1. − 1. − 1. − 1 :: . − 1 that has the nonminimal 12-crossing algebraic representation (2, −2), −2, 2, (2, −2) is a member of the family (2, −2), −2k, 2k, (2, −2) with trivial Alexander and Conway polynomial. Hence, for n ≤ 10 all links with trivial Alexander and Conway polynomial, except one, are members of the families with the same property. For the link 2 0. − 2. − 2 0.2 0 we have not succeeded to find its corresponding family of links with trivial Alexander and Conway polynomial. Let t1 ,t2 ,. . .,tk and −t1 ,−t2 ,. . .,−tk (k ≥ 2) be rational tangles not beginning nor ending with 1. All pretzel (stellar) links composed from them have trivial Alexander and Conway polynomial. Such links are: • 2, 2, −2, −2 for n = 8; • −3, 3, 2, −2 and −3, 2, 3, −2 for n = 10; • −4, 4, 2, −2, −4, 2, 4, −2, −3, 3, 3, −3, −3, 3, −3, 3, −2 2, 2 2, 2, −2, −2 2, 2, 2 2, −2, and 2, 2, 2, −2, −2, −2 for n = 12, etc. First knots with trivial Alexander and Conway polynomial appear for n = 11 crossings: famous Kinoshita-Terasaka mutants (Kinoshita-Terasaka knot . − (3, 2).2 and Conway knot . − (2, 3).2, that also can be written as .−(2 1, 2).2 0 and .−(2, 2 1).2 0, i.e., knots 11n34 and 11n42 in the Knotscape notation). Both of them have non-minimal 12-crossing algebraic representations: (3, −2), (2, −3), 2 and (3, −2), (−3, 2), 2. They are members of the families of knots with n = 4k + 2l + 1 crossings that have trivial Alexander and Conway polynomial, given by their non-minimal representations((2k + 1), −2k), (2k, −(2k + 1)), 2l and ((2k + 1), −2k), (−(2k + 1), 2k), 2l, respectively.

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Questions such as: is there any non-trivial knot with trivial Jones polynomial, or is there any non-trivial link with trivial Kauffman polynomial, are still open. The invention of Vassiliev’s invariants brought a new hope that finite-order invariants will be able to classify KLs. Till now, that hope is neither realized nor disproved. Definition 2.10. A Laurent polynomial in some (commutative) variables x1 , x2 ,. . ., xn is a polynomial in xi ’s and their inverses x−1 i . Under the term “polynomial” usually we mean Laurent polynomial. All polynomial invariants are based on a KL surgery and skein relations (Conway, 1970). The first surgery is a crossing change that turns an overcrossing to an undercrossing or vice versa (Fig. 1.58a). It is also known as a flip. Conway’s second surgical operation, called smoothing, is the same as an ∞-unknotting (unlinking) operation (Fig. 1.69). Both operations were known and often used by knot theorists well before Conway, and the second is crucial in obtaining “mirror curves” (see Subsection 3.2.3). Conway’s main contribution was showing that skein relation can serve as a basis for defining a new polynomial invariant– the Conway polynomial. To every KL diagram D, polynomial ∇(D) is associated. It satisfies the following identities: (1) Normalization The polynomial of the unknot (unlink) is equal to 1 ∇( ) = 1; (2) Conway’s skein relation Suppose that three KL diagrams D+ , D− , D0 are identical outside the neighborhood of one crossing, and that in the crossing they have forms as indicated in Fig. 2.7. Then the Laurent polynomials of the three KLs are related as follows: ∇(D+ ) − ∇(D− ) = x∇(D0 ). √ Substituting x in the Conway polynomial by t − √1t yields the Alexander polynomial. HOMFLYPT is an acronym for the eight researchers working in four different independent groups, who discovered this polynomial almost at the same time (in 1984). HOMFLYPT comes from the names: Hoste, Onceanu, Millett, Freyd, Lickorish, and Yetter, J. Przytycki and P. Traczyk.2 2 The

first six of them published their papers together in 1985 (Freyd, Yetter, Hoste, Lickorish, Millett, Onceanu, 1985), and J. Przytycki and P. Traczyk published their work somewhat later, because it didn’t arrive by mail on time.

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The simplest way to define the HOMFLYPT polynomial P (l, m) in two variables l and m is to use the Conway rules with P in the place of ∇ and with the modified skein relations lP (D+ ) − mP (D− ) = P (D0 ). The HOMFLYPT polynomial (or two-variable Jones polynomial) is more sensitive than the Conway polynomial, but it is still an incomplete invariant. For example, for two different eleven-crossing mutant knots .2.(3,2) and .2.(2,3), all their polynomial invariants are equal (Fig. 2.10). In general, mutant KLs can not be distinguished by any polynomial invariant. However, any pair of alternating KLs (including mutants) can easily be distinguished using the LinKnot function SameAltConKL that compares minimal Dowker codes. For example, the minimal Dowker code of the knot .2.(3,2) is {{11}, {6, 8, −12, 2, 20, −18, −4, −10, 22, −14, 16}}, and the minimal Dowker code of the knot .2.(2,3) is {{11}, {6, 8, −12, 2, −18, 16, −4, 20, −22, 14, −10}}.

Fig. 2.10

Knots (a) .2.(3,2); (b) .2.(2,3)

In 1984 V. Jones discovered a new polynomial invariant of KLs. He got the idea from operator algebras, an area of mathematics previously unrelated to knot theory. In his work on von Neumann algebras he observed the relations strongly resembling to the algebraic expression of the relations in braid group (see page 160, Theorem 1.56). The significance of this famous polynomial extends beyond the scope of knot theory as it is applicable to the

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other branches of mathematics (operator algebra, braid theory) and physics (statistical models and quantum groups) (Jones, 2005). In the case of the Jones polynomial the first Conway rule V ( ) = 1 remains unchanged, and the Conway skein relation is replaced by the modified skein relation √ 1 1 V (D+ ) − tV (D− ) = ( t − √ )V (D0 ). t t The Jones polynomial can be obtained from the HOMFLYPT polynomial 1 1 by substituting l by it−1 and m by i(t− 2 − t 2 ). Conway’s approach is generalized by J. Przytycki and P. Traczyk (Przytycki and Traczyk, 1987; 1987a): Definition 2.11. An algebra A consisting from a fixed sequence of elements (constants) a1 , a2 , ..., an and two binary operations ◦ and / is called Conway algebra if the following conditions are satisfied: 1) Initial conditions: • an /an+1 = an ; • an ◦ an+1 = an ; 2) Transposition properties: • (a/b)/(c/d) = (a/c)/(b/d); • (a/b) ◦ (c/d) = (a ◦ c)/(b ◦ d); • (a ◦ b) ◦ (c ◦ d) = (a ◦ c) ◦ (b ◦ d); 3) Inverse operation properties: • (a ◦ b)/b = a; • (a/b) ◦ b = a For each link diagram L let us construct W (L) as follows: denote by an element of A corresponding to the n-component trivial link, and require for any Conway triple L+ , L− , L0 the following Conway skein relations to be satisfied: W (L+ ) = W (L− ) ◦ W (L0 ), W (L+ ) = W (L0 ) ◦ W (L− ).

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Theorem 2.10. For each Conway algebra, there exists a unique function W (L) on link diagrams that has a value an on the n-component unlink diagrams and satisfies Conway skein relations. This function is invariant on oriented links (Manturov, 2004, Theorem 5.3). Let A be an arbitrary commutative ring with the unit element, a1 ∈ A, and with invertible elements α, β. Define binary operations ◦ and / with: x ◦ y = αx + βy, x/y = α−1 x − α−1 βy, satisfying the initial conditions: an = (β −1 (1 − α))

n−1

a1 ,

n ≥ 1.

The following proposition holds: Proposition 2.1. For any choice of invertible elements α, β and element a1 , the ring A with operations ◦ and / defined above, and with the initial conditions an is a Conway algebra. Conway algebra provides different realizations: • let A be a ring of polynomials of variable x with integer coefficients. Conway polynomial is obtained for α = 1, β = x, a1 = 1 (page 211); • let A be the ring of Laurent polynomials with integer coefficients in l, m. HOMFLYPT polynomial is obtained for α = − ml , β = 1l , a1 = 1 (page 211); √ • let A be the ring of Laurent polynomials in t. Jones polynomial is √ 1 2 √ obtained for α = t , β = t( t − t ), a1 = 1 (page 212). L. Kauffman defined bracket polynomial in the summer of 1985. The Kauffman bracket polynomial is the Laurent polynomial uniquely defined by the following axioms: (1) (2) (3)

h i = 1; hL ∪ i = (−a2 − a−2 )hLi; h0i = ah1i + a−1 hHi.

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The third rule is the fundamental one. It is analogous to the Conway skein relation (Fig. 2.7), but significantly different (Fig. 2.11), so KLs considered by Kauffman are not oriented. Consequently, there is only one type of crossing (not two of in Conway skein relation), and two possible ways of smoothing. In order to prove that all the mentioned polynomials are KL invariants, we need to show that they are invariant under Reidemeister moves. The interested reader can find proofs in the books by L. Kauffman (1987a), C. Adams (1994), K. Murasugi (1996), A. Kawauchi (1996), A. Sossinsky (2002), and J. Przytycki (2004). Figure 2.12 gives a sketch of the proof for the invariance of the bracket polynomial with regard to the second and third Reidemeister moves. In the case of Kauffman bracket polynomial, in order to have the invariance with regard to the first Reidemeister move we need an additional “correction factor” given by the equality X(L) = (−a)−3w(L) h|L|i, where w(L) is a writhe of L. To define X(L) and w(L) we have to orient the diagram L. The writhe w(L) is defined to be the sum of signs over all crossings of L. The original Jones polynomial can be obtained from X(L) by replacing 1 each a in X(L) by t− 4 .

Fig. 2.11

Kauffman skein relations with labelled regions.

We will try to give an elementary explanation of the connection between the Kauffman bracket polynomial and statistical models in physics, that is, theoretical models of regular atomic structures that can adopt a variety of states, each state being determined by the distribution of spins in atoms. In the model with two spins, they can be represented by arrows associated to every atom, pointing up and down, respectively. Four regions of the projection plane come together at each crossing. Rotate the overstrand counterclockwise, passing over two of the regions, label them by A, and label the remaining two regions by B. A two-sided mirror placed in the crossing

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Fig. 2.12 Invariance of the Kauffman bracket polynomial with regard to the second and third Reidemeister move.

can have two positions: it can be an A-mirror or B-mirror, inducing an A-split or B-split (Fig. 2.11b,c). The analogy with “mirror curves” present in Celtic, Tamil or Tchokwe knot-art is complete (see Chapter 3). A knot or link projection with n crossings gives 2n possible distributions of mirrors, this means, 2n possible states of the KL. Every state is a split link with unknotted components. If |S| is the number of components in a state L′ , the bracket polynomial of L′ is (−a2 + a−2 )|S|−1 . The total contribution to the bracket polynomial by the state S is aA(S) a−B(S) (−a2 − a−2 )|S|−1 , where A(S) is the number of A-splits, and B(S) the number of B-splits in S. The bracket polynomial of a link L will depend on the bracket polynomials of all the possible states of the projection of L. It will be a sum over all the possible states given by the formula X aA(S) a−B(S) (−a2 − a−2 )|S|−1 . S

After defining Kauffman bracket, we can prove its invariance under the second and third Reidemeister moves and show that it is the only bracket polynomial satisfying the given set of axioms. Figure 2.13 illustrates all the states of a trefoil knot. Its bracket polynomial will be a7 − a3 − a−5 . By substituting the coefficients in the axioms for the Kauffman bracket polynomial by the following terms, we obtain the so called square bracket polynomial. We get the function, well defined on link diagrams, but not invariant under Reidemeister moves. However, this function is worth studying on graphs. It satisfies the following axioms:

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Fig. 2.13

(1) (2) (3)

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All the states of a trefoil knot.

1

[ ] = q 2 ; 1 [L ∪ ] = q 2 [L]; − 12 [0] = q v[1] + [H].

A square bracket polynomial is not necessarily an invariant of KLs, but it is related to the dichromatic polynomial ZG (q, v) corresponding to the graph of a KL (page 24). The dichromatic polynomial is defined by three axioms: (1) Z(•) = q (2) Z(•G) = qZ(G) (3) Z( –•— • ) = Z( –• • –• ) ) + vZ( The first rule is just an initial condition for a graph with a single vertex. According to the second rule, adding new isolated vertex to a graph, causes the polynomial of the graph to be multiplied by q. The third rule says that if we pick a particular edge of a graph G, then the polynomial for G is obtained by adding the polynomial of the graph with that edge deleted to v times the polynomial of the graph with that edge collapsed down to a single vertex (see, e.g., Adams, 1994). The square bracket polynomial [L(G)] and the dichromatic polynomial ZG (q, v) are connected by the relation: n

ZG (q, v) = q 2 [L(G)] where n is the number of vertices of the graph G.

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LinKnot function fGraphKL produces the graph of a KL given by its Conway symbol, Dowker code, or P -data. The graph is given as a list of unordered pairs, followed by a drawing of the graph. For every graph obtained you can calculate its dichromatic polynomial ZG (q, v). A coloring of a graph G is an assignment of one of possible z colors to each vertex of G such that no two adjacent vertices have the same color. Definition 2.12. The chromatic polynomial of a graph G is a function of the graph G and of the number of colors z that gives the number of different colorings of a graph G by z colors (Birkhoff, 1912; Birkhoff and Lewis, 1946). Chromatic polynomial of a graph of KL obtained using the LinKnot function fGraphKL can be computed with the Mathematica function ChromaticPolynomial. For example, the underlying graph of the trefoil knot is a triangle, so its chromatic polynomial is z(z − 1)(z − 2). The program Knot 2000 (K2K) provides functions for computing polynomial invariants of KLs from P -data or from a braid word. Depending on the type, the function SkeinPolynomial computes the Jones polynomial, HOMFLYPT polynomial (or two-variable Jones polynomial), Alexander polynomial by the Conway relation, and the Conway polynomial. In order to compute polynomial invariants from a braid word, or from the Burau representation for a braid word, you can use the Knot 2000 functions JonesPolynomialbyBraid, AlexanderPolynomialbyBurauRep, and ThreeParallelPolynomialInvariant. The LinKnot function fAlexPoly calculates the multi-variable Alexander polynomial of a KL from its Conway symbol using the Wirtinger presentation of KL. The functions KauffmanPolynomial and RedKauffmanPolynomial compute Kauffman polynomial and reduced Kauffman polynomial of a KL given by P data. Some new constructed polynomial invariants (e.g., quantum invariants, Khovanov polynomial obtained as a categorification of Jones polynomial (Shumakovitch, 2004), and Links-Gould invariant (De Wit, Kauffman and Links, 1999; De Wit, 2000)) are more sensitive then previous. For the possibility to compute them, special thanks are due to Dror Bar Natan and David De Wit. Khovanov polynomial, A2 invariant, and colored Jones polynomial can be computed with the functions Kh, A2, and ColouredJones from the program Knot Theory (http://katlas.math.toronto.edu/wiki/The Mathematica Package KnotTheory), which are included in LinKnot.

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Links-Gould invariant can be computed using the additional functions LinksGould and LinksGouldInv from the program Links-Gould Explorer written by David de Wit (http://www.maths.uq.edu.au/∼ddw/Links– GouldExplorer/Links–GouldExplorer.htm). The Linknot functions fCon wayToPD, fKnotscapeDowToPD, fDowkerToPD, and fPdataToPD are conversion functions from LinKnot to KnotTheory. They convert LinKnot input data (Conway symbols, Dowker codes in Knotscape format, Dowker codes, and P -data) to P D-notation used by D. Bar Natan. Those functions enable compatibility of the programs LinKnot and KnotTheory, so the programs can be used simultaneously. 2.3

Vassiliev invariants

Vassiliev invariants (or finite-order invariants) are the most general invariants, in the sense that many KL-invariants can be deduced from them. In a crossing change from an overcrossing to an undercrossing Vassiliev introduced an intermediate phase: a catastrophe, when one part of a KL cuts another part transversely. Besides overcrossings and undercrossings knots or links have double points, where KL cuts itself. KLs with double points are called singular. If we denote the set of all singular KLs by F , KLs without special crossings form a subset of F denoted by Σ0 . In the same sense, the remaining part of the set F can be divided into strata Σ1 , Σ2 , Σ3 ,. . . consisting of singular KLs with 1,2,3,. . . double points, respectively. With every crossing change from overcrossing to undercrossing or vice versa, a KL becomes singular, passing through an intermediate phase– a catastrophe. In the same way as for ordinary KLs, we can define an ambient isotopy for singular KLs: Definition 2.13. Two singular knots or links L1 and L2 are ambient isotopic if there is an orientation-preserving homeomorphism of ℜ3 that sends L1 to L2 preserving the arrows indicating orientation and the cyclic order of the branches with double points. In order to represent ambient isotopy of singular KLs by Reidemeister moves we need to introduce additional Reidemeister move Ω for special crossings (Fig. 2.14). As an equivalent of the skein relation for singular KLs we introduce the following relation that defines the derivative v ′ of the invariant v: v ′ (h i) = v(h!i) − v(h"i).

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Fig. 2.14

Reidemeister move Ω for special crossings.

From this relation we deduce, so called, one-term relation and four-term relation (Fig. 2.15a). The sketch of the proof of four-term relation is given in Fig. 2.15b. Every Vassiliev invariant assigns a numerical value to each KL (in particular to singular KLs). Definition 2.14. A singular KL invariant satisfying the preceding relation is called a Vassiliev invariant of order (at most n) (or of finite type) if for any singular KL with n + 1 vertices v(L) = 0. In other words, a Vassiliev invariant of order less or equal to n satisfies the skein relation for special KLs and vanishes for all KLs with n+1 double points or more. In particular, if v is of order at most n, but not of order n − 1, i.e., if there exists a singular KL with exactly n vertices for which v is non-zero, then v is called a Vassiliev invariant of order (exactly) n. The set Vn of all Vassiliev invariants of order at most n has a vector space structure with the inclusions V0 ⊂ V1 ⊂ V2 ⊂ V3 . . .. A Vassiliev invariant is essentially different from all previously considered KL invariants that associate a single mathematical entity (number, polynomial, etc.) to a KL. Instead of considering KLs individually, Vassiliev considered the space of all KLs, defining the whole space of invariants. The second defining relation is a choice of the beginning point in the space F , and after that we consider the position of that point with respect to the stratification in the space F : F ⊃ Σ0 ∪ Σ1 ∪ Σ2 . . .. Let v be a Vassiliev invariant of order 0. For any singular knot K, v(K) = V ( ), and there is essentially one Vassiliev invariant of order 0, constant for any singular knot. There is no Vassiliev invariant of order 1. The theory becomes non-trivial from the second order onward. Let us distinguish among the elements of V2 a specific invariant, denoted v0 , satisfying the relation v0 ( ) = 0, that is equal to 1 for a trefoil knot with two special vertices. The calculation of v0 for a trefoil knot 3 (Fig. 2.16) shows that v0 (3) = 1. Most of the KL invariants are not flexible and

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(a) The one-term and four-term relations; (b) the proof of four-term relation.

allow no improvement if they do not detect some property (e.g., Alexander polynomial can not distinguish left from and there is no way to remedy it. In the case of a single Vassiliev invariant (which is also not a complete invariant and can not tell all knots apart) we have a possibility to define another Vassiliev invariant that distinguishes them, and have an unlimited number of possibilities to do that. For Vassiliev invariants, the following Conjecture is still open: Conjecture 2.1. Finite-order invariants classify KLs; in other words, for each pair of non-equivalent knots or links L1 and L2 there is a natural number n ∈ N and an invariant v ∈ Vn such that v(L1 ) 6= v(L2 ) (Sossinsky, 2002, page 104). The importance of Vassiliev invariants lies in their universality: many other invariants, including coefficients of the Alexander, Jones and Kauffman polynomial can be expressed in terms of Vassiliev invariants or their limits. However, not all KL invariants can be derived from Vassiliev in-

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Fig. 2.16

The calculation of a V2 invariant for the trefoil knot.

variants: for example, the crossing number, unknotting (unlinking) number, braid index, or signature of a KL are not Vassiliev invariants, because they are not of finite order. Analogous to the way we considered states of a KL shadow with regard to overcrossings and undercrossings, we can deal with special projections by giving three possible choices for every vertex (overcrossing, undercrossing, or double vertex). For a KL shadow with n vertices, the number of possible variations is 3n , but this can be considerably reduced using symmetry of a KL in question. We can consider Vassiliev invariants in terms of purely geometrical combinatorial theory by introducing Gauss diagrams (or chord diagrams) of the order n (n = 1, 2, 3, . . .). Definition 2.15. A graph consisting of a circle and n chords joining 2n different points on it is called chord diagram of order n, or shortly n-diagram. Planar diagrams of oriented knots are characterized by their Gauss diagrams. A Gauss diagram of a classical knot projection is an oriented circle considered as the preimage of the immersed circle with chords connecting the preimages of each crossing. To mentain information about overcrossings and undercrossings, the chords are oriented toward the undercrossing point and signed in accordance with crossing signs. Recall that crossing changes do not change the value of an nth -order invariant of a KL with n double points. Independent on knotting, it depends only on the order in which double points appear when tracing KL. Chord diagrams of singular knots contain only chords corresponding to double points. In order to draw a Gauss diagram (or chord diagram) of a singular knot K with n double points we draw a circle and label on it in a cyclic order all double points visited while tracing our oriented knot K. At the end, join each pair of points having the same label by a chord.

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Definition 2.16. The resulting diagram is called a Gauss diagram (or chord diagram) of a singular knot K.

Fig. 2.17 The Gauss diagram of the special projection of a trefoil knot with three double points and one loop.

The Gauss diagram of the special projection of a trefoil knot with three double points and one loop is given in Fig. 2.17. In the same way as before, using the one-term relation, we can delete all double points with a loop and continue to work with proper (or reduced) special projections and corresponding proper (or reduced) Gauss diagrams. All non-singular knots have the same diagram– a circle without any chords. In general, many knots correspond to the same diagram. We can construct all the different Gauss diagrams Dn of order n (i.e., with n double points) (Fig. 2.18). Then we can rewrite the one-term relation and four-term relation (Fig. 2.15a) in the language of chord diagrams (Fig. 2.19a).

Fig. 2.18

Gauss diagrams Dn of order n ≤ 3.

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Fig. 2.19 (a) The one-term and four-term relations (Fig. 2.15a) in the language of chord diagrams; (b) the actuality tables for n ≤ 3.

Using the one-term relation, we can reduce the list Dn to one diagram for n = 2 and two diagrams for n = 3, by eliminating the diagrams d11 , d22 , d34 , d35 and reducing d33 to d21 . If we think of Gauss diagrams as vectors, from the four-term relation we obtain d31 = 2d32 , so dim(D1 ) = 0, dim(D2 ) = 1, dim(D3 ) = 1, etc. Recalling that dim(D0 ) = 1, and continuing the calculation for n = 3, 4, . . . , 9, we obtain the sequence 1, 0, 1, 1, 3, 4, 9, 14, 27, 44 of the dimensions of the spaces Dn for n = 0, 1, 2, . . . , 9 (Bar Natan, 1995). Denote the space of chord diagrams with n chords modulo the one-term and four-term relations by An . The main result of the combinatorial theory of Gauss diagrams and Vassiliev invariants is expressed in Kontsevich’s theorem: Theorem 2.11. Kontsevich’s Theorem The vector space Vn /(Vn−1 ) of nth -order Vassiliev invariants is isomorphic to the space An . As the final result, for every n we can obtain the actuality table An : the list of all independent Gauss diagrams with n chords (Fig. 2.19b). As the value of n increases, the amount of computer resources necessary for the computation of actuality table grows exponentially. Vassiliev invariants provide the universal language to talk about quan-

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tum invariants. For the study of quantum invariants we recommend the book Quantum Invariants by T. Ohtsuki (2001) and the papers by V.G. Drinfeld (1987) and V.G. Turaev (1994).

2.4

Experimenting with KLs

The program LinKnot can also be used for educational purposes. Experimenting with KLs and computing their polynomials, students can guess, recognize, or rediscover some well known KL properties. For example, to every alternating KL corresponds an Alexander polynomial with alternating signs. Hence, the Alexander polynomial of any alternating KL can not be equal to 1, but there are many non-trivial KLs with the Alexander and Conway polynomials equal to 1. In fact, there are families of KLs with this property (see page 209). Many different KLs have equal Alexander polynomials. Alexander polynomial does not distinguish a KL from its mirror image. Alexander polynomial satisfies the relation A(K1 #K2 ) = A(K1 )A(K2 ), and so do Jones and HOMFLYPT polynomials. Jones polynomials of a link L and its mirror image L satisfy the relation VL (t) = VL (t−1 ). The HOMFLYPT polynomial of L is obtained by substituting each l in the HOMFLYPT polynomial of L by l−1 . Jones polynomial (or HOMFLYPT) polynomial of a KL remains the same after changing orientations of KL components. The bracket polynomial of an achiral KL must be palindromic, etc. Despite all nice properties and the increasing sensitivity of the polynomial invariants, they are still useless in some cases. For example, the non-trivial link 9∗ 3.1. − 1. − 1.2. − 1. − 1.1. − 3

has a trivial Jones polynomial (see page 208), and the same is true for a whole family of links (Eliahou, Kauffman and Thistlethwaite, 2003). Even combined together, polynomials sometimes fail in detecting chirality: for instance, Jones, HOMFLYPT, and Kauffman polynomials of the knots 2 2, 3, −2 (942 ), (2 1, 2+) (3, −2), or 10∗∗ 2 0.2.2.2 0.2 0.2 0 and their mirror images are equal, although these knots are chiral. Moreover, this property holds for the families of alternating chiral KLs 10∗∗∗ p :: .p 0, 10∗∗∗ p :: .p 0 : .q 0.q, 10∗∗∗ p :: .p 0 : .q 0.q, 10∗∗∗ p :: .p 0 : .q.q 0, 10∗∗∗ p : .q 0.q.p 0, 10∗∗∗ p.q 0 : .q : .p 0.r 0.r, 10∗∗∗ p : .q 0.q.p 0 : .r.r 0, etc. LinKnot can be used for extensive computations which may lead to interesting discoveries. For example, after computing that the Alexander

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and Conway polynomial of a non-trivial 15-crossing knot 7, 5, −3 are 1, one can check some members of the family (2k + 1), 3, −3. The real surprise is that both Alexander and Conway polynomial are independent of k, so they remain the same: 2-5t+2t2 and 1 − 2x, respectively. Moreover, all knots in the family (4k+3), (4k+1), −(2k+1) with n = 10k+5 crossings have trivial Alexander and Conway polynomials, and this property also holds for a more general three-parameter family of pretzel knots (2p + 1), (2q + 1), −(2r + 1), where integers p, q, r satisfy the condition pq = pr + qr + r. It is not difficult to find a few more families of knots with trivial Alexander and Conway polynomial: • ((2p) (q+1), −(q+1)), (−(2p) (q+1), (q+1)), 2r with n = 4p+4q+2r+3 crossings, • ((2p) 1 (2q), −(2q + 1)), (−(2p) 1 (2q), (2q + 1)), 2r with n = 4p + 8q + 2r + 3 crossings, or even the family of knots given by their minimal representation • (2k + 1, −2k), (−(2k + 1), 2k), . . . , (2k + 1, −2k), (−(2k + 1), 2k), 2l where (2k + 1, −2k), (−(2k + 1), 2k) repeats an arbitrary number of times. All mutant knots derived from such families also have trivial Alexander and Conway polynomials. So, maybe there still are some undiscovered properties of Alexander and Conway polynomial? The important open question is: why are Alexander and Conway polynomial unable to distinguish knots belonging to the same family, or even unable to distinguish some families of knots from the unknot? In an arbitrary family of KLs, their corresponding polynomials are well ordered. For example, for the family of knots (2k + 1), (k ≥ 1) we obtain the sequence of Alexander polynomials 1-t+t2 , 1-t+t2 -t3 +t4 , etc., so for the knot family (2k + 1) we have the general formula for the Alexander polynomial 2k X ∆(p) = (−1)i ti . i=0

Following the same idea, for the knot families that originated from the general Conway symbol p q, for (m, n ≥ 1, m ≥ n) we obtain ∆((2m) (2n)) = mn − (2mn + 1)t + mnt2

∆((2m + 1) (2n)) = (m + 1) + (2m + 1)

2n−1 X

(−1)i ti + (m + 1)t2n

i=1

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2m−1 X

(−1)i ti + (n + 1)t2m .

i=1

For the knot families derived from the general Conway symbol p 1 q (m, n ≥ 1, m ≥ n) we have

∆((2m+1) 1 (2n+1)) = (m+1)(n+1)−(2mn+2m+2n+1)t+(m+1)(n+1)t2 ∆((2m) 1 (2n + 1)) = m + (2m + 1)

2n+2 X

(−1)i ti + mt2n+3

i=1

∆((2m + 1) 1 (2n)) = n + (2n + 1)

2m+2 X

(−1)i ti + nt2m+3 ,

i=1

etc. For the mentioned family of pretzel knots (2p + 1), (2q + 1), −(2r + 1) ∆((2p + 1), (2q + 1), −(2r + 1)) = pq − pr − qr − r + (−2pq + 2pr + 2qr + 2r + 1)t + (pq − pr − qr − r)t2 , and the Conway polynomial is ∇((2p + 1), (2q + 1), −(2r + 1)) = 1 + (pq − pr − qr − r)x. For pq = pr + qr + r it follows ∆ = 1 and ∇ = 1. These particular results imply a more general conclusion: a general formula for Alexander polynomial one can derive for every family of KLs, where its coefficients are expressed by parameters from the Conway symbol of the family. Before we revisit this question and pose a series of similar questions about families and the regular distribution of KL invariants (as signatures, symmetry groups, chirality, unknotting and unlinking numbers, etc.) we continue with the derivation of KLs in Conway notation, that will enable us to work with all KLs, and not only the rational ones. 2.5

Derivation and classification of KLs

The first world we have derived is the linear world (or L-world) that consists of KLs given by a general Conway symbol p (p ≥ 1). For an odd p we obtain knots, and for an even p two-component links. All knots of the L-world are periodic with graph symmetry group G = [2, p], and knot symmetry

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group G′ = [2, p]+, generated by a p-rotation and 2-rotation. For every knot in this family the unknotting number is u(p) = p−1 2 . Moreover, the Alexander polynomial ∆(t) for knots, or reduced two-variable Alexander polynomial ∆(t, t) for 2-component links, is a sign-alternating polynomial of order p + 1, with all coefficients equal to 1. The rational world consists of alternating knots and two-component links. Theorem 2.12. Every rational link has a two-bridge presentation. For the proof, see e.g., Cromwell (2004). Rational KLs are also known as two-bridge KLs (Definition 1.64) since they all share this property. Because of the connection between rational KLs and continued fractions, all calculations with rational KLs are simple (including the recognition of KLs, checking equality, computing BJunknotting and unlinking numbers, etc.) The next world is the stellar or prismatic world, abbreviated as S-world. The name stellar, introduced by A. Caudron, comes from the properties of corresponding graphs, and we propose the more geometrical name prismatic. Stellar KLs are also known as pretzel KLs, or Montesinos KLs. In the S-world we can distinguish the source links of the type 2, 2, . . . , 2 (S-links), from the source links of the type 2, 2, . . . , 2 + k (S + -links), where +k denotes a sequence of k pluses (k = 1, 2, ...) (Fig. 2.20). Source links are the base for the derivation of all generating KLs, their corresponding families, and all different particular KLs belonging to them. For every even n (n ≥ 6) we have an ( n2 )-component source link 2, 2, . . . , 2. Its shadow is an n-gonal prism with colored lateral edges denoting bigons. The symmetry group of this graph G = [2, n] is generated by an n-fold rotation, vertical, and horizontal plane reflection (Gr¨ unbaum and Shephard, 1985). Hence, we conclude that every source link 2, 2, . . . , 2 remains invariant after cyclic permutations of bigons (cyclic rotations), where every permutation is identified with its reverse (because of a vertical reflection), or if all bigons are reverted (because of a horizontal reflection). We can obtain all stellar KLs from the source KLs mentioned, substituting bigons with chains of bigons and using symmetry. In fact, in the source KLs of the stellar world we replace bigons by rational tangles 3, 4, . . ., from the linear world (L-world). Because the L-world is a subworld of the rational world (R-world), we will continue with the derivation of stellar-rational KLs, treating stellar-linear KLs as a subworld of the RS-world. In this way, the whole S-world will be included in the SR-world.

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Fig. 2.20

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Source links of the stellar world.

The next world is the arborescent world. Its members are multiple combinations of KLs belonging to the preceding worlds. For example, we have the following combinations: stellar-rational subworld (S(R)-subworld) obtained by replacing bigons in stellar source KLs by rational tangles which do not begin with 1, rational-stellar subworld (R(S)-subworld) obtained by replacing the first and last bigon in rational source KLs by stellar KLs, (R(S))(R)-subworld obtained by replacing bigons belonging to stellar parts of the source KLs in the R(S)-subworld by rational tangles which do not begin with 1, stellar-stellar subworld (S(S)-subworld) obtained by replacing bigons in stellar source KLs by stellar KLs, etc. In a certain sense, the structure of KLs looks like Chinese nested spheres, where every sphere is placed inside the preceding one. Definition 2.17. Every additive expression of a natural number n as an ordered sequence of natural numbers is called a composition of n (or an ordered partition of n). Denote by n the set of all compositions of n which do not begin with 1. For example, 3 is the set {3, 2 1},

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4 is {4, 3 1 2 2, 2 1 1}, 5 is {5, 4 1, 3 2, 3 1 1, 2 3, 2 2 1, 2 1 2, 2 1 1 1}, etc. From the stellar source links 2, 2, . . . , 2, we derive all KLs of the stellarrational subworld (S(R)-subworld) by replacing bigons by R-tangles– rational tangles which do not begin with 1. Stellar-rational KLs (or arborescent stellar KLs, according to Caudron), known also as Montesinos KLs, are derived from the source links 2, 2, . . . , 2 for 6 ≤ n ≤ 12. They are given in Table 1. Table 1 n = 6

2, 2, 2

n = 7

3, 2, 2

n = 8

4, 2, 2 3, 3, 2

2, 2, 2, 2

n = 9

5, 2, 2 4, 3, 2 3, 3, 3

3, 2, 2, 2

n = 10

6, 2, 2 5, 3, 2 4, 4, 2 4, 3, 3

4, 2, 2, 2 3, 3, 2, 2 3, 2, 3, 2

n = 11

7, 2, 2 6, 3, 2 5, 4, 2 5, 3, 3 4, 4, 3

5, 2, 2, 2 4, 3, 2, 2 4, 2, 3, 2 3, 3, 3, 2

3, 2, 2, 2, 2

n = 12

8, 2, 2 7, 3, 2 6, 4, 2 6, 3, 3 5, 5, 2 5, 4, 3

6, 2, 2, 2 5, 3, 2, 2 5, 2, 3, 2 4, 4, 2, 2 4, 2, 4, 2 4, 3, 3, 2

4, 2, 2, 2, 2 3, 3, 2, 2, 2 3, 2, 3, 2, 2

2, 2, 2, 2, 2, 2

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4, 3, 2, 3 3, 3, 3, 3

A pretzel KL a1 , a2 , a3 , . . . remains the same after cyclic permutation or reversal of its sequence of tangles. Hence, from a symbol 3, 2, 2 we obtain (3, 2, 2) and (2 1, 2, 2), from 4, 2, 2 we obtain (4, 2, 2), (3 1, 2, 2), (2 2, 2, 2), (2 1 1, 2, 2), from 3, 3, 2 we obtain (3, 3, 2), (3, 2 1, 2) = (2 1, 3, 2) and (2 1, 2 1, 2), and then eliminate duplicates. In the same way, we obtain all alternating stellar-rational KLs. Thus, the stellar world is completely included in the SR-world. Instead of repeating the derivation of S + links from the source links of the type 2, 2, . . . , 2 + k, (k = 1, 2, . . .), they can be obtained directly from the stellar-rational KLs, by adding an appropriate number of pluses. For n = 7 we add one plus to stellar-rational KLs with 6 crossings, for n = 8 we derive S + links from SR-links with 6 crossings by adding two pluses, and from SR-links with 7 crossings by adding one plus, etc. In this way, for every n we derive KLs with k pluses from SR-links with n − k, . . ., n − 2, n − 1 crossings (n ≥ k + 6, k = 1, 2, . . .). The LinKnot functions fStellarBasic, fStellar, and fStellarPlus calculate the number and Conway symbols of stellar and stellar-rational KLs without and with pluses for a given number of crossings n, respectively. If we try to calculate the number of different classes in the first column of Table 1 (beginning with series 1, 1, 2, 3, 4, 5, 7 for n ≤ 12), the number of source KLs, or even the number of all KLs derived from some generating KL, we encounter different combinatorial problems. For example, the number of different classes mentioned n above is equal to the coefficient corresponding to q n−6 in n−1 , where 3 r is the Gauss polynomial n (1 − q n ) . . . (1 − q n−r+1 ) . = r (1 − q r ) . . . (1 − q) These problems belong to the theory of partitions with a given symmetry group (P -partitions). Let P be a permutation group P on k objects and n ≥ k be an integer. A natural number ni is assigned to every object ki P (1 ≤ ki ≤ k), where ki=1 ni = n. Two partitions defined by signed (or weighted) objects are equal iff there is a permutation from P transforming one to another. We want to find and enumerate different P -partitions. In some special cases, we can reduce problems of P -partitions to classical partition theory, but in general, this enumeration is an open problem. We could also analyze families, reflecting a “vertical” structure of every world.

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Fig. 2.21

Family p,q,r.

For example, considering source KLs from the second column of Table 1, we obtain the families: p, q, r p q, r, s p q 1, r, s p 1 1 1, q, r p 1, q 1, r 1

(p ≥ q ≥ r ≥ 2) (p, q ≥ 2, r ≥ s ≥ 2) (p, q ≥ 2, r ≥ s ≥ 2) (p ≥ 2, q ≥ r ≥ 2) (p ≥ q ≥ r ≥ 2)

p 1, q, r p 1, q 1, r p 1 q, r, s p q, r 1, s

(p ≥ 2, q ≥ r ≥ 2) (p ≥ q ≥ 2, r ≥ 2) (p ≥ q ≥ 2, r ≥ s ≥ 2) (p, q, r, s ≥ 2)

In the family p,q,r we have three-component links for p = q = r = 0 (mod 2), two-component links if exactly one of the numbers p, q, r is odd, and knots if at least two of them are odd (Fig. 2.21). Let us consider the knot family (2k + 1), 3, 3, (k ≥ 2). The next table contains Jones polynomials for the first six knots in this family.

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5, 3, 3 7, 3, 3 9, 3, 3 11, 3, 3 13, 3, 3 15, 3, 3

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1, −2, 3, −4, 5, −5, 6, −4, 4, −3, 1, −1 1, −2, 3, −4, 5, −5, 6, −6, 6, −4, 4, −3, 1, −1 1, −2, 3, −4, 5, −5, 6, −6, 6, −6, 6, −4, 4, −3, 1, −1 1, −2, 3, −4, 5, −5, 6, −6, 6, −6, 6, −6, 6, −4, 4, −3, 1, −1 1, −2, 3, −4, 5, −5, 6, −6, 6, −6, 6, −6, 6, −6, 6, −4, 4, −3, 1, 1 1, −2, 3, −4, 5, −5, 6, −6, 6, −6, 6, −6, 6, −6, 6, −6, 6, −4, 4, −3, 1, −1

We conjecture that the signature of all knots in this family is is 2, and unknotting number is 3. Non-alternating KLs first appear in the SR-world. Here is the list of non-alternating stellar KLs up to 10 crossings:

n=6 2, 2, 2− No. of KLs: 1 n=7 3, 2, 2− 2 1, 2, 2− No. of KLs: 2 n=8 4, 2, 2− 2 2, 2, 2− 2, 2, 2, 2 − −

3, 2 1, 2− 2 1 1, 2, 2−

3, 3, 2− 3 1, 2, 2−

2 1, 2 1, 2− 2, 2, 2, 2−

No. of KLs: 9 n=9 5, 2, 2− 2 1 1, 3, 2− 2 2, 2 1, 2− 3 1 1, 2, 2− 3, 3, 2 1− 3,2,2,2−

4, 3, 2− 4 1, 2, 2− 3, 3, 3− 2 2, 3, 2− 2 1 1, 2 1, 2− 3 2, 2, 2− 2 2 1, 2, 2− 3 1, 3, 2− 2 1, 2 1, 2 1− 2 1 2,2,2− 2 1,2,2,2− 3,2,2,2−− No. of KLs: 23

6,2,2− 4 1,3,2− 4 1,2 1,2− 4 1 1,2,2− 3 2,3,2− 2 3,3,2− 2 2,2 2,2−

4,4,2− 4,3,3− 4 2,2,2− 4,2 1 1,2− 3 2,2 1,2− 2 3,2 1,2− 2 2,2 1 1,2−

4, 2 1, 2− 3, 2 1, 2 1− 23, 2, 2− 3 1, 2 1, 2− 2 1 1 1,2,2−

n = 10 5,3,2− 4,3,2 1− 2 4,2,2− 5 1,2,2− 3 1 1,3,2− 2 2 1,3,2− 2 1 1,2 1 1,2−

5,2 1,2− 4,2 1,2 1− 4,2 2,2− 4,3 1,2− 3 1 1,2 1,2− 2 2 1,2 1,2− 3 1,3,3−

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3 1,3,2 1− 2 1 1,2 1,2 1− 2 1 3,2,2− 3 1,2 2,2− 2 1 1 1,3,2− 2 1 1,3,2 1− 2 2 1 1,2,2− 3,2 1,2,2− 2 1,2,2 1,2− 2,2,2,2,2− 3,2,3,2−−

3 1,2 1,2 1− 2 2,3,2 1− 3 1 2,2,2− 3 1 1 1,2,2− 2 1 2 1,2,2− 2 1 1 2,2,2− 3 1,2 1 1,2− 2 1 2,3,2− 2 1 1 1,2 1,2− 2 2,3,3− 3 3,2,2− 3 2 1,2,2− 3 1,3 1,2− 4,2,2,2− 2 1,2 1,2,2− 3,2,3,2− 3 1,2,2,2− 2 2,2,2,2− 4,2,2,2−− 3,3,2,2−− 3,2,2 1,2−− 2 2,2,2,2−− No. of KLs: 72

2 1 1,3,3− 2 3 1,2,2− 2 1 1 1 1,2,2− 2 1 2,2 1,2− 2 2,2 1,2 1− 2 2 2,2,2− 3,3,2,2− 3,2,2 1,2− 2 1 1,2,2,2− 3,2 1,2,2−− 2,2,2,2,2−−

The principle of their derivation is simple. We add minuses to all alternating stellar and stellar-rational KLs with n crossings according to the following rule: if an alternating stellar or stellar-rational KL consists of k rational tangles connected by the operation of ramification (denoted by a comma) we may add at most [ k2 ] minuses. This means that we add to alternating stellar or stellar-rational KLs formed from three rational tangles one minus, to KLs consisting of four or five rational tangles we add one or two minuses, etc. In order to write them as KLs with a minimal number of crossings, we can write −2 instead of 2−; −2, −2 instead of 2, 2 − −; −2 1 instead of 3−; −3 instead of 2 1−; −2 1, −2 instead of 3, 2 − −, etc. The same non-alternating KLs can be obtained only when we add k minuses to a stellar-rational KL expressed by a sequence of the length 2k. Two KLs of this kind will be equal iff one can be obtained from the other by the following rules applied to rational tangles that are the parts of those KLs: (1) every single rational tangle 2 remains unchanged; (2) replace every rational tangle p1 p2 . . . pi 1 by p1 p2 . . . (pi + 1); (3) replace every rational tangle p1 p2 . . . pi (pi ≥ 2) by p1 p2 . . . (pi − 1) 1. For example, for n = 10, among the KLs consisting of four parts and with two minuses there are the following pairs of equal KLs: 4, 2, 2, 2 − − = 3 1, 2, 2, 2 − − 2 2, 2, 2, 2 − − = 2 1 1, 2, 2, 2 − − 3, 3, 2, 2 − − = 2 1, 2 1, 2, 2 − − 3, 2, 3, 2 − − = 2 1, 2, 2 1, 2 − −

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where the first KL is always the mirror-image of the other. The LinKnot function fStellarNalt calculates the number and Conway symbols of all non-alternating stellar-rational KLs with a given number of crossings n. The next subworld is the rational-stellar R(S)-subworld. In the same way as before, we can derive KLs with pluses directly from KLs without them by adding an appropriate number of pluses in certain positions. Therefore, we will first consider the derivation of rational-stellar (or arborescent generalized, according to Caudron) KLs without pluses. As the basis of the derivation of rational-stellar (RS-links) we use rational source KLs and source links 2, . . . , 2 from the stellar world, including the link 2, 2. Then we substitute the first and last bigon in every rational source KL by a stellar source link. By k we denote all rational tangles that can be obtained from k (as all compositions of the number k). For example, 2 gives {2, 1 1}, 3 gives {3, 2 1, 1 2, 1 1 1}, 4 gives {4, 3 1, 2 2, 1 3, 2 1 1, 1 2 1, 1 1 2, 1 1 1 1}, etc.

Fig. 2.22

Rational-stellar source links.

The list of rational-stellar source KLs (Fig. 2.22) for n ≤ 12 is given in Table 2:

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Table 2 n = 8

(2, 2) (2, 2) (2, 2) 1 (2, 2)

n = 9 n = 10

(2, 2, 2) (2, 2)

n = 11 n = 12

(2, 2, 2) (2, 2, 2) (2, 2, 2, 2) (2, 2)

(2, 2) 2 (2, 2) (2, 2, 2) 1 (2, 2)

(2, 2) 3 (2, 2)

(2, 2, 2) 2 (2, 2)

(2, 2) 4 (2, 2)

As before, the main tool in derivation of KLs from source KLs is symmetry. Therefore, among the source KLs from the first column of Table 2 we distinguish links with the graph symmetry group G = [2+ , 4] ((2, 2)(2, 2) and (2, 2, 2)(2, 2, 2)), links with the symmetry group G = [2] ((2, 2, 2)(2, 2), (2, 2, 2, 2)(2, 2), etc.), and delete duplicates. For example, from symmetry reasons, it is clear that (2, 2) 2 1 (2, 2) = (2, 2) 1 2 (2, 2), etc. In the same way, we continue with the derivation of S(R)-links, by substituting bigons that belong to stellar parts of rational-stellar source KLs by rational tangles which do not begin with 1, denoted by k. RSR-links derived are given in the following table: Table 3 n = 8

(2, 2) (2, 2)

n = 9

(3, 2) (2, 2)

n = 10

(2, 2, 2) (2, 2) (3, 2) (3, 2) (4, 2) (2, 2)

n = 11

(5, 2) (2, 2) (4, 3) (2, 2) (4, 2) (3, 2) (3, 3)(3, 2)

n = 12

(2, 2) (2, 2) (2, 2) (2, 2, 2) (2, 2, 2) (2, 2, 2, 2) (2, 2) (4, 2) (4, 2) (3, 3)(3, 3)

(2, 2) 1 (2, 2) (3, 3) (2, 2) (2, 2) 2 (2, 2) (3, 2) 1 (2, 2) (3, 2, 2) (2, 2) (2, 3, 2)(2, 2) (2, 2, 2) (3, 2)

(2, 2) 3 (2, 2) (3, 2) 2 (2, 2) (4, 2) 1 (2, 2) (3, 3) 1 (2, 2) (3, 2) 1 (3, 2) (3, 2) 1 (2, 3)

(2, 2, 2) 1 (2, 2)

(4, 2, 2) (2, 2) (2, 4, 2) (2, 2)

(5, 2) 1 (2, 2) (4, 3) 1 (2, 2)

(3, 2, 2) 1 (2, 2) (2, 3, 2) 1 (2, 2)

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(6, 2) (2, 2) (5, 3)(2, 2) (5, 2) (3, 2) (4, 4)(2, 2) (4, 3)(3, 2) (4, 2)(3, 3)

(2, 2, 2) (4, 2) (3, 3, 2)(2, 2) (3, 2, 3) (2, 2) (3, 2, 2)(3, 2) (2, 3, 2)(3, 2) (2, 2, 2)(3, 3)

(4, 2) 1 (3, 2) (4, 2) 1 (2, 3) (4, 2) 2 (2, 2) (3, 3) 2 (2, 2) (3, 2) 2 (3, 2) (3, 2) 2 (2, 3) (3, 2) 3 (2, 2) (2, 2) 4 (2, 2)

237

(2, 2, 2) 1 (3, 2) (2, 2, 2) 2 (2, 2)

where we need to eliminate duplicates, e.g., (2, 2) 2 1 (2, 2) = (2, 2) 1 2 (2, 2), (2, 2) 3 1 (2, 2) = (2, 2) 1 3 (2, 2), etc. All centro-antisymmetric KLs derived from 3-component link (2, 2) (2, 2) (or 834 ) inherit 2-antirotation, hence they are all achiral. For example, for n ≤ 12 we have following achiral knots: (3, 2) (3, 2) (2 1, 2) (2 1, 2) (2 2, 2) (2 2, 2) (2 1 1, 2) (2 1 1, 2) (3, 3) (3, 3) (2 1, 2 1) (2 1, 2 1). All of them are non-invertible (Fig. 2.23). From (2, 2) (2, 2) we also derive 3-component achiral links (4, 2) (4, 2) and (3 1, 2) (3 1, 2). Hence, (2, 2) (2, 2) generates an infinite series of achiral KLs of the form (p, q) (p, q) (p, q ≥ 2) composed from two identical terms (p, q).

Fig. 2.23 Centro-antisymmetric presentations of achiral knots (3, 2) (3, 2) and (2 1, 2) (2 1, 2).

From the source KLs without pluses we directly obtain the corresponding KLs with pluses. For example, from the source link without pluses (2, 2) (2, 2) we obtain source links with pluses: (2, 2+) (2, 2) for n = 9, (2, 2 + +) (2, 2), (2, 2+) (2, 2+) for n = 10, (2, 2 + +) (2, 2+) for n = 11, (2, 2 + +) (2, 2 + +) for n = 12, etc.

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Among them, for n ≤ 12 following KLs are achiral: (2, 2+) (2, 2+), (3, 2+) (3, 2+), (2 1, 2+) (2 1, 2+), (2, 2 + +) (2, 2 + +), etc. Their common origin is the achiral 3-component link (2, 2) (2, 2) whose achirality is preserved under symmetric addition of pluses. It generates the families of achiral KLs of the form (p, q + k) (p, q + k) (p, q ≥ 2, k ≥ 1).

Fig. 2.24

Some (S(R))(S) source links.

The third subworld of the arborescent world is the stellar-stellar subworld (S(S)-subworld). We obtain the source KLs substituting bigons in the source KLs of the stellar world (Table 1) by stellar tangles of the form 2, . . . , 2, where the number 2 appears at least two times. KLs obtained by this replacement must contain at least two stellar tangles. Instead of these substitutions, since stellar world is contained in the

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stellar-rational world, we can make analogous substitutions beginning from the S(R)-world. As the result, for n ≤ 12 we obtain (S(R))(S)-links (Fig. 2.24) given in Table 4: Table 4 n = 10

(2, 2), (2, 2), 2

n = 11

(2, 2) 1, (2, 2), 2

n = 12

(2, 2), (2, 2), (2, 2) (2, 2, 2), (2, 2), 2 (2, 2)2, (2, 2), 2 (2, 2) 1 1, (2, 2), 2 (2, 2) 1, (2, 2) 1, 2

(2, 2), 2, (2, 2), 2 (2, 2), (2, 2), 2, 2

In the same manner, replacing the first bigon in the S-tangle (2, 2) of the source link (2, 2), (2, 2), 2 by the tangle (2, 2), we obtain the first source link of the next subworld ((2, 2), 2), (2, 2), 2 (or ((2, 2), 2)((2, 2), 2) in a more symmetric form) (Fig. 2.25). From the source KLs derived we obtain remaining KLs of this world by using rational compositions r, r, and adding pluses.

Fig. 2.25

The link ((2, 2), 2), (2, 2), 2 = ((2, 2), 2) ((2, 2), 2).

The choice of the steps in the proposed derivation and the stratification of the worlds is made to maximally avoid overlapping and occurrence of duplicates. Can we be sure that no duplicates remained and that the derivation is exhaustive? The answer to the first question gives the graphtheoretical approach used by A. Caudron (1982) (see page 65, Fig. 1.49). To solve the problem of possible duplicates we can use graph-transformation

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method developed by the same author, and also the LinKnot function SameAltConKL which compares two alternating KLs and determines their equality. For example, for the two source links ((2, 2), 2), (2, 2), 2 and ((2, 2), 2) ((2, 2), 2) shown in Fig. 2.25 the result is: yes, they are equal. With non-alternating KLs given in Conway notation, the situation is somewhat different. In principle, it is possible to implement exhaustive derivation algorithms for all non-alternating algebraic KLs (similar to that used for the derivation of non-alternating KLs of the stellar world). From the alternating KLs belonging to the arborescent world for n = 8 we obtain the following non−alternating KLs: n=8 (2,2) (2,2−) (2,2) −(2,2) No. of KLs: 2 n=9 (3,2−) (2,2) (2,2+) (2,2−)

(3,2) (2,2−) (2 1,2−) (2,2) (2,2+) −(2,2) No. of KLs: 6

(2 1,2) (2,2−)

n = 10 (4,2) (2,2−) (3,3−) (2,2) (2 1,2 1−) (2,2) (2 2,2−) (2,2) (2 1,2) (2 1,2−) (3,2) −(2 1,2) (3,2) −(3,2) (2,2,2−) (2,2−) (2,2) −2 (2,2) (2,2),2,−(2,2) (2,2+) (2 1,2−) (2,2+) −(2 1,2)

(2,2) (4,2−) (4,2) −(2,2) (3,2 1) (2,2−) (2 1,3−) (2,2) (3,2) (2 1,2−) (2 1,2) (3,2−) (2 1 1,2) (2,2−) (2 1 1,2−) (2,2) (3,3) −(2,2) (3,2 1) −(2,2) (3 1,2) −(2,2) (2 2,2) −(2,2) (2 1,2) −(2 1,2) (2,2,2) (2,2−) (2,2,2−−) (2,2) (2,2,2) −(2,2) (2,2) 2 −(2,2) (2,2),2,(2,2−) (3,2+) (2,2−) (2 1,2+) (2,2−) (3,2+) −(2,2) (2 1,2+) −(2,2) (2,2++) (2,2−) (2,2++) −(2,2) No. of KLs: 48

(3,3) (2,2−) (2 1,2 1) (2,2−) (2 2,2) (2,2−) (3,2) (3,2−) (2 1,2 1) −(2,2) (2 1 1,2) −(2,2) (2,2,2−) (2,2) (2,2) 2 (2,2−) (2,2),−2,(2,2) (2,2+) (3,2−) (2,2+) −(3,2) (2,2,2) (2,2−−)

In order to represent them as KLs with a minimal number of crossings, we can write −2 instead of 2−; −2, −2 instead of 2, 2 − −, −21 instead of 3−, 2 1− as −3 etc.. For example, pretzel knot 2 1, 4, 3− is given by minimal representations −3, 4, 3 = 2 1, −3 1, 3 = 2 1, 4, −2 1. Derivation of KLs in Conway notation is inseparable from KL classification: families (or even classes) of KLs could be described by their general symbols. For the derivation of alternating KLs we can use general guidelines. For example, every rational KL is defined by a sequence of natural numbers,

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which do not begin or end with 1, where every sequence is identified with its reverse. Every alternating stellar (pretzel) KL consisting from three rational tangles is given by the following Conway symbols, where t1 , t2 , t3 are mutually different R-tangles: t1 , t1 , t1

t1 , t1 , t2

t1 , t2 , t3

Nine possible Conway symbols of pretzel KLs consisting of four R-tangles are given by: t1 , t1 , t1 , t1 t1 , t2 , t1 , t3

t1 , t1 , t1 , t2 t1 , t2 , t3 , t4

t1 , t1 , t2 , t2 t1 , t2 , t4 , t3

t1 , t2 , t1 , t2 t1 , t3 , t2 , t4

t1 , t1 , t2 , t3

From five R-tangles we obtain 28 pretzel KLs, from six 144, from seven 832, from eight 5942, etc. From the source link (2, 2) (2, 2) we obtain 9 arborescent KLs: (t1 , t1 ) (t1 , t1 ) (t1 , t2 ) (t1 , t3 )

(t1 , t1 ) (t1 , t2 ) (t1 , t2 ) (t3 , t4 )

(t1 , t1 ) (t2 , t2 ) (t1 , t3 ) (t2 , t4 )

(t1 , t2 ) (t1 , t2 ) (t1 , t4 ) (t2 , t3 )

(t1 , t1 ) (t2 , t3 )

Because of symmetry, the same holds for the source link (2, 2+) (2, 2+). From the source link (2, 2+) (2, 2) we obtain 16 KLs, and the same holds for (2, 2 + +) (2, 2): (t1 , t1 +) (t1 , t1 ) (t1 , t2 +) (t1 , t2 ) (t1 , t3 +) (t1 , t2 ) (t1 , t4 +) (t2 , t3 )

(t1 , t1 +) (t1 , t2 ) (t2 , t2 +) (t1 , t1 ) (t2 , t3 +) (t1 , t1 ) (t2 , t3 +) (t1 , t4 )

(t1 , t2 +) (t1 , t1 ) (t1 , t1 +) (t2 , t3 ) (t1 , t2 +) (t3 , t4 ) (t2 , t4 +) (t1 , t3 )

(t1 , t1 +) (t2 , t2 ) (t1 , t2 +) (t1 , t3 ) (t1 , t3 +) (t2 , t4 ) (t3 , t4 +) (t1 , t2 )

From (2, 2, 2) (2, 2) we obtain 76 arborescent KLs, etc. 2.6

Basic polyhedra and polyhedral KLs

Polyhedral world or P -world is significantly different from the previous ones. The main task and the first problem is the derivation of the basic polyhedra: 4-valent, 4-edge connected, at least 2-vertex connected graphs without bigons (Definition 1.43). This problem was solved for n ≤ 12 crossings by T.P. Kirkman (1885a,b) with one omission: the basic polyhedron 12E. Kirkman derived basic polyhedra by eliminating all bigons in KL diagrams. For the derivation he used KL diagrams which satisfy the following necessary condition: each

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Fig. 2.26

Kirkman’s method for the derivation of basic polyhedra.

diagram contains at most three bigons belonging to the same face. Eliminating bigons is achieved by inscribing a triangle to this face, with the vertices belonging to the face edges (e.g., coinciding with their midpoints) (Fig. 2.26). Table 5 contains KL projections with n crossings satisfying this necessary condition, Dowker codes of the derived basic polyhedra, and their list (Figs. 2.27-2.28). Table 5 n=3

3

462

6∗

n=5

212

6 8|2 10 4

8∗

n=6

312 6∗

4 8 10 12 2 6 6 8|10 12|2 4

9∗ 9∗

n=7

2 1 1 1 2′ .2

4 8 10 12 2 14 6 6 8|10 12 14 2 4

10∗ , 10∗∗ 10∗∗ , 10∗∗∗

n=8

31112 2 1 2 1 2′ 2 1 2 1 2′′′ .3 .21 .2.20 8∗ 3#2 1 2

4 4 8 6 6 6 6

11∗ 11∗∗ 11∗ 11∗ , 11∗∗ 11∗ , 11∗∗ 11∗ , 11∗∗∗ 11∗ , 11∗∗ 11∗∗

n=9

31212 2 1 3 1 2′ 2111112

4 12 10 16 14 2 18 6 8 4 10 12 14 18 2 16 6 8 4 10 12 14 2 18 16 8 6

10 12 14 2 16 8 6 10 12 14|8 2 16 6 10 14|2 16 4 6 12 8|12 14 16|10 2 4 8|10 14 12 16 2 4 8 14 12 4 16 2 10 8 10 12 14 16 2 4

12D 12D 12A, 12B, 12F

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2 1 1 1 1 1 2′′′ 2 1, 2 1, 2 1 .4 .3 1 .2 2 .2 1 1 .3.20 .2 1.2′ 2:2:2 .(2, 2) 8∗ 2 8∗ 20 9∗ 2 1 2#1#3 6∗ #3

4 12 10 16 18 2 8 6 14 8 12 16|2 18 4 10 6 14 6 8|12 14 16 18 2 4 10 6 8|10 14 16 18 2 4 12 6 8|16 14 12 18 2 4 10 6 8|12 14 18|16 2 4 10 8 10 12|14 2 16 18 6 4 4 8 14 12 2 16 18 10 6 8 12 16|2 14 4 18 6 10 10 12|14 18|6 16 8 2 4 8 10 12|6 14 16 18 2 4 6 8 10 16 14 18 4 2 12 6 16 14 12 4 2 18 10 8

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12B, 12F 12G 12E 12J, 12L 12E 12B, 12H, 12I, 12J, 12K 12D 12B, 12F, 12H 12C 12I 12B, 12F, 12G, 12H, 12I 12F, 12I, 12K 12D, 12H, 12L 12E 12J

Complete results were obtained by T.P. Kirkman for n ≤ 8. For n = 9 two links from which basic polyhedra with n = 12 crossings can be derived were omitted by Kirkman, but even his incomplete list is sufficient for the derivation of all basic polyhedra with n ≤ 12 crossings. The missing basic polyhedron 12E is the only polyhedron that can be derived from the projection of the link .2 2, denoted by Kirkman as 9 Bn. Maybe his omission was deliberate: the polyhedron 12E is the only 2-vertex connected graph, and all the others are 3-vertex connected (Fig. 2.29). It is also the first basic polyhedron with two different projections: one with, and the other without a bigon. The alternating link obtained from the basic polyhedron 12E by a flype, the second projection of 12E, is the link 11∗∗∗ 2 (Fig. 1.46a). Hence, some KLs that can be derived from the basic polyhedron 12E can be derived from the basic polyhedron 11∗∗∗ . The complete list of the basic polyhedra with n = 12 crossings was obtained by A. Caudron (1982) by composing hyperbolic tangles, and we refer to his list and notation (Fig. 2.28, Figs. 2.46-2.47). Our idea is to generalize Kirkman’s method: in order to eliminate bigons in the KL diagrams we introduce not only triangular, but also p-gonal faces (p > 3). In this way we can derive all basic polyhedra with 6 ≤ n ≤ 12 vertices from the KLs belonging to the L-world or from their direct products (Fig. 2.30). This method has a particularly nice description by the corresponding partitions: 6∗ =3+3 10∗ =5+5 11∗ = 8∗ + 3 =4+3 + 3 11∗∗∗ =3#3+5

8∗ =4+4 10∗∗ =4+3 + 3 11∗ = 8∗ + 3 =4+3 + 3 12A =6+6

9∗ = 6∗ + 3 =3+3 + 3 10∗∗∗ =4+3 + 3 11∗∗ = 8∗ + 3 =4+3 + 3 12B =3#3+3 + 3

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Fig. 2.27

Basic polyhedra with n ≤ 11 crossings.

12C = 8∗ + 4 =4+4 + 4 12D = 9∗ + 3 =3+3 + 3 + 3 12E =6+3 + 3 12F =5+3 + 4 12G =5+3 + 4 12H = 9∗ + 3 =3+3 + 3 + 3 12I = 5 + 3 + 4 12J = 3#3+3 + 3 12K = 3#3+3 + 3 12L = 9∗ + 3 =3+3 + 3 + 3

where bold numbers denote the corresponding links p (p ≥ 3) from the L-world. For the larger values of n, the question of completeness of this derivation remains open. Conjecture 2.2. All basic polyhedra can be derived from diagrams of KLs belonging to L-world (KLs given by Conway symbol n, n ≥ 2, denoted in

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Basic polyhedra with n = 12 crossings.

classical notation as 221 , 31 , 421 , 51 ,...) and their direct products, by recursive inscribing of p-gons (p ≥ 3). The program LinKnot contains basic polyhedra up to n ≤ 20 crossings

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Fig. 2.29

The missing basic polyhedron 12E.

(see page 57). Lists of basic polyhedra with 12 ≤ n ≤ 20 crossings were first introduced in the program LinKnot (in the form of databases with more than 80 000 KLs) and we hope that the proposed notation will become a standard for denoting basic polyhedra with a large number of crossings. The first basic polyhedron is the octahedron 6∗ or .1, with the graph symmetry group G = [3, 4] of order 48, generated by the 4-rotation S = (1)(2, 3, 5, 6)(4), 2-rotation T = (1, 3)(2, 5)(4, 6) and inversion Z = (1, 4)(2, 5)(3, 6) (Fig. 2.31). It is a 3-component alternating link, the famous Borromean rings, the first non-trivial Brunnian 3-component link, with the link symmetry group G′ = [3+ , 4]. Introducing orientation results in the antisymmetry group that contains a rotational antireflection, which means that Borromean rings are achiral. This is the only basic polyhedron given by two symbols: 6∗ or .1 (Conway, 1970). From the basic polyhedron 6∗ (or .1) we derive source links by replacing its vertices by bigons. First we make all different symmetry choices of n − 6 vertices (7 ≤ n ≤ 12), i.e., all different vertex bicolorings of the octahedron. The number of vertex bicolorings can be computed using the Polya Enumeration Theorem (PET) (P´olya, 1937; Harary and Palmer, 1973; P´olya and Read, 1987). For G = [3, 4], 1 6 ZG = (t + 3t41 t2 + 9t21 t22 + 6t21 t4 + 7t32 + 6t2 t4 + 8t23 + 8t6 ), 48 1 and the coefficients of ZG (x, 1) = 1 + x + 2x2 + 2x3 + 2x4 + x5 + x6

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Fig. 2.30

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The generalization of Kirkman’s method.

give, respectively, the number of different choices of n − 6 vertices for 6 ≤ n ≤ 12. For 7 ≤ n ≤ 12, these vertex bicolorings are: {1}; {1, 2}, {1, 4}; {1, 2, 3}, {1, 2, 4}; {1, 2, 4, 5}, {1, 2, 3, 4}; {1, 2, 3, 4, 5}; {1, 2, 3, 4, 5, 6},

and here are the corresponding source links

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Fig. 2.31

The basic polyhedron 6∗ .

.a, .a.b, .a : b, .a.b.c, a : b : c, .a.b.c.d, a.b.c.d, a.b.c.d.e, a.b.c.d.e.f , given in Conway notation. Next thing is making one of two possible substitutions in every chosen vertex (2 or 2 0, Fig. 2.32a,b), based on symmetry of the vertex bicolored octahedron (Fig. 2.32c). In terms of colorings, this is the other bicoloring: bicoloring of the chosen vertices. For n ≤ 12, the source KLs obtained from 6∗ by the vertex substitutions are given in Table 6. Among them, for n = 11, there is the 3-component link 2.2 0.2.2 0.2, omitted by A. Caudron (1982) (Fig. 2.33). The complete list of obtained source KLs is given in Table 6. Table 6 n = 7

.2

n = 8

.2.2 .2.2 0

n = 11

2.2.2.2.2 2.2.2.2.2 0 2.2.2.2 0.2 2.2.2.2 0.2 0 2.2 0.2.2.2 0 2 0.2.2.2.2 0 2.2 0.2.2 0.2

n = 12

2.2.2.2.2.2 2.2.2.2.2.2 0 2.2.2.2.2 0.2 0 2.2.2.2 0.2.2 0 2.2.2 0.2.2.2 0 2.2.2.2 0.2 0.2 0 2.2 0.2.2 0.2.2 0

.2 : 2 .2 : 2 0 n = 9

.2.2.2 .2.2.2 0 .2.2 0.2

2:2:2 2 : 2 : 20 2 : 20 : 20 20 : 20 : 20

n = 10

.2.2.2.2 .2.2.2.2 0 .2.2.2 0.2 0 .2.2 0.2.2 0

2.2.2.2 2.2.2.2 0 2.2.2 0.2 2.2.2 0.2 0 2.2 0.2.2 0 2 0.2.2.2 0 2 0.2.2 0.2 0

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Fig. 2.32

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Derivation of source KLs from the basic polyhedron 6∗ .

If we want to avoid the ambiguity of notation that results from two different symbols .1 and 6∗ used for the same basic polyhedron, we can use only 6∗ . In this case, the KLs from the first column of Table 6 will be: .2 = 6∗ 2 .2 : 2 = 2 : .2 0 .2.2.2 0 = 2.2 : 2 .2.2.2.2 0 = 2.2 : 2.2 0

.2.2 = 2.2 .2 : 2 0 = 2 : .2 .2.2 0.2 = 2.2 0 : 2 0 .2.2.2 0.2 0 = 2.2 : 2.2

.2.2 0 = 2.2 0 .2.2.2 = 2.2 : 2 0 .2.2.2.2 = 2.2 : 2 0.2 0 .2.2 0.2.2 0 = 2.2 0 : 2 0.2

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Fig. 2.33

The link 2.2 0.2.2 0.2 omitted by A. Caudron.

The polyhedral rational subworld (P (R)-subworld) of the P -world consists of the KLs obtained by replacing bigons in the source KLs by rational tangles not beginning with 1. Derivation of KLs is completely determined by partitions of number of crossings and the symmetries of the source KL, i.e., partitions with the given permutation group P . Two permutation groups are equivalent iff their permutation representations are isomorphic. Equivalent permutation groups produce the same number of P -partitions. Hence, we will classify source KLs from Table 6 with respect to the P equivalence, and then derive generating KLs from one representative of each class. For 7 ≤ n ≤ 11, we have the following classes: .2 with P ≃ {(1)}; .2.2, .2.20, .2:2, .2:2 0 with P ≃ {(1, 2)}; .2.2.2, .2.20.2, 2:2:2 0, 2:2 0:2 0 with P ≃ {(1, 3)(2)}; .2.2.20 with P ≃ {(1)(2)(3)}; 2:2:2, 20:2 0:2 0 with P ≃ {(1, 2, 3)}; .2.2.2.2, .2.20.2.20 with P ≃ {(1, 2, 4, 5)}; 2.2.2.20 with P ≃ {(1)(2)(4)(5)}; .2.2.20.20, with P ≃ {(1, 2)(4, 5), (1, 4)(2, 5)}; 2.2.2.2, 20.2.2.20 with P ≃ {(1, 4)(2, 3)}; 2.2.20.2, .2.2.2.20, 2.2.20.20, 2.20.2.20, 20.2.20.20 with P ≃ {(1)(2, 3)(4)}; 2.2.2.2.2, 2.2.2.20.20, 2.20.2.2.20 with P ≃ {(1, 2)(3)(4, 5)}; 2.2.2.2.20, 2.2.2.20.2 with P ≃ {(1)(2)(3)(4)(5)}; 20.2.2.2.20, 2.20.2.20.2 with P ≃ {(1, 2)(3)(4, 5), (1, 5)(2, 4)(3)}. Taking the first element as a representative of each class, we obtain the list of P (R)-links (Table 7) derived from these representatives for 7 ≤ n ≤

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12. We can obtain the complete list of P (R)-links derived from 6∗ (or .1) for 7 ≤ n ≤ 12 directly from Table 7, using the isomorphism mentioned above and working with source links for n = 12 (Table 6). After that, by replacing every k by the corresponding rational compositions we obtain all such KLs. The symbol k ⋄ has the similar meaning as k, and it is used to denote partitions that are not mutually equivalent according to symmetry. For example, .4 ⋄ .4 ⋄ denotes .2 2.2 2 and .2 2.2 1 1 (=.2 1 1.2 2), .3 ⋄ .3 ⋄ denotes .3.3, .3.2 1 (= .2 1.3) and .2 1.2 1, 3 ⋄ : 3 ⋄ : 3 ⋄ denotes 3 : 3 : 3, 3 : 2 1 : 2 1 (= 2 1 : 3 : 2 1 = 2 1 : 2 1 : 3) and 2 1 : 2 1 : 2 1, etc. Table 7 n = 7

.2

n = 8

.3

.2.2

n = 9

.4

.3.2

.2.2.2

.2.2.2 0

2:2:2

n = 10

.5

.4.2 .3 ⋄ .3 ⋄

.3.2.2 .2.3.2

.3.2.2 0 .2.3.2 0 .2.2.3 0

3:2:2

n = 11

.6

.5.2 .4.3

.4.2.2 .2.4.2 .3.3.2 .3 ⋄ .2.3 ⋄

.4.2.2 0 .2.4.2 0 .2.2.4 0 .3.3.2 0 .3.2.3 0 .2.3.3 0

4:2:2 3⋄ : 3⋄ : 2

n = 12

.7

.6.2 .5.3 .4 ⋄ .4 ⋄

.5.2.2 .2.5.2 .4.3.2 .4.2.3 .3.4.2 .3 ⋄ .3.3 ⋄

.5.2.2 0 .2.5.2 0 .2.2.50 .4.3.2 0 .4.2.3 0 .3.4.2 0 .3.2.4 0 .2.4.3 0 .2.3.4 0 .3.3.3

5:2:2 4:3:2 3⋄ : 3⋄ : 3⋄

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n = 10

.2.2.2.2

2.2.2.2 0

.2.2.2 0.2 0

2.2.2.2

2.2.2 0.2

n = 11

.3.2.2.2

3.2.2.2 0 2.3.2.2 0 2.2.3.2 0 2.2.2.3 0

.3.2.2 0.2 0

3.2.2.2 2.3.2.2

3.2.2 0.2 2.3.2 0.2 2.2.2 0.3

n = 12

.4.2.2.2 .3.3.2.2 .3 ⋄ .2.3 ⋄ .2

4.2.2.2 0 2.4.2.2 0 2.2.4.2 0 2.2.2.4 0 3.3.2.2 0 3.2.3.2 0 3.2.2.3 0 2.3.3.2 0 2.3.2.3 0 2.2.3.3 0

.4.2.2 0.2 0 .3 ⋄ .3 ⋄ .2 0.2 0 .3 ⋄ .2.3 ⋄ 0.2 0 .3 ⋄ .2.2 0.3 ⋄ 0

4.2.2.2 2.4.2.2 3.3.2.2 3.2.3.2 3 ⋄ .2.2.3 ⋄ 2.3 ⋄ .3 ⋄ .2

4.2.2 0.2 2.4.2 0.2 2.2.2 0.4 3.3.2 0.2 3.2.2 0.3 2.3 ⋄ .3 ⋄ 0.2 2.3.2 0.3

n = 11

2.2.2.2.2

2.2.2.2.2 0

20.2.2.2.2 0

n = 12

3.2.2.2.2 2.3.2.2.2 2.2.3.2.2

3.2.2.2.2 0 2.3.2.2.2 0 2.2.3.2.2 0 2.2.2.3.2 0 2.2.2.2.3 0

3 0.2.2.2.2 0 20.2.3.2.2 0

Fig. 2.34

The basic polyhedron 8∗ .

The next basic polyhedron 8∗ is a 4-antiprism, with the graph symmetry group G = [2+ , 8] of order 16, generated by the rotational reflection S˜ = (1, 2, 3, 4, 5, 6, 7, 8)

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and reflection R = (1, 3)(5, 7)(4, 8)(2)(6) containing its axis (Fig. 2.34). We can use PET to find the number of different symmetry choices of the vertices (i.e., vertex bicolorings of 8∗ ). In this case, 1 8 ZG = (t + 4t21 t32 + 5t42 + 2t24 + 4t8 ), 16 1 and the coefficients of ZG (x, 1) = 1 + x + 4x2 + 5x3 + 8x4 + 5x5 + 4x6 + x7 + x8 represent, respectively, the number of choices of n − 8 vertices for 8 ≤ n ≤ 16. For 9 ≤ n ≤ 12, these choices are: {1}; {1, 2}, {1, 3}, {1, 4}, {1, 5}; {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 5}, {1, 4, 7}; {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 4, 7}, {1, 2, 5, 6}, {1, 3, 5, 7} corresponding, respectively, to the source KLs of the form 8∗ a; 8∗ a.b, 8∗ a : b, 8∗ a : .b, 8∗ a :: b; 8∗ a.b.c, 8∗ a.b : c, 8∗ a.b : .c, 8∗ a : b : c, 8∗ a : .b : .c; 8∗ a.b.c.d, 8∗ a.b.c : d, 8∗ a.b.c : .d, 8∗ a.b : c.d, 8∗ a.b : c : d, 8∗ a : b.c : d, 8∗ a.b : .c.d, 8∗ a : b : c : d, given in the Conway notation. The coefficients of ZG (x, x, 1) = 1 + 2x + 12x2 + 34x3 + 87x4 + 124x5 + 136x6 + 72x7 + 30x8 give the number of different source KLs derived from 8∗ for 8 ≤ n ≤ 16. We can divide all the obtained vertex bicolorings into equivalence classes, with regard to their symmetry groups, and then consider only their representatives. According to this, for n = 9 we have the representative 8∗ a giving 2 source links; for n = 10 the representative 8∗ a.b (8∗ a : b, 8∗ a : .b, 8∗ a :: b) giving 3 source links; for n = 11 two representatives: 8∗ a.b.c (8∗ a : b : c, 8∗ a : .b : .c) giving 6 source links and 8∗ a.b : c (8∗ a.b : .c) giving 8 source links; for n = 12 five representatives: 8∗ a.b.c.d (8∗ a.b : c.d, 8∗ .a : b.c : d) giving 10 source links, 8∗ a.b.c : d (8∗ a.b : c : d) giving 16 source links, 8∗ a.b.c : .d giving 12 source links, 8∗ a : b : c : d giving 6 source links, and 8∗ a.b : .c.d giving 7 source links, where the other members of equivalence classes are given in parentheses. The list of source links derived from these representatives is given in Table 8:

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Table 8 n = 9

8∗ 2 8∗ 2 0

n = 10

8∗ 2.2 8∗ 2.2 0 8∗ 2 0.2 0

n = 12

8∗ 2.2.2.2 8∗ 2.2.2.2 0 8∗ 2.2.2 0.2 8∗ 2.2.2 0.2 0 8∗ 2.2 0.2.2 0 8∗ 2.2 0.2 0.2 8∗ 2 0.2.2.2 0 8∗ 2.2 0.2 0.2 0 8∗ 2 0.2.2 0.2 0 8∗ 2 0.2 0.2 0.2 0

n = 11

8∗ 2.2.2:2 8∗ 2.2.2 : 2 0 8∗ 2.2.2 0 : 2 8∗ 2.2 0.2 : 2 8∗ 2 0.2.2 : 2 0 8∗ 2.2.2 0 : 2 0 8∗ 2.2 0.2 : 2 0 8∗ 2 0.2.2 : 2 0 8∗ 2.2 0.2 0 : 2 8∗ 2 0.2.2 0 : 2 8∗ 2 0.2 0.2 : 2 8∗ 2.2 0.2 0 : 2 0 8∗ 2 0.2.2 0 : 2 0 8∗ 2 0.2 0.2 : 2 0 8∗ 2 0.2 0.2 0 : 2 8∗ 2 0.2 0.2 0.2 0

8∗ 2.2.2 8∗ 2.2.2 0 8∗ 2.2 0.2 8∗ 2.2 0.2 0 8∗ 2 0.2.2 0 8∗ 2 0.2 0.2 0

8∗ 2.2:.2 8∗ 2.2 : .2 0 8∗ 2.2 0 : .2 8∗ 2 0.2 : .2 8∗ 2.2 0 : .2 0 8∗ 2 0.2 : .2 0 8∗ 2 0.2 0 : .2 8∗ 2 0.2 0 : .2 0

8∗ 2.2.2:.2 8∗ 2.2.2 : .2 0 8∗ 2.2.2 0 : .2 8∗ 2.2 0.2 : .2 8∗ 2.2.2 0 : .2 0 8∗ 2.2 0.2 : .2 0 8∗ 2 0.2.2 0 : .2 8∗ 2 0.2 0.2 : .2 8∗ 2 0.2.2 0 : .2 0 8∗ 2 0.2 0.2 : .2 0 8∗ 2 0.2 0.2 0 : .2 8∗ 2 0.2 0.2 0 : .2 0

8∗ 2:2:2:2 8∗ 2 : 2 : 2 : 2 0 8∗ 2 : 2 : 2 0 : 2 0 8∗ 2 : 2 0 : 2 : 2 0 8∗ 2 : 2 0 : 2 0 : 2 0 8∗ 2 0 : 2 0 : 2 0 : 2 0

8∗ 2.2:.2.2 8∗ 2.2 : .2.2 0 8∗ 2.2 : .2 0.2 0 8∗ 2.2 0 : .2.2 0 8∗ 2.2 0 : .2 0.2 8∗ 2.2 0 : .2 0.2 0 8∗ 2 0.2 0 : .2 0.2 0

In the similar way we obtain P (R)-links derived from 8∗ . For n = 9 we have the representative 8∗ 2 (8∗ 2 0) with P ≃ {(1)}; for n = 10 two representatives: 8∗ 2.2 (8∗ 2 0.2 0) with P ≃ {(1, 2)} and 8∗ 2.2 0 with P ≃ {(1)(2)}; for n = 11 two representatives: 8∗ 2.2.2 (8∗ 2.2 0.2, 8∗ 2 0.2.2 0, 8∗ 2 0.2 0.2 0) with P ≃ {(1, 3)(2)}, and 8∗ 2.2.2 0 (8∗ 2.2 0.2 0, and all source links derived from 8∗ 2.2 : .2) with P ≃ {(1)(2)(3)}, where the other members of equivalence classes are given in parentheses. The permutation groups P are already considered in the derivation of source links from 6∗ , so it will not be repeated. For every even n ≥ 8 (n = 2k) there is an k-antiprism, the basic polyhedron of the form n∗ = (2 × k)∗ (Conway, 1970; Gr¨ unbaum and Shephard, 1985) with the graph symmetry group G = [2+, n] of order 2n generated by rotational reflection S˜ and reflection R, so all the results obtained for 8∗ can be generalized to k-antiprisms. The graph symmetry group G = [2, 3] of order 12, corresponding to the basic polyhedron 9∗ is generated by the 3-rotation S = (1, 4, 7)(2, 5, 8)(3, 6, 9) and two reflections, R = (1)(2, 8)(3, 6)(4, 7)(5)(9) containing the rotation axis and R1 = (1, 9)(2)(3, 4)(5)(6, 7)(8) perpendicular to it (Fig. 2.35). Hence, 1 9 ZG = (t + 4t31 t32 + 3t1 t42 + 2t33 + 2t3 t6 ), 12 1

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The basic polyhedron 9∗ .

the coefficients of ZG (x, 1) = 1 + 2x + 6x2 + 12x3 + 16x4 + 16x5 + 12x6 + 6x7 + 2x8 + x9 represent, respectively, the number of different symmetry choices of n − 9 vertices for 9 ≤ n ≤ 18, and the coefficients of

ZG (x, x, 1) = 1+4x+20x2+76x3 +202x4+388x5 +509x6 +448x7 +228x8 +4x9

the number of source KLs derived from 9∗ for 9 ≤ n ≤ 18. For n ≤ 12, these vertex choices are divided into symmetry equivalence classes and given by their representatives. For n = 10 we have one representative 9∗ a ({1}, {2}) generating 2 source links; for n = 11 two representatives: 9∗ a.b ({1, 2}, ({1, 5}) generating 4 source links, 9∗ a : b ({1, 3}, {1, 4}, {1, 9}, {2, 5}) generating 3 source links; for n = 12 three representatives: 9∗ a.b.c ({1, 2, 3}, {1, 2, 4}, {1, 2, 8}, {1, 2, 9}, {1, 4, 6}, {1, 5, 9}) generating 6 source links, 9∗ a.b : .c ({1, 2, 5}, {1, 2, 6}, {1, 3, 4}, {1, 4, 5}) generating 8 source links, 9∗ a : .b : .c ({1, 4, 7}, {2, 5, 8}) generating 4 source links. The list of the source links derived from these representatives is given in Table 9: Table 9 n = 10

9∗ .2 9∗ .2 0

n = 11

9∗ 2.2 9∗ 2.2 0 9∗ 2 0.2 9∗ 2 0.2 0

n = 12 9∗ 2:2 9∗ 2 : 2 0 9∗ 2 0 : 2 0

9∗ 2.2.2 9∗ 2.2.2 0 9∗ 2.2 0.2 9∗ 2.2 0.2 0 9∗ 2 0.2.2 0 9∗ 2 0.2 0.2 0

9∗ 2.2:.2 9∗ 2.2 : .2 0 9∗ 2.2 0 : .2 9∗ 2 0.2 : .2 9∗ 2.2 0 : .2 0 9∗ 2 0.2 : .2 0 9∗ 2 0.2 0 : .2 9∗ 2 0.2 0 : .2 0

9∗ 2:.2:.2 9∗ 2 : .2 : .2 0 9∗ 2 : .2 0 : .2 0 9∗ 2 0 : .2 0 : .2 0

The KLs of the P (R)-subworld derived from 9∗ are obtained by replacing bigons in the source KLs by R-tangles which do not begin with 1.

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Using the symmetry equivalents, we reduce the derivation to the derivation from the corresponding representatives. For n = 10 we have the representative 9∗ 2 (9∗ 2 0) with P ≃ {(1)}; for n = 11 two representatives: 9∗ 2 : 2 (9∗ 2 0 : 2 0) with P ≃ {(1, 2)}, 9∗ 2.2 (9∗ 2 : 2 0, and all source links derived from 9∗ 2.2) with P ≃ {(1)(2)}. Their permutation groups P have already been considered.

Fig. 2.36

The basic polyhedron 10∗ .

The next member (2 × 5)∗ of the infinite class (2 × k)∗ is the basic polyhedron 10∗ – 5-antiprism, with the graph symmetry group G = [2+ , 10] of order 20, generated by the rotational reflection S˜ = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and by reflection R = (1)(2, 5)(3, 4)(6)(7, 10)(8, 9) (Fig. 2.36). According to PET, ZG =

1 10 (t + 6t52 + 5t21 t42 + 4t25 + 4t10 ), 20 1

ZG (x, 1) = 1 + x + 5x2 + 8x3 + 16x4 + 16x5 + 16x6 + 8x7 + 5x8 + x9 + x10 , ZG (x, x, 1) = 1 + 2x + 15x2 + 56x3 + 194x4 + 428x5 +728x6 + 800x7 + 636x8 + 272x9 + 78x10 (10 ≤ n ≤ 20). For n = 11 we have the representative 10∗ a ({1}) generating 2 source links 10∗ 2, 10∗ 2 0; for n = 12 the representative

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Fig. 2.37

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The basic polyhedron 10∗∗ .

10∗ a.b ({1, 2}, {1, 3}, {1, 6}, {1, 7}, {1, 9}) generating 3 source links 10∗ 2.2, 10∗ 2.2 0, 10∗ 2 0.2 0. Taking for n = 11 the representative 10∗ 2 (10∗ 2 0) with P ≃ {(1)}, we can obtain for n ≤ 12 all links derived from 10∗ . The graph symmetry group G = [2, 2]+ of order 4 corresponds to the basic polyhedron 10∗∗ (Fig. 2.37). This group is generated by two perpendicular 2-rotations S = (1, 6)(2, 7)(3, 8)(4, 9)(5, 10) and S1 = (1, 6)(2, 5)(3, 4)(7, 10)(8, 9). According to PET, ZG =

1 10 (t + t21 t42 + 2t52 ), 4 1

ZG (x, 1) = 1+3x+15x2 +32x3 +60x4 +66x5 +60x6 +32x7 +15x8 +3x9 +x10 , ZG (x, x, 1) = 1 + 6x + 53x2 + 248x3 + 874x4 + 2040x5 +3432x6 + 3872x7 + 2956x8 + 1296x9 + 288x10 (10 ≤ n ≤ 20). For n = 11 we have one representative 10∗∗ a generating 2 source links 10∗∗ 2 and 10∗∗ 20; for n = 12 we have one asymmetric representative 10∗∗ a.b of the equivalence class consisting of eight 2-vertex choices, generating 4 source links 10∗∗ 2.2, 10∗∗ 2.2 0, 10∗∗ 20.2, 10∗∗ 20.2 0 and one symmetric representative 10∗∗ a : b of the equivalence class that consists of seven 2-vertex choices, generating 3 source links 10∗∗ 2 : 2, 10∗∗ 2 : 2 0, 10∗∗ 20 : 2 0.

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Fig. 2.38

The basic polyhedron 10∗∗∗ .

The graph symmetry group G = [2, 4] of order 16 corresponds to the basic polyhedron 10∗∗∗ (Fig. 2.38). This group is generated by the 4rotation S = (1)(2, 3, 4, 5)(6, 7, 8, 9)(10), by the reflection R = (1)(2, 3)(4, 5)(6, 7)(8, 9)(10) containing the rotation axis, and by the reflection R1 = (1, 10)(2, 6)(3, 7)(4, 8)(5, 9) perpendicular to it. For the basic polyhedron 10∗∗∗ , the formula 1 10 ZG = (t + 2t21 t24 + 3t21 t42 + 2t61 t22 + 6t52 + 2t2 t24 ), 16 1 and ZG (x, 1) = 1 + 2x + 8x2 + 13x3 + 25x4 + 25x5 + 25x6 + 13x7 + 8x8 + 2x9 + x10 gives the number of different vertex bicolorings of n − 10 vertices of 10∗∗∗ for 10 ≤ n ≤ 20. Because the axis of 4-rotation contains two vertices of 10∗∗∗ , we can not use PET to obtain the number of source KLs derived from 10∗∗∗ . For n = 11 we have two representatives of equivalence classes: 10∗∗∗ a ({1}) giving 1 source link 10∗∗∗ 2, and 10∗∗∗ .a ({2}) giving 2 source links 10∗∗∗ .2, 10∗∗∗ .2 0; for n = 12 we have three representatives of equivalence classes: 10∗∗∗ a.b ({1, 2}, {1, 6}) generating 2 source links, 10∗∗∗ 2.2 and 10∗∗∗ 2.2 0, 10∗∗∗ .a : b ({2, 4}, {1, 10}, {2, 6}, {2, 8}) generating 2 source

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Fig. 2.39

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The basic polyhedron 11∗ .

links 10∗∗∗ .2 : 2 and 10∗∗∗ .2 : 2 0, 10∗∗∗ .a.b ({2, 3}, {2, 7}) generating 3 source links 10∗∗∗ .2.2, 10∗∗∗ .2.2 0, 10∗∗∗ .2 0.2 0. For n = 12, from 10∗∗∗ 2 we derive the generating link 10∗∗∗ 3, from 10∗∗∗ .2 the generating link 10∗∗∗ .3, and from 10∗∗∗ .2 0 the generating link 10∗∗∗ 3. The graph symmetry group G = [1] of order 2 corresponds to the basic polyhedron 11∗ (Fig. 2.39). This group is generated by the reflection R = (1, 5)(2, 4)(3)(6, 10)(7, 9)(8)(11). According to PET, ZG =

1 11 (t + t31 t42 ), 2 1

ZG (x, 1) = 1 + 7x + 31x2 + 89x3 + 174x4 + 242x5 + 242x6 +174x7 + 89x8 + 31x9 + 7x10 + x11 , ZG (x, x, 1) = 1 + 14x + 120x2 + 688x3 + 2700x4 + 7496x5 + 14944x6 +21312x7 + 21320x8 + 14256x9 + 5728x10 + 1088x11 , so we have the number of vertex choices and the number of source KLs derived from 11∗ for 11 ≤ n ≤ 22. For n = 12 there are 7 vertex choices, each giving 2 source links. The graph symmetry group G = [2] of order 4, generated by two mutually perpendicular reflections R = (1, 11)(2)(3, 4)(5)(6, 7)(8)(9, 10)

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Fig. 2.40

The basic polyhedron 11∗∗ .

and R1 = (1, 10)(2, 8)(3, 6)(4, 7)(5)(9, 11) corresponds to the basic polyhedron 11∗∗ (Fig. 2.40). According to PET, ZG =

1 11 (t + 2t1 t52 + t31 t42 ), 4 1

ZG (x, 1) = 1+4x+18x2+47x3 +92x4 +126x5+126x6 +92x7 +47x8 +18x9 +4x10 +x11 , ZG (x, x, 1) = 1 + 8x + 65x2 + 354x3 + 1370x4 + 3788x5 + 7512x6 +10736x7 + 10700x8 + 7208x9 + 2880x10 + 576x11 . For n = 12, from each of 4 vertex choices, we derive 2 source links. The graph symmetry group G = [2] generated by perpendicular reflections R = (1)(2, 3)(4)(5, 6)(7, 9)(8)(10, 11) and R1 = (1)(2, 10)(3, 11)(4, 8)(5, 7)(6, 9) corresponds to the basic polyhedron 11∗∗∗ (Fig. 2.41). Since the permutation representations of graph symmetry groups of 11∗∗ and 11∗∗∗ are isomorphic, we obtain the same enumeration result, and particular KLs can be mapped from one basic polyhedron to the other using this isomorphism.

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Fig. 2.41

The basic polyhedron 11∗∗∗ .

Next subworld of the P -world is the P RS-subworld. KLs of this kind are obtained by replacing bigons in source KLs of the P -world by RStangles. The symmetry rules determining such substitutions are the same as in the P R-subworld. The representatives of source KLs derived from 6∗ are: .2 .2.2.2.2

.2.2 .2.2.2.2 0

.2.2.2 .2.2.2 0.2 0

.2.2.2 0 2.2.2.2

2:2:2 2.2.2 0.2

The generating P RS-links derived from them are given in Table 10: Table 10 n = 9

.(2, 2)

n = 10

.(3, 2) .(2, 2)1 .(2, 2+)

.(2, 2).2

n = 11

.(2, 2, 2) .(4, 2) .(3 ⋄ , 3 ⋄ ) .(3, 2)1 .(2, 2)2 .(3, 2+) .(2, 2+)1 .(2, 2 + +)

.(3, 2).2 .(2, 3).2 .(2, 2).3 .(2, 2+).2

.(2, 2).2.2 .2.(2, 2).2

.(2, 2).2.2 0 .2.(2, 2).2 0 .2.2.(2, 2) 0

(2, 2) : 2 : 2

n = 12

.(3, 2, 2) .(2, 2, 2)1 .(2, 2, 2+) .(5, 2) .(4, 3) .(4, 2)1 .(3 ⋄ , 3 ⋄ )1

.(2, 2, 2).2 .(4, 2).2 .(2, 4).2 .(3 ⋄ , 3 ⋄ ).2 .(3, 2)1.2 .(2, 3)1.2 .(2, 2)2.2

.(3, 2).2.2 .(2, 3).2.2 .2.(3, 2).2 .(2, 2).3.2 .(2, 2).2.3 .3.(2, 2).2 .(2, 2)1.2.2

.(3, 2).2.2 0 .(2, 3).2.2 0 .2.(3, 2).2 0 .2.(2, 3).2 0 .2.2.(3, 2) 0 .2.2.(2, 3) 0 .(2, 2)1.2.2 0

(3, 2) : 2 : 2 (2, 2) : 3 : 2 (2, 2)1 : 2 : 2 (2, 2+) : 2 : 2

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n = 12

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.(3, 2)2 .(2, 2)3 .(4, 2+) .(3 ⋄ , 3 ⋄ +) .(3, 2+)1 .(2, 2+)2 .(3, 2 + +) .(2, 2 + +)1

.(3, 2).3 .(2, 3).3 .(2, 2).4 .(2, 2).(2, 2) .(3, 2+).2 .(2, 3+).2 .(2, 2+).3 .(2, 2 + +).2

.2.(2, 2)1.2 .(2, 2+).2.2 .2.(2, 2+).2

.2.(2, 2)1.2 0 .2.2.(2, 2)1 0 .(2, 2).3.2 0 .(2, 2).2.3 0 .3.(2, 2).2 0 .2.(2, 2).3 0 .3.2.(2, 2) 0 .2.3.(2, 2) 0 .(2, 2+).2.2 0 .2.(2, 2+).2 0 .2.2.(2, 2+) 0

.(2, 2).2.2.2

.(2, 2).2.2.2 0 .2.(2, 2).2.2 0 .2.2.(2, 2).2 0 .2.2.2.(2, 2) 0

.(2, 2).2.2 0.2 0

(2, 2).2.2.2 2.(2, 2).2.2

(2, 2).2.2 0.2 2.(2, 2).2 0.2 2.2.2 0.(2, 2)

In the same way, from the representatives 8∗ 2, 8∗ 2.2 and 8∗ 2.2 0 (Table 8) for n = 11 we derive the generating link 8∗ (2, 2), and for n = 12 the generating links 8∗ (3 ⋆ , 2), 8∗ (2, 2)1, 8∗ (2, 2+), 8∗ (2, 2).2, 8∗ (2, 2).2 0, 8∗ 2.(2, 2) 0. From the representative 9∗ 2 (Table 9) for n = 12 we derive the generating link 9∗ (2, 2). For n = 12 there exist 12 basic polyhedra (Caudron, 1982), illustrated in Fig. 2.28, given in 3D-form showing their symmetry, not always directly visible from their graphs or Schlegel diagrams.

Fig. 2.42

The basic polyhedra 1413∗ and 1451∗ .

As we have seen, the basis of derivation for every class of KLs are source KLs. In the case of polyhedral world, from every basic polyhedron we first derive source KLs, and then continue the derivation by different tangle substitutions. For the basic polyhedra that do not have rotation axes of order 4 incident with some vertices, we used the PET (Polya Enumeration

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Theorem) for the computation of the number of source KLs. If such a 4-rotation exists, it interchanges two possible positions of a bigon (2 and 2 0), so the application of the PET is not possible. For n ≤ 14, the basic polyhedra with such 4-rotation are 6∗ , 10∗∗∗ , 1413∗ and 1451∗ (Figs. 2.31, 2.38, 2.42). For all the other basic polyhedra with n ≤ 14 crossings we computed the number of source KLs, given in Table 11, where the first datum is the ordering number of the polyhedron, followed by the number of source KLs generated from it, with 1,2,. . ., n bigons. Table 11 B.P.

1

2

3

4

5

6

7

8

9

10

11

8∗ 9∗ 10∗ 10∗∗ 11∗ 11∗∗ ∗∗∗ 11 121∗ 122∗ 123∗ 124∗ 125∗ 126∗ 127∗ 128∗ 129∗ 1210∗ 1211∗ 1212∗ 131∗ 132∗ 133∗ 134∗ 135∗ 136∗ 137∗ 138∗ 139∗ 1310∗ 1311∗ 1312∗ 1313∗ 1314∗ 1315∗ 1316∗ 1317∗ 1318∗ 1319∗ 141∗ 142∗ 143∗ 144∗ 145∗ 146∗ 147∗ 148∗ 149∗ 1410∗ 1411∗ 1412∗ 1414∗ 1415∗ 1416∗ 1417∗ 1418∗ 1419∗ 1420∗ 1421∗ 1422∗ 1423∗

2 4 2 6 14 8 8 12 2 4 2 12 14 14 12 5 6 4 6 10 16 16 10 16 14 14 14 26 14 16 10 16 26 26 16 26 14 14 10 16 28 14 16 16 14 14 14 28 14 8 12 28 18 8 28 14 14 28 18 18

12 20 15 53 120 65 65 138 12 34 18 138 139 139 138 22 75 34 41 92 167 167 92 167 162 162 162 312 162 167 92 167 312 312 167 312 162 126 107 190 364 189 190 190 189 189 189 364 189 102 128 364 199 102 364 189 189 364 199 199

34 76 56 248 688 354 354 880 54 178 84 880 900 900 880 65 440 172 196 612 1178 1178 612 1178 1156 1156 1156 2288 1156 1178 612 1178 2288 2288 1178 2288 1156 792 768 1480 2912 1456 1480 1480 1456 1456 1456 2912 1456 740 876 2912 1512 740 2912 1456 1456 2912 1512 1512

87 202 194 874 2700 1370 1370 3990 206 746 368 3990 4000 4000 3990 139 2025 738 746 2955 5800 5800 2955 5800 5750 5750 5750 11440 5750 5800 2955 5800 11440 11440 5800 11440 5750 3589 4126 8062 16016 8050 8062 8062 8050 8050 8050 16016 8050 4073 4503 16016 8156 4073 16016 8050 8050 16016 8156 8156

124 388 428 2040 7496 3788 3788 12672 596 2260 1096 12672 12752 12752 12672 218 6336 2224 2130 10486 20752 20752 10486 20752 20652 20652 20652 41184 20652 20752 10486 20752 41184 41184 20752 41184 20652 12154 16236 32152 64064 32032 32152 32152 32032 32032 32032 64064 32032 16076 17244 64064 32352 16076 64064 32032 32032 64064 32352 32352

136 509 728 3432 14944 7512 7512 29648 1356 5219 2573 29648 29688 29688 29648 252 14904 5170 4697 27776 55192 55192 27776 55192 54992 54992 54992 109824 54992 55192 27776 55192 109824 109824 55192 109824 54992 31240 48518 96296 192192 96236 96296 96296 96236 96236 96236 192192 96236 48288 50684 192192 96696 48288 192192 96236 96236 192192 96696 96696

72 448 800 3872 21312 10736 10736 50688 2256 8792 4304 50688 50848 50848 50688 218 25344 8704 7772 55392 110224 110224 55392 110224 109984 109984 109984 219648 109984 110224 55392 110224 219648 219648 110224 219648 109984 61168 110464 219968 439296 219648 219968 219968 219648 219648 219648 439296 219648 109984 114304 439296 220608 109984 439296 219648 219648 439296 220608 220608

30 228 636 2956 21320 10700 10700 63480 2836 11032 5451 63480 63560 63560 63480 139 31860 10932 9668 82948 165256 165256 82948 165256 164856 164856 164856 329472 164856 165256 82948 165256 329472 329472 165256 329472 164856 90856 193212 384824 768768 384664 384824 384824 384664 384664 384664 768768 384664 192692 198888 768768 385704 192692 768768 384664 384664 768768 385704 385704

74 272 1296 14256 7208 7208 56320 2504 9772 4776 56320 56480 56480 56320 65 28160 9680 8642 92200 183600 183600 92200 183600 183280 183280 183280 366080 183280 183600 92200 183600 366080 366080 183600 366080 183280 100848 257296 512992 1025024 512512 512992 512992 512512 512512 512512 1025024 512512 256496 264304 1025024 514112 256496 1025024 512512 512512 1025024 514112 514112

78 288 5728 2880 2880 33888 1568 2208 2960 33888 33968 33968 33888 22 17040 5920 5465 73760 146928 146928 73760 146928 146528 146528 146528 292864 146528 146928 73760 146928 292864 292864 146928 292864 146528 81392 257520 513088 1025024 512848 513088 513088 512848 512848 512848 1025024 512848 256880 264712 1025024 514128 256880 1025024 512848 512848 1025024 514128 514128

1088 576 576 12288 576 436 1056 12288 12352 12352 12288 5 6144 2176 2132 40448 80288 80288 40448 80288 80064 80064 80064 159744 80064 80288 40448 80288 159744 159744 80288 159744 80064 45472 187264 373120 745472 372736 373120 373120 372736 372736 372736 745472 372736 186560 193216 745472 374144 186560 745472 372736 372736 745472 374144 374144

12

13

14

2208 4224 4224 2208 4224 4160 4160 4160 8192 4160 4224 2208 4224 8192 8192 4224 8192 4160 2688 28992 57472 114688 57344 57472 57472 57344 57344 57344 114688 57344 28736 30784 114688 57856 28736 114688 57344 57344 114688 57856 57856

4320 8320 16384 8256 8320 8320 8256 8256 8256 16384 8256 4224 4704 16384 8448 4224 16384 8256 8256 16384 8448 8448

2080 144 224 2080 2112 2112 2080 1 1072 430 462 13520 26816 26816 13520 26816 26656 26656 26656 53248 26656 26816 13520 26816 53248 53248 26816 53248 26656 15808 94016 186784 372736 186592 186784 186784 186592 186592 186592 372736 186592 93616 97856 372736 187392 93616 372736 186592 186592 372736 187392 187392

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1424∗ 1425∗ 1426∗ 1427∗ 1428∗ 1429∗ 1430∗ 1431∗ 1432∗ 1433∗ 1434∗ 1435∗ 1436∗ 1437∗ 1438∗ 1439∗ 1440∗ 1441∗ 1442∗ 1443∗ 1444∗ 1445∗ 1446∗ 1447∗ 1448∗ 1449∗ 1450∗ 1452∗ 1453∗ 1454∗ 1455∗ 1456∗ 1457∗ 1458∗ 1459∗ 1460∗ 1461∗ 1462∗ 1463∗ 1464∗

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8 16 16 28 14 18 18 16 18 8 28 28 28 18 16 28 28 16 28 16 14 28 28 14 6 2 14 28 8 10 14 14 8 28 14 6 24 18 10 10

102 190 190 364 189 199 199 190 199 102 364 364 364 199 190 364 364 190 364 190 189 364 364 189 60 21 189 364 102 107 189 189 102 364 145 49 274 175 107 107

740 1480 1480 2912 1456 1512 1512 1480 1512 740 2912 2912 2912 1512 1480 2912 2912 1480 2912 1480 1456 2912 2912 1456 398 116 1456 2912 740 768 1456 1456 740 2912 1012 252 1976 1152 768 768

4073 8062 8062 16016 8050 8156 8156 8062 8156 4073 16016 16016 16016 8156 8062 16016 16016 8062 16016 8062 8050 16016 16016 8050 2112 623 8050 16016 4073 4126 8050 8050 4073 16016 5079 1069 10050 5550 4126 4126

16076 32152 32152 64064 32032 32352 32352 32152 32352 16076 64064 64064 64064 32352 32152 64064 64064 32152 64064 32152 32032 64064 64064 32032 8198 2348 32032 64064 16076 16236 32032 32032 16076 64064 19132 3500 38024 20340 16236 16236

48288 96296 96296 192192 96236 96696 96696 96296 96696 48288 192192 192192 192192 96696 96296 192192 192192 96296 192192 96296 96236 192192 192192 96236 24447 7044 96236 192192 48288 48518 96236 96236 48288 192192 55148 9314 109896 57714 48518 48518

109984 219968 219968 439296 219648 220608 220608 219968 220608 109984 439296 439296 439296 220608 219968 439296 439296 219968 439296 219968 219648 439296 439296 219648 55472 15850 219648 439296 109984 110464 219648 219648 109984 439296 123008 19604 245376 127488 110464 110464

192692 384824 384824 768768 384664 385704 385704 384824 385704 192692 768768 768768 768768 385704 384824 768768 768768 384824 768768 384824 384664 768768 768768 384664 97009 27836 384664 768768 192692 193212 384664 384664 192692 768768 212312 32888 423744 218868 193212 193212

256496 512992 512992 1025024 512512 514112 514112 512992 514112 256496 1025024 1025024 1025024 514112 512992 1025024 1025024 512992 1025024 512992 512512 1025024 1025024 512512 129048 36848 512512 1025024 256496 257296 512512 512512 256496 1025024 281440 42942 561920 289568 257296 257296

256880 513088 513088 1025024 512848 514128 514128 513088 514128 256880 1025024 1025024 1025024 514128 513088 1025024 1025024 513088 1025024 513088 512848 1025024 1025024 512848 129254 37088 512848 1025024 256880 257520 512848 512848 256880 1025024 282048 43299 562944 290280 257520 257520

186560 373120 373120 745472 372736 374144 374144 373120 374144 186560 745472 745472 745472 374144 373120 745472 745472 373120 745472 373120 372736 745472 745472 372736 93984 26816 372736 745472 186560 187264 372736 372736 186560 745472 207232 32464 413696 214208 187264 187264

93616 186784 186784 372736 186592 187392 187392 186784 187392 93616 372736 372736 372736 187392 186784 372736 372736 186784 372736 186784 186592 372736 372736 186592 47322 13648 186592 372736 93616 94016 186592 186592 93616 372736 106144 17568 211456 110592 94016 94016

28736 57472 57472 114688 57344 57856 57856 57472 57856 28736 114688 114688 114688 57856 57472 114688 114688 57472 114688 57472 57344 114688 114688 57344 14624 4160 57344 114688 28736 28992 57344 57344 28736 114688 33920 6144 67584 36096 28992 28992

4224 8320 8320 16384 8256 8448 8448 8320 8448 4224 16384 16384 16384 8448 8320 16384 16384 8320 16384 8320 8256 16384 16384 8256 2244 687 8256 16384 4224 4320 8256 8256 4224 16384 5248 1176 10240 5760 4320 4320

Obtained combinatorial results can serve for a double check of computer derivation of source KLs from basic polyhedra. In order to work with KLs or for further derivation of polyhedral KLs from source KLs, their number is not enough– the complete list is needed. To obtain this list, we can use an external LinKnot program with the functions fForSourceLinks and fSourceDow. The first function creates all possible Conway symbols of source KLs that can be derived from a basic polyhedron. The other function computes the Dowker codes of all source links derived before, and takes only the first representatives of classes of equal source links. For example, all the source links 6∗ 2, 6∗ .2, 6∗ : 2, 6∗ : .2, 6∗ :: 2, 6∗ :: .2, 6∗ 2 0, 6∗ .2 0, 6∗ : 2 0, 6∗ : .2 0, 6∗ :: 2 0, 6∗ :: .2 0 are isomorphic, so we take the first of them, the source link 6∗ 2, as the representative. All source KLs derived from the basic polyhedron 6∗ (or .1) are given in Table 6. The source links with n ≤ 12 crossings (i.e., with at the most four bigons) derived from the basic polyhedron 8∗ are given in Table 8. For the basic polyhedron 8∗ and for 12 ≤ n ≤ 16 crossings we use the LinKnot external function fSourceDow to obtain the following source links: 124 source KLSs with 5 bigons 8∗ 2.2 0.2 0:.2.2 0

8∗ 2.2 0.2 0.2:2 0

8∗ 2.2 0:2 0.2:2 0

8∗ 2.2 0:2.2 0:2 0

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8∗ 2.2 0:2 0.2 0:2 0 8∗ 2.2 0.2 0:2 0.2 8∗ 2.2 0:2 0.2 0.2 0 8∗ 2.2 0.2:2 0.2 0 8∗ 2.2 0:2.2 0:2 8∗ 2.2.2 0.2 0:.2 8∗ 2.2:2 0.2 0.2 0 8∗ 2.2.2 0.2.2 8∗ 2.2.2 0:.2.2 8∗ 2.2.2 0:2:.2 0 8∗ 2.2.2:2 0.2 0 8∗ 2.2.2 0.2 0:2 8∗ 2.2.2 0:2:2 8∗ 2.2 0.2 0:.2:2 0 8∗ 2.2.2 0.2 0:.2 0 8∗ 2.2.2 0:2 0:.2 0 8∗ 2.2 0.2 0:2:.2 0 8∗ 2.2.2 0.2.2 0 8∗ 2.2 0:2:2:2 0 8∗ 2.2.2 0.2:.2 8∗ 2.2 0:2:2 0.2 0 8∗ 2.2:2.2:2 8∗ 2.2 0.2:2.2 0 8∗ 2.2.2 0.2 0.2 8∗ 2.2:2 0.2:2 0 8∗ 2.2.2 0:2.2 0 8∗ 2.2 0:2 0.2:2 8∗ 2.2 0:2 0.2 0:.2 0 8∗ 2.2 0.2.2 0::2 0 8∗ 2.2 0.2.2 0.2 0

8∗ 2.2 0.2 0:2 0:2 0 8∗ 2.2 0.2:2 0:2 0 8∗ 2.2 0.2 0.2 0:2 0 8∗ 2 0.2 0.2 0:2 0:2 0 8∗ 2.2 0.2 0:2 0.2 0 8∗ 2.2.2.2 0:2 0 8∗ 2.2.2.2:2 0 8∗ 2.2.2 0:2.2 8∗ 2.2 0.2 0.2 0.2 8∗ 2.2.2 0.2 0::2 0 8∗ 2:2:2 0.2 0.2 0 8∗ 2.2:2.2 0:2 0 8∗ 2.2.2:2:2 0 8∗ 2.2 0.2 0:.2 0.2 0 8∗ 2.2 0.2 0::2 0.2 0 8∗ 2.2.2 0:.2 0.2 0 8∗ 2.2.2.2 0:.2 0 8∗ 2.2.2 0:.2 0.2 8∗ 2.2 0.2.2 0.2 8∗ 2.2:2.2 0:2 8∗ 2.2 0.2 0:2:2 0 8∗ 2.2.2:2:2 8∗ 2.2.2 0.2::2 0 8∗ 2.2.2:2 0.2 8∗ 2.2.2 0:2 0.2 0 8∗ 2.2.2.2 0::2 0 8∗ 2.2:2 0.2 0.2 8∗ 2.2 0.2 0:2 0:.2 0 8∗ 2.2 0.2 0:.2 0:2 0 8∗ 2.2 0.2 0:.2 0.2

8∗ 2.2 0:2 0:2 0.2 0 8∗ 2.2 0.2 0:2 0:2 8∗ 2.2 0.2 0:2.2 0 8∗ 2 0.2 0:2 0.2 0:2 0 8∗ 2.2 0.2 0.2:2 8∗ 2.2.2.2 0.2 8∗ 2.2.2.2.2 0 8∗ 2.2.2 0:.2.2 0 8∗ 2.2.2 0.2 0.2 0 8∗ 2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2:2.2 0 8∗ 2.2:2:2.2 0 8∗ 2.2:2:2 0.2 0 8∗ 2 0.2 0.2 0:2 0.2 0 8∗ 2.2 0:2 0:2 0:2 0 8∗ 2.2 0.2.2 0:2 0 8∗ 2.2 0.2 0.2 0:.2 0 8∗ 2.2:2 0.2.2 0 8∗ 2.2.2.2:2 8∗ 2:2 0.2 0:2 0.2 0 8∗ 2.2.2 0:2:2 0 8∗ 2.2 0.2 0.2 0.2 0 8∗ 2.2.2.2 0.2 0 8∗ 2.2:2.2 0.2 0 8∗ 2.2.2 0.2 0:2 0 8∗ 2.2.2 0.2:2 0 8∗ 2.2.2.2.2 8∗ 2.2 0:2.2 0:.2 0 8∗ 2.2 0:2:2 0:2 0 8∗ 2.2 0.2 0.2.2 0

265

8∗ 2.2.2 0:2 0:2 0 8∗ 2.2 0:2 0.2 0:2 8∗ 2.2 0.2:2 0.2 8∗ 2.2 0.2:2:2 0 8∗ 2.2 0.2 0:2:2 8∗ 2.2:2.2 0.2 8∗ 2.2.2 0:2 0.2 8∗ 2.2.2.2 0:.2 8∗ 2.2.2 0.2:.2 0 8∗ 2.2 0.2 0.2 0:.2 8∗ 2:2 0.2 0.2 0.2 0 8∗ 2.2.2 0.2:2 8∗ 2.2:2.2:2 0 8∗ 2 0.2 0.2 0.2 0:2 0 8∗ 2.2 0:.2 0.2 0.2 0 8∗ 2.2 0:2 0.2.2 0 8∗ 2.2 0.2 0.2 0::2 0 8∗ 2.2 0.2.2 0:.2 8∗ 2.2.2.2 0:2 8∗ 2:2 0.2 0.2 0:2 0 8∗ 2.2 0.2:2:2 8∗ 2.2 0:2 0:2 0.2 8∗ 2.2.2:2.2 8∗ 2.2:2 0.2 0:2 0 8∗ 2.2 0.2 0.2 0:2 8∗ 2.2.2 0:2 0:2 8∗ 2.2.2:2 0:2 0 8∗ 2.2 0.2.2 0:.2 0 8∗ 2.2 0.2.2 0:2 8∗ 2.2 0:2:2 0.2

136 source KLs with 6 bigons: 8∗ 2.2.2 0.2 0.2:2 0 8∗ 2.2 0.2 0.2 0:2.2 0 8∗ 2.2 0.2 0.2 0:2 0.2 0 8∗ 2.2.2 0:2 0.2 0.2 8∗ 2.2.2.2 0:2 0.2 0 8∗ 2.2.2 0:2 0.2 0.2 0 8∗ 2.2.2 0:2.2 0.2 8∗ 2 0.2 0.2 0.2 0:2 0.2 0 8∗ 2.2:2.2 0.2 0.2 8∗ 2.2.2 0.2.2:2 0 8∗ 2.2.2 0.2:.2.2 0 8∗ 2.2.2 0.2 0.2:.2 0 8∗ 2.2.2.2 0:2.2 8∗ 2.2.2 0.2 0.2:2 8∗ 2.2.2.2.2.2 8∗ 2.2 0.2 0:2.2 0:2 0 8∗ 2.2 0.2 0.2.2 0.2 0 8∗ 2.2 0.2.2 0.2 0.2

8∗ 2.2.2 0:2.2 0:2 0 8∗ 2.2 0.2 0.2 0.2 0:2 0 8∗ 2.2 0.2 0.2:2 0.2 0 8∗ 2.2.2 0.2 0:2 0.2 8∗ 2.2.2 0.2:2 0.2 0 8∗ 2.2.2.2 0.2:2 0 8∗ 2 0.2 0.2 0.2 0.2 0:2 0 8∗ 2 0.2 0.2 0:2 0.2 0.2 0 8∗ 2.2.2.2:2.2 0 8∗ 2.2.2 0.2 0.2 0.2 8∗ 2.2.2.2 0.2.2 8∗ 2.2 0.2 0.2 0.2 0.2 0 8∗ 2.2:2.2 0.2 0.2 0 8∗ 2.2.2.2 0.2 0:2 8∗ 2.2.2.2:2.2 8∗ 2.2 0.2 0.2 0:2 0:2 0 8∗ 2.2 0.2 0:2 0:2 0.2 0 8∗ 2.2 0.2.2 0.2 0:2 0

8∗ 2.2.2 0:2.2 0.2 0 8∗ 2.2 0:2 0.2 0.2 0.2 0 8∗ 2.2 0.2 0:2.2 0.2 0 8∗ 2.2 0.2:2 0.2 0.2 0 8∗ 2.2.2 0.2 0:2 0:2 0 8∗ 2.2.2 0.2:2 0:2 0 8∗ 2.2.2 0.2:2 0.2 8∗ 2.2.2 0.2 0:2 0.2 0 8∗ 2.2.2 0:2 0.2.2 8∗ 2.2.2.2.2.2 0 8∗ 2.2.2 0:2.2.2 0 8∗ 2.2.2 0.2:.2 0.2 0 8∗ 2.2.2.2.2 0:2 0 8∗ 2.2.2:2.2 0.2 0 8∗ 2.2.2 0.2 0:.2 0.2 0 8∗ 2.2 0.2 0:2 0.2 0:2 0 8∗ 2.2 0:2 0.2 0.2 0:2 0 8∗ 2.2 0.2 0.2.2 0:2

8∗ 2.2 0.2 0.2 0.2:2 0 8∗ 2.2 0.2 0:2 0.2 0.2 0 8∗ 2.2.2.2 0.2 0:2 0 8∗ 2.2.2 0.2 0.2 0:2 0 8∗ 2.2.2 0:2 0.2 0:2 0 8∗ 2.2.2 0.2 0:2.2 0 8∗ 2.2.2.2 0:2.2 0 8∗ 2.2.2.2 0.2 0.2 8∗ 2.2.2 0.2:2.2 8∗ 2.2.2.2.2 0.2 8∗ 2.2.2.2 0.2 0.2 0 8∗ 2.2 0.2 0.2 0.2 0.2 8∗ 2.2 0.2 0.2 0.2 0:2 8∗ 2.2.2 0.2:2:2 0 8∗ 2.2 0.2 0.2.2 0:2 0 8∗ 2 0.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2 0:2 0.2.2 0 8∗ 2.2 0.2.2 0.2 0:2

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8∗ 2.2 0.2.2 0:2 0.2 0 8∗ 2.2 0.2.2 0.2 0.2 0 8∗ 2.2.2.2 0.2.2 0 8∗ 2.2 0.2.2 0.2:2 0 8∗ 2:2 0.2 0.2 0.2 0.2 0 8∗ 2.2 0.2 0:2 0.2 0.2 8∗ 2.2.2 0.2:2.2 0 8∗ 2.2.2 0.2 0.2.2 8∗ 2.2.2.2 0.2:.2 0 8∗ 2.2.2 0.2 0:2.2 8∗ 2.2.2.2:2 0.2 0 8∗ 2.2 0.2.2 0.2.2 0 8∗ 2.2 0.2.2 0.2:2 8∗ 2.2.2 0.2 0.2.2 0 8∗ 2.2.2:2 0.2.2 0 8∗ 2.2.2:2.2.2

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8∗ 2.2 0.2 0:2 0.2.2 0 8∗ 2.2 0.2 0.2 0.2.2 0 8∗ 2.2.2.2.2 0:.2 0 8∗ 2.2 0.2:2 0.2.2 0 8∗ 2.2.2.2 0.2:2 8∗ 2.2 0.2 0.2 0:2 0.2 8∗ 2.2:2 0.2 0.2 0.2 0 8∗ 2.2.2 0:2.2:2 0 8∗ 2.2.2 0.2.2.2 0 8∗ 2.2.2.2.2:2 0 8∗ 2.2 0.2 0.2 0.2 0:.2 0 8∗ 2.2 0.2 0.2 0:.2 0.2 0 8∗ 2.2.2 0.2.2 0:2 8∗ 2.2.2 0.2.2 0.2 0 8∗ 2.2.2.2 0:2:2 0 8∗ 2.2 0.2.2 0.2 0:.2 0

8∗ 2.2.2 0.2 0.2 0:.2 0 8∗ 2.2 0.2 0:2 0.2:2 0 8∗ 2.2:2.2 0.2.2 0 8∗ 2.2.2.2.2 0:2 8∗ 2.2.2 0.2.2:2 8∗ 2.2 0.2:2.2 0.2 0 8∗ 2.2.2.2.2 0.2 0 8∗ 2.2.2 0.2 0.2 0.2 0 8∗ 2.2.2.2 0:2 0.2 8∗ 2.2 0.2 0.2 0.2:2 8∗ 2.2 0.2.2 0:2 0:2 0 8∗ 2.2 0:2 0.2.2 0:2 0 8∗ 2.2 0.2 0:2:2 0.2 0 8∗ 2.2 0.2.2 0:2 0.2 8∗ 2.2.2.2.2:2 8∗ 2.2 0.2 0.2.2 0:.2 0

8∗ 2.2.2 0.2.2 0:2 0 8∗ 2.2.2 0.2.2 0.2 8∗ 2.2 0.2.2 0:2.2 0 8∗ 2.2.2:2.2 0.2 8∗ 2.2.2 0.2 0:2:2 0 8∗ 2.2 0.2 0.2:2.2 0 8∗ 2.2.2:2.2.2 0 8∗ 2.2.2.2 0:2 0:2 0 8∗ 2.2.2:2 0.2 0.2 0 8∗ 2.2.2 0.2 0.2 0:2 8∗ 2.2.2 0.2.2 0:.2 0 8∗ 2.2 0.2 0.2 0:2:2 0 8∗ 2.2.2.2 0.2 0:.2 0 8∗ 2.2:2 0.2.2 0.2 0 8∗ 2.2 0.2:2.2 0.2 8∗ 2.2 0.2.2 0:2:2 0

72 source KLs with 7 bigons: 8∗ 2.2.2 0.2 0.2 0.2:2 0 8∗ 2.2.2.2.2 0.2 0:2 0 8∗ 2.2.2.2.2 0.2:2 0 8∗ 2.2.2.2.2 0.2 0.2 8∗ 2.2.2 0.2.2.2 0.2 8∗ 2.2.2.2 0.2 0.2 0.2 0 8∗ 2.2.2 0.2.2.2 0:2 0 8∗ 2.2.2 0.2 0.2 0.2.2 8∗ 2.2.2 0.2 0.2.2 0:2 0 8∗ 2.2 0.2 0.2 0.2.2 0.2 0 8∗ 2.2 0.2 0.2 0:2 0.2 0.2 0 8∗ 2.2.2 0.2 0.2.2 0.2 8∗ 2.2.2.2 0.2.2 0.2 0 8∗ 2.2.2 0.2.2 0:2 0.2 0 8∗ 2.2.2 0.2.2 0.2:2 0 8∗ 2.2.2.2.2.2 0.2 8∗ 2.2 0.2 0.2 0.2 0.2 0.2 8∗ 2.2.2.2 0.2 0.2 0:2 0 8∗ 2.2.2.2 0.2.2:2 0 8∗ 2.2 0.2.2 0.2 0.2 0:2 0 8∗ 2.2 0.2.2 0.2.2 0.2 0 8∗ 2.2 0.2 0.2.2 0.2 0:2 0 8∗ 2.2 0.2.2 0.2 0.2 0.2 8∗ 2.2.2.2 0.2:2.2 0

8∗ 2.2 0.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2 0.2:2 0.2 0.2 0 8∗ 2 0.2 0.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2 0.2 0.2.2.2 0 8∗ 2.2.2 0.2.2.2 0.2 0 8∗ 2.2.2.2.2 0.2 0.2 0 8∗ 2.2.2 0.2 0.2 0.2 0.2 8∗ 2.2.2.2.2.2.2 0 8∗ 2.2 0.2 0.2 0.2 0.2 0:2 0 8∗ 2.2 0.2 0.2.2 0.2 0.2 0 8∗ 2.2.2.2 0.2 0.2.2 0 8∗ 2.2 0.2.2 0.2 0.2 0.2 0 8∗ 2.2 0.2 0.2 0.2 0.2.2 0 8∗ 2.2.2 0.2.2 0.2 0:2 0 8∗ 2.2.2 0.2.2 0:2.2 0 8∗ 2.2.2.2.2.2.2 8∗ 2.2.2 0.2:2.2 0.2 0 8∗ 2.2.2.2 0.2:2 0.2 0 8∗ 2.2.2.2.2.2 0:2 0 8∗ 2.2 0.2.2 0.2 0:2 0.2 0 8∗ 2.2.2 0.2.2 0.2.2 0 8∗ 2.2 0.2.2 0.2 0.2.2 0 8∗ 2.2 0.2 0.2.2 0.2 0.2 8∗ 2.2 0.2.2 0.2.2 0:2 0

8∗ 2.2.2 0.2 0.2:2 0.2 0 8∗ 2.2.2 0.2 0.2 0.2 0:2 0 8∗ 2.2.2 0.2 0.2 0:2 0.2 0 8∗ 2.2.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2.2 0.2 0.2.2 8∗ 2.2.2.2 0.2.2.2 0 8∗ 2.2.2.2.2.2 0.2 0 8∗ 2.2.2.2.2 0.2.2 8∗ 2.2.2 0.2 0.2.2 0.2 0 8∗ 2.2 0.2 0.2 0.2 0:2 0.2 0 8∗ 2.2.2 0.2.2 0.2 0.2 8∗ 2.2.2 0.2 0.2 0.2.2 0 8∗ 2.2.2 0.2.2 0.2 0.2 0 8∗ 2.2.2.2.2 0.2.2 0 8∗ 2.2.2 0.2.2 0.2.2 8∗ 2.2.2.2 0.2.2.2 8∗ 2.2.2.2 0.2 0:2 0.2 0 8∗ 2.2.2.2 0.2 0.2:2 0 8∗ 2.2.2.2 0.2 0.2 0.2 8∗ 2.2 0.2.2 0.2 0.2:2 0 8∗ 2.2 0.2 0.2 0.2.2 0:2 0 8∗ 2.2.2.2 0.2.2 0.2 8∗ 2.2.2.2 0.2.2 0:2 0 8∗ 2.2 0.2.2 0.2.2 0.2

and 30 source KLs with 8 bigons: 8∗ 2.2.2 0.2.2.2 0.2 0.2 0 8∗ 2.2.2.2.2 0.2 0.2 0.2 0 8∗ 2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2.2.2 0.2.2 0.2 0 8∗ 2.2.2.2.2.2.2.2 0 8∗ 2.2.2.2.2.2 0.2 0.2 0

8∗ 2.2.2 0.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2.2.2.2.2 0.2 0 8∗ 2.2.2 0.2 0.2.2 0.2 0.2 0 8∗ 2.2.2 0.2.2 0.2 0.2 0.2 0 8∗ 2.2.2.2 0.2 0.2 0.2 0.2 0 8∗ 2.2 0.2.2 0.2 0.2 0.2 0.2 0

8∗ 2.2.2 0.2 0.2.2.2 0.2 0 8∗ 2.2.2.2.2 0.2.2.2 0 8∗ 2.2 0.2 0.2 0.2 0.2 0.2 0.2 0 8∗ 2.2.2 0.2.2.2 0.2.2 0 8∗ 2.2.2.2 0.2.2.2 0.2 0 8∗ 2.2.2 0.2.2 0.2.2 0.2 0

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Recognition and Generation of Knots and Links

8∗ 2.2 0.2 0.2.2 0.2 0.2 0.2 0 8∗ 2.2.2.2.2.2 0.2.2 0 8∗ 2.2.2.2.2.2.2.2 8∗ 2.2 0.2.2 0.2 0.2.2 0.2 0

8∗ 2.2 0.2 0.2 0.2.2 0.2 0.2 0 8∗ 2.2.2.2 0.2.2 0.2 0.2 0 8∗ 2.2.2.2 0.2.2.2.2 0 8∗ 2.2.2.2 0.2.2 0.2.2 0

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8∗ 2.2.2 0.2.2 0.2 0.2.2 0 8∗ 2.2.2.2 0.2 0.2.2 0.2 0 8∗ 2.2 0.2.2 0.2.2 0.2 0.2 0 8∗ 2.2 0.2.2 0.2.2 0.2.2 0

As we already computed using the PET (see Table 10), from the basic polyhedron 8∗ we derive 2 source KLs with 1 bigon, 12 with 2 bigons, 34 with 3 bigons, 87 with 4 bigons, 124 with 5 bigons, 136 with 6 bigons, 72 with 7 bigons, and 30 source KLs with 8 bigons. We can use these results to continue with the derivation of KLs belonging to the P (R)-subworld and to all other subworlds for larger values of n substituting bigons by R-tangles etc). Using the external LinKnot programs for the derivation of polyhedral source KLs we can try to fill in the missing data in Table 10, for the basic polyhedra 10∗∗∗ , 1413∗, and 1451∗. The main obstacles for derivation are the size of input data: 2n source KLs for a basic polyhedron with n crossings, and the complexity of algorithm for finding representatives of classes of non-equivalent source KLs. So far, we were able to get the following source KLs from the basic polyhedron 10∗∗∗ : 2 source KLs with 1 bigon, 18 with 2 bigons, 66 with 3 bigons, and 237 with 4 bigons. The LinKnot function fGenKL generates all different alternating KLs with n crossings from a given source KL by rational tangle substitutions. For example, from each of P -equivalent source links 2.2.20.2, .2.2.2.20, 2.2.20.20, 2.20.2.20, 20.2.20.20 we obtained 6, 27, 100, 334, 1032, 3020, KLs with n = 11, . . . , 16 crossings, respectively. We are interested not only for particular polyhedral KLs with a given number of crossings, but for their general classes, derived, e.g., from the basic polyhedron 6∗ . From the source link 6∗ 2 we derive 6∗ t1 ; from the source links 6∗ 2.2, ∗ 6 2.2 0, 6∗ 2 : .2 0, 6∗ 2 : .2 we obtain, respectively: 6∗ t1 .t1 6∗ t1 .t1 0 6∗ t1 : .t1 0 6∗ t1 : .t1

6∗ t1 .t2 6∗ t1 .t2 0 6∗ t1 : .t2 0 6∗ t1 : .t2

From the source links 6∗ 2.2 0.2, 6∗ 2.2 : 2 0, 6∗ 2.2 0 : 2 0, 6∗ 2.2.2 0, 6∗ 2.2 : 2, 6∗ 2.2 0 :: 2 0, 6∗ 2.2.2 we obtain, respectively: 6∗ t1 .t1 0.t1 6∗ t1 .t1 6∗ t1 .t3

: t1 0 : t2 0

6∗ t1 .t1 0.t2 6∗ t1 .t1 6∗ t2 .t1

: t2 0 : t3 0

6∗ t1 .t2 0.t3 6∗ t1 .t2 : t1 0

6∗ t1 .t2 : t3 0

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6∗ t1 .t1 0 : t1 0 6∗ t1 .t3 0 : t2 0

6∗ t1 .t1 0 : t2 0 6∗ t2 .t1 0 : t3 0

6∗ t1 .t2 0 : t1 0

6∗ t1 .t2 0 : t3 0

6∗ t1 .t1 .t1 0 6∗ t2 .t1 .t3 0

6∗ t1 .t1 .t2 0 6∗ t3 .t1 .t2 0

6∗ t2 .t1 .t1 0

6∗ t1 .t2 .t3 0

6∗ t1 .t1 : t1 6∗ t1 .t2 : t3 6∗ t3 .t1 : t2 6∗ t1 .t1 0 :: t1 0

6∗ t1 .t1 : t2 6∗ t1 .t3 : t2 6∗ t3 .t2 : t1 6∗ t1 .t1 0 :: t2 0

6∗ t1 .t2 : t1 6∗ t2 .t1 : t3

6∗ t2 .t1 : t1 6∗ t2 .t3 : t1

6∗ t1 .t1 .t1 6∗ t1 .t3 .t2

6∗ t1 .t1 .t2 6∗ t2 .t1 .t3

6∗ t1 .t2 0 :: t3 0 6∗ t1 .t2 .t1

6∗ t1 .t2 .t3

Source link 6∗ 2.2.2.2 0 generates 47 classes, 6∗ 2.2 : 2.2 0, 6∗ 2.2.2 0.2, 6 2.2 0.2.2 0, 6∗ 2.2.2 : 2 0, and 6∗ 2.2.2 : 2 generate 27 classes each, 6∗ 2.2.2.2 and 6∗ 2.2.2 0 : 2 generate 25 classes each, 6∗ 2.2 2.2 generates 14 classes, 6∗ 2.2 0 : 2 0.2 and 6∗ 2.2 : 2 0.2 0 generate 9 classes each. Source links 6∗ 2.2.2.2.2 0, 6∗ 2.2.2.2 0.2 generate 246 classes each, ∗ 6 2.2.2 0.2.2, 6∗ 2.2.2.2 0.2 0, 6∗ 2.2.2.2.2 generate 126 classes each, and 6∗ 2.2.2.2 0 : 2 0, 6∗ 2.2 0.2.2 0.2 generate 66 classes each. Source link 6∗ 2.2.2.2.2.2 0 generates 814 classes, 6∗ 2.2.2.2 0.2.2 0 generates 420, 6∗ 2.2.2.2.2 0.2 0 generates 412, 6∗ 2.2.2.2 0.2 0.2 0 generates 217, 6∗ 2.2.2 0.2.2.2 0, 6∗ 2.2.2.2.2.2 generate 144 classes each, and 6∗ 2.2.2.2.2.2 generates 78 classes. KLs mentioned above are all P R-classes which can be derived from the basic polyhedron 6∗ . ∗

2.7

Basic polyhedra and non-algebraic tangles

Basic polyhedra were considered first by T.S. Kirkman (who called them “solid knots”) (1885a,b), then by J. Conway (1970), and A. Caudron (1982). In the first part of his paper (June 2, 1984) T.S. Kirkman wrote: “Of solid knots we are not treating. If the apparent dignity of knots so maintains itself as to make a treatise on these n-acra desirable, it will be no difficult thing to show in a future memoir how to enumerate and construct them to any value of n without omission or repetition. The beginner can amuse himself with the regular 8-hedron, which is trifilar, or with the unifilar of eight crossings made by drawing within a square askew, and filling up with eight triangles.” and in its Postcript (September 1, 1984): “As it is a brief matter, it may be worth the wile to show how all solid knots can be constructed without omission and repetition.”

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Fig. 2.43

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The basic polyhedra 6∗ , 8∗ , and 9∗ .

A century after, the “missing” basic polyhedron 12E with n = 12 crossings was discovered by A. Caudron (1982). A. Caudron derived basic polyhedra with n ≤ 12 crossings by combining non-algebraic (hyperbolic) tangles. Every basic polyhedron is a 4-regular, 4-edge connected, at least 2vertex connected graph without bigons (Definition 1.43). Polyhedral KLs can be derived from basic polyhedra by substituting vertices by algebraic tangles. Unfortunately, some kind of a data base is unavoidable: for each basic polyhedron we need to know the order of vertices and orientation of tangles. This info is similar to the classical knot tables, but now for basic polyhedra (Figs. 2.43-2.47). In the program LinKnot the list (data base) of basic polyhedra is extended to n ≤ 20 crossings. Applying flypes to basic polyhedra with n ≤ 11 crossings and source links derived from them gives nothing new. For n ≤ 11 there is oneto-one correspondence between basic polyhedra and their corresponding alternating KLs. Unfortunately, this does not hold for the basic polyhedra with n ≥ 12 crossings. This probably explains the “mystery of the missing basic polyhedron” 12E: among the basic polyhedra with n ≤ 12 crossings, 12E is the only basic polyhedron which is a two vertex-connected graph and

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Fig. 2.44

The basic polyhedra 10∗ , 10∗∗ , and 10∗∗∗ .

the first basic polyhedron with two different projections. It has two nonisomorphic alternating diagrams– its other projection is the link 11∗∗∗ 2 (Fig. 2.48). Definition 2.18. A non-algebraic tangle (or hyperbolic tangle) is a tangle that can not be obtained from elementary tangles 0, 1 and −1 by using three operations (page 52), sum, product, and ramification. We can consider two infinite series of non-algebraic tangles. The first series is 5⋆ , 8⋆1 , 11⋆1 , 14⋆1 , 17⋆1 , . . . with n = 3k + 2 crossings, giving the link 2 1 2 and the basic polyhedra 8∗ , 11∗∗ , 141∗ , 171225∗, . . . as the numerator closures. The second is 7⋆ , 9⋆1 , 11⋆2 , 13⋆1 , . . . with n = 2k + 5 crossings. They are illustrated in Fig 2.49. Let 5⋆ and 7⋆ (Fig. 2.49) denote non-algebraic tangles with n = 5 and n = 7 crossings, respectively. The numerator closures of the products 5⋆ 1, 7⋆ 1, 9⋆1 1 are the basic polyhedra 6∗ , 8∗ , 10∗ , respectively. We can distinguish elementary basic polyhedra and composite basic polyhedra: Definition 2.19. A basic polyhedron is called elementary basic polyhedron if it contains at most one non-algebraic tangle, and composite basic polyhedron if it contains at least two non-algebraic tangles.

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Fig. 2.45

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The basic polyhedra 11∗ , 11∗∗ , and 11∗∗∗ .

In this way, for example, the basic polyhedron 10∗∗∗ can be represented as 5⋆ 5⋆ , 11∗∗∗ as 5⋆ 1 5⋆ , etc. Applying flypes to the basic polyhedra with n ≤ 11 crossings yields nothing new– they have only one minimal alternating diagram. The first exception, basic polyhedron 12E, can be denoted by 5⋆ , 1, 5⋆ , 1 (Fig. 2.48a). If we apply one flype, we obtain another projection 5⋆ 2 5⋆ , corresponding to the link 11∗∗∗ 2 (Fig. 2.48b). There is a complete analogy between the first rational link 2 2 2, that has two projections, and the first basic polyhedron 12E, expressed as 5⋆ 2 5⋆ , with the same property. In the same way, we can obtain other basic polyhedra having more then one minimal alternating diagram, e.g., 1318∗ and 136∗ that are two non-isomorphic projections of 5⋆ 1 1 1 5⋆ (Fig. 2.50), where 1318∗ corresponds to the projection 5⋆ 1 1 1 5⋆. Some KLs can be derived from both, but there are KLs that can be derived only from one, but not from some other projection of the same basic polyhedron. For example, the link 125∗ 2 0 ::: 2 0 = 5∗ , 2, 5∗ , 2 can not be obtained from the basic polyhedron 11∗∗∗ (this means, from 11∗∗∗ 2). Among the basic polyhedra with n = 12 crossings, tree are composite: 12E = 5⋆ , 1, 5⋆ , 1 ∼ 5⋆ 2 5⋆ = 11∗∗∗ 2, 12I = 7⋆ 5⋆ , and 12J = 5⋆ 1 1 5⋆. For n = 13 composite basic polyhedra are: 131∗ = 8⋆1 5⋆ , 135∗ = 8⋆2 5⋆ , 1318∗ = 5⋆ 1 1 1 5⋆ ∼ 136∗ , 139∗ = 7⋆ , 1, 5⋆ , 1313∗ = 7⋆ 1 5⋆ ∼ 1311∗, 1319∗ = 5⋆ 1, 1, 5⋆, 1 ∼ (5⋆ , 1) 2 5⋆ = 1210∗2. In other words, the links

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Fig. 2.46

The basic polyhedra 12A-12F .

12E ∼ 11∗∗∗ 2, 1318∗ ∼ 136∗ , 1313∗ ∼ 1311∗, and 1319∗ ∼ 1210∗2 have two different minimal diagrams each. All other KLs corresponding to the basic polyhedra with n ≤ 13 crossings have a single alternating diagram. In order to derive all composite basic polyhedra with n ≤ 15 crossings, 21 hyperbolic tangles (Fig. 2.51) we need: one with n = 5, one with n = 7, two with n = 8, six with n = 9, and eleven with n = 10 crossings. Nonalgebraic tangles with n = 11 crossings are given in Fig. 2.52. Among basic polyhedra with n = 14 crossings, there are 27 composite polyhedra, 18 of their corresponding KLs permit flypes, and among them 15 will have more then one minimal projection. Links associated to the basic polyhedra with n = 14 crossings satisfy following equalities: 1429∗ ∼ 1430∗, 1434∗ ∼ 1445∗, 1435∗ ∼ 1439∗, 1455∗ ∼ 1456∗ ∼ 1458∗, 1463∗ ∼ 1464∗, and the links 1459∗ - 1464∗ have other projections that contain bigons. Among 76 composite basic polyhedra with n = 15 crossings, 59 of their corresponding alternating

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Fig. 2.47

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The basic polyhedra 12G-12L.

KLs permit flypes, and among 257 composite basic polyhedra with n = 16 crossings, 201 of their corresponding alternating KLs permit flypes. Representations of composite basic polyhedra can be used for determining properties of their corresponding alternating KLs. For example, KLs corresponding to the “palindromic” basic polyhedra 10∗∗∗ = 5⋆ 5⋆ , 12J = 5⋆ 1 1 5⋆ , 1420∗ = 7⋆ 7⋆ , 16160∗ = 7⋆ 1 1 7⋆ are achiral from the same reason as their analogous rational KLs from the class p p, where p is an arbitrary tangle. The same holds for all “palindromic” basic polyhedra of the form p′ p′ , where p′ is any non-algebraic tangle.

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Fig. 2.48

(a) The basic polyhedron 12E; (b) the source link 11∗∗∗ 2.

Basic polyhedra representations are not unique: the same composite basic polyhedron (e.g., 16268∗) can be represented by different hyperbolic tangle decompositions (Fig. 2.53). As we have seen, for n ≥ 13 two or more different basic polyhedra (e.g., 1318∗ and 136∗) can be obtained as non-isomorphic projections of the same alternating KL, so even the term “basic polyhedron of an alternating KL” must be reconsidered. For example, the Conway symbols 125∗ 2 0 and 11∗∗∗ 2 1 represent the same alternating link. In the same way, 136∗ 2 = 1318∗2, etc. The same holds for source KLs with n ≥ 12 crossings: two or more mutually non-isomorphic source links can be projections of the same alternating KL. For example, two non-isomorphic source links 11∗∗∗ .2 and 11∗∗∗ :: 2 are two different projections of the same alternating link (Fig. 2.54). The same property holds for source links derived from the basic polyhedra 12E, 12J,. . . In order to avoid ambiguity, instead of defining a basic polyhedron and source link as a graph, it is possible to define it as an alternating KL corresponding to this graph and introduce extended Conway notation for composite basic polyhedra. The LinKnot functions fProdTangles and fSumTangles calculate P data of the product and sum of non-algebraic tangles (denoted by m5∗ , m7∗ , m81∗ − m82∗ , m91∗ − m96∗ , m101∗ − m1011∗, m111∗ − m1138∗) composed with algebraic tangles placed in the basic polyhedron obtained by the product or sum.

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Fig. 2.49 7⋆ .

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Infinite classes of non-algebraic tangles (a) starting from 5⋆ ; (b) starting from

The LinKnot function fCompositePoly detects composite basic polyhedra, and the function fPolyFlype finds those with the corresponding KLs that permit flypes. From the works of J. Conway (1970) and A. Caudron (1982) it is known that the basic polyhedra 6∗ and 10∗∗∗ , and some classes of alternating KLs derived from them are algebraic (according to Definition 1.46, (2)), i.e., they have non-minimal algebraic representations. For example, this holds for the basic polyhedron 6∗ and all alternating KLs of the form

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Fig. 2.50

The basic polyhedra 136∗ and 1318∗ .

.t1 , .t1 .t2 , .t1 : t2 , .t1 .t2 .t3 , .t1 .t2 .t3 .t4 where ti (i = 1, 2, 3, 4) are algebraic tangles (with or without zeros after the tangles). This property of the basic polyhedra 6∗ and 10∗∗∗ is based on the fact that the non-algebraic tangle 5∗ (Fig. 2.51) has the algebraic representation ((2, −2), −2) 1, so the basic polyhedron 6∗ can be represented in the form 5∗ , 1, i.e., as the algebraic link ((2, −2), −2)1, 1. In the same way, 10∗∗∗ can be represented as 5∗ 5∗ , i.e., as the algebraic link (((2, −2), −2) 1) (((2, −2), −2) 1). Definition 2.20. A representation of a basic polyhedron obtained from an algebraic KL diagram by replacing its elementary tangles 1 by 5∗ is called 5∗ -representation. Definition 2.21. Collapse of the non-algebraic alternating tangle 5∗ into the elementary tangle 1 is called 5∗ -collapse. A 5∗ -collapse can be visualized in a simple way: a circular component is deleted, and the central crossing changes the sign (switches from an overcrossing to undercrossing) (Fig. 2.55). Theorem 2.13. Alternating link L corresponding to a basic polyhedron is algebraic if it can be reduced to a KL with an algebraic representation by 5∗ -collapses. Since we have the algebraic representation of the tangle 5∗ , the proof follows immediately. Since the algebraic representation ((2, −2), −2)1, 1 of the tangle 5∗ permits the variation of three parameters, it is clear that algebraic alternating KLs derived from the basic polyhedron 6∗ can have at most four algebraic tangles. For example, if t,t1 ,t2 ,t3 are algebraic tangles and n1 ,n2 ,n3 are

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Fig. 2.51

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Non-algebraic tangles with n ≤ 10 crossings.

n-tangles, alternating KLs of the form .t1 n1 .t2 n2 0.t3 n3 0.t have the algebraic representation ((t1 (n1 + 1), −t3 − (n3 + 1)), −t2 − (n2 + 1)) 1, t Algebraic representations of the basic polyhedra 6∗ and 10∗∗∗ and alternating algebraic KLs with n ≤ 11 crossings derived from them are given by A. Caudron (1982). They can easily be generalized for biger values of n. An example is the basic polyhedron 11∗∗∗ , alternating KLs derived from it, and their algebraic representations. If t,t1 ,...,t6 are algebraic tangles and

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Fig. 2.52

Non-algebraic tangles with n = 11 crossings.

n1 ,...,n6 are n-tangles, alternating KLs of the form 113∗ t 0 : t1 n1 0.t2 n2 0.t3 n3 : .t4 n4 0.t5 n5 0.t6 n6 have the algebraic representation ((t3 (n3 + 1), −t1 − (n1 + 1)), −t2 − (n2 + 1)) 1, t, ((t6 (n6 + 1), −t5 − (n5 + 1)), t4 − (n4 + 1)) 1

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Fig. 2.53

Two different decompositions of the basic polyhedron 16268∗ .

Fig. 2.54

The source links 11∗∗∗ .2 and 11∗∗∗ :: 2.

Fig. 2.55

5∗ -collapse.

For example, the alternating knot 113∗ (2 1, 3) 0 : 3 0.7 2 0.4 : .(3, 2) 3 0.3 0.4 with n = 41 crossings has the algebraic representation ((5, −4), −7 − 3) 1, (2 1, 3), ((5, −4), −(3, 2) − 4) 1 with n = 45 crossings.

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Fig. 2.56

Basic polyhedra with n ≤ 13 crossings which have algebraic representations.

The following list contains the basic polyhedra with n ≤ 13 crossings which have an algebraic representation, each given with its 5∗ representation (Fig. 2.56). n=6 n = 10 n = 11 n = 12 n = 13

6∗ 10∗∗∗ 11∗∗∗ 125∗ 1210∗ 136∗ 1318∗ 1319∗

5∗ 1 5∗ 5∗ 5∗ 1 5∗ 5∗ , 1, 5∗ , 1 5∗ 1 1 5∗ ∗ (5 , 1) 1 (5∗ , 1) 5∗ 1 1 1 5∗ (1, 5∗ , 1) (5∗ , 1)

The following table contains basic polyhedra with 14 ≤ n ≤ 16 crossings which have 5∗ -representations and offer a possibility for further derivation of alternating KLs having an algebraic representation. n = 14 n = 15

n = 16

1414∗ 1463∗ 157∗ 15130∗ 15154∗ 16145∗ 16339∗ 16384∗ 16432∗ 16457∗ 16492∗ 16502∗

1451∗ 1464∗ 1543∗ 15134∗ 15155∗ 16162∗ 16347∗ 16388∗ 16437∗ 16461∗ 16494∗ 16503∗

1455∗

1456∗

1458∗

1460∗

1461∗

1548∗ 15138∗

1551∗ 15146∗

1552∗ 15147∗

1553∗ 15152∗

15115∗ 15153∗

16181∗ 16369∗ 16391∗ 16442∗ 16464∗ 16495∗ 16504∗

16221∗ 16373∗ 16393∗ 16448∗ 16465∗ 16496∗ 16509∗

16256∗ 16377∗ 16402∗ 16451∗ 16473∗ 16498∗

16272∗ 16380∗ 16428∗ 16452∗ 16484∗ 16500∗

16298∗ 16381∗ 16431∗ 16456∗ 16491∗ 16501∗

Among them, one of the most interesting is the basic polyhedron 1451∗,

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Fig. 2.57 (a) A nested collapsing: basic polyhedra 1451∗ and 10∗∗∗ ; (b) knot 1451∗ : 3.2.2.2.2 0 :: .2 0 : 2 0 with n = 22 crossings and its algebraic representation ((2, −3), −(((4, −3), −3) 1) − 1) 1, ((3, −3), −3) 1 with n = 28 crossings.

which reduces into the basic polyhedron 10∗∗∗ by one 5∗ -collapse. The next two 5∗ -collapses reduce it to a Hopf link, so this is an example of a nested 5∗ -collapsing (Fig. 2.57a). The knot 1451∗ : 3.2.2.2.2 0 :: .2 0 : 2 0 with n = 22 crossings and its algebraic representation ((2, −3), −(((4, −3), −3) 1) − 1) 1, ((3, −3), −3) 1 with n = 28 crossings is illustrated in Fig. 2.57b. An example of a double 5∗ -collapse is the alternating knot K = 16442∗ : .2 0 :: 2 ::: .2.2 0 with n = 20 crossings, given by the algebraic representation with n = 26 crossings (Fig. 2.58a) ((2, (−1, −1, −(((2, −3), −3) 1))), (−1, −1, −(((2, −3), −3) 1))) 1, 1

Its basic polyhedron 16442∗ is 5∗ -collapse reducible to the knot 6∗ 2.2 = .2.2 (Fig. 2.58b), which has the algebraic representation ((2, −3), −3) 1, 1= ((2, (−1, −1, −1)), (−1, −1, −1)) 1, 1. The algebraic representation of K

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Fig. 2.58 (a) Knot K = 16442∗ : .2 0 :: 2 ::: .2.2 0 and its algebraic representation; (b) basic polyhedron 16442∗ which by two 5∗ -collapses reduces to .2.2.

is obtained replacing the last −1 in both tangles (−1, −1, −1) by −(((2, −3), −3) 1). 2.7.1

Generalized tangles

Generalized n-tangles with 2n instead of 4 (n = 2) emerging arcs are mentioned in Conway’s paper (1970) without a further elaboration. The same holds for Murasugi (1996, page 172): after defining n-tangles (called there (n, n)-tangles) and giving an example of a (3,3)-tangle, the author continued to work exclusively with 2-tangles (or (2,2)-tangles). Even examples of particular n-tangles (n ≥ 3) and their use is hard to find. However, we are very familiar with an elementary 3-tangle: it is the standard illustration of the third Reidemeister move (Fig. 2.59). A. Caudron (1982) used hyperbolic n-tangles for the construction of basic polyhedra, and H. Moriuchi (2004) enumerated theta-curves (i.e., non-algebraic 2-tangles) with up to 7

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crossings. Non-algebraic 2-tangles with n ≤ 11 crossings are considered in preceding section.

Fig. 2.59

Third Reidemeister move as a 3-tangle.

Generalized tangles will be used for extending Conway notation to nonalgebraic tangles and basic polyhedra, describing non-algebraic tangle types and type-algebra for computing the number of components of non-algebraic KLs. 2.7.2

n-tangles and basic polyhedra

Every n-tangle will be denoted by a regular 2n-gon with 2n arcs emerging from its vertices. For every n-tangle we can distinguish 2n possible positions obtained rotating the tangle by the angle πk n (k = 0, 1, . . . , 2n − 1), and 2n positions of the tangle obtained by a mirror-reflection in a horizontal reflection line and then rotated by the angle πk n (k = 1, . . . , 2n − 1) (Fig. 2.60).

Fig. 2.60

Positions of 3-tangle.

A closure of n-tangle is obtained by joining the remaining free arcs in pairs, without introducing new crossings. For 2-tangles there are two closures: numerator and denominator (N

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and D) closure, and for 3-tangles there are two basic types of closures: a closure where only adjacent vertices are joined (A-closure), and a closure where two opposite vertices are joined (O-closure) (Fig. 2.61). Since the result of a closure depends on a position of a 3-tangle, for every 3-tangle there are two possible A-closures and three possible O-closures. The number of closures, i.e., the number of ways of joining 2n points on a circle by n non-intersecting chords is known as Catalan number (or Segner number). For n ≤ 10 Catalan numbers (the sequence A000108 from OnLine Encyclopedia of Integer Sequences) are n Catalan no.

2 2

3 5

4 14

5 42

6 132

7 429

9 1430

9 4862

10 16796

2n! In general, Catalan number is given by the formula C(n) = n!(n+1)! . LinKnot function fAllClosures gives the list of all closures of a n-tangle (n ≥ 3).

Fig. 2.61

Fig. 2.62

A- and O-closures of 3-tangle.

Elementary non-algebraic 3-tangles |2| and |3|.

An elementary n-tangle with n − 1 vertices (Fig. 2.62) is denoted by |n − 1| or |1 1 . . . 1|, where 1 occurs n − 1 times. As the basic position

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of elementary tangle we take the one where one strand is horizontal and remaining n−1 strands are vertical. An elementary n-tangle |n−1| induces a coordinate system of concentric regular 2n-gons and corresponding regions (Fig. 2.63), where the first lower middle or right region with two vertices is denoted by 1, and other regions (from 1 to 2n) are given in a clockwise order. Every n-tangle placed in this coordinate system can be denoted by |t1 t2 . . . tn−1 | (tn )r1 (tn+1 )r2 . . . (tn+k−1 )rk , where ti (i = 1, 2, . . . , n − 1) is an algebraic tangle placed in the corresponding vertex of |n − 1| (in the order from the right to the left), and tj rj−n+1 (rj−n+1 ∈ {1, . . . , 2n}, j = n, . . . , n + k − 1, k = 1, 2, . . .) is an algebraic tangle tj placed in the region rj−n+1 , between k th and (k + 1)th concentric regular 2n-gon (at the k th level). Since our primary interest is the derivation of basic polyhedra we start adding with algebraic tangles 1 in such a way that no bigons are created. Therefore, all pairs of adjacent regions must have different indexes. If all algebraic tangles are 1, in order to simplify notation, instead of 1rj−n+1 we write just rj−n+1 . (Figure 2.64). In the initial state, all (potential) algebraic tangles have the same orientation (Fig. 2.65). In our notation the symbol 0 has the same meaning as in the Conway notation for polyhedral KLs.

Fig. 2.63

Coordinate system of the tangle |2|.

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Fig. 2.64

Fig. 2.65

Tangle |2| 1 2 3 2 5.

Orientation of algebraic tangles in the coordinate system.

From every open region two arcs emerge, and adjacent regions share the same arc. We can distinguish open regions with one, two, or more vertices, and denote their type by 1,2,3, respectively. Placing new 1-tangle in an open region changes its type and the types of adjacent regions. If its original type was 1, the addition of new 1-tangle is forbidden, because a

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bigon will be obtained. If the type of a region is 2 or 3, it will be changed to 1, and types of its adjacent regions increase by 1. The process of weaving non-algebraic n-tangles, followed by the changes of the types of open regions is illustrated in Fig. 2.66. Thus we obtain non-algebraic n-tangles without bigons. Note that obtained tangles are not necessarily different, i.e., a tangle may be obtained in different ways.

Fig. 2.66

Change of the types of regions.

A closure of an n-tangle is a basic polyhedron, if connecting free arcs yields no bigons. Notice that joining free arcs either closes a region, or merges two regions into one. This means that the region type of a closed region must be greater then 2, and the sum of region types of the two joined regions must be greater then 2. In the case of 3-tangles and A-closures we need three non-adjacent regions of the type 3, and for an O-closure two opposite regions of the type 3 and a pair of opposite regions with the sum of region types greater then 2. In both cases we close regions of the type 3 by connecting emerging arcs. The closure giving a basic polyhedron is unique (up to symmetry). The main purpose of the Crazy Spider Algorithm described above is the derivation of basic polyhedra, and as a side result we get all non-algebraic

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n-tangles without bigons (Fig. 2.64). The introduction of 2n-gonal coordinate system gives new notation for n-tangles derived from the elementary n-tangle |n − 1| and for the basic polyhedra obtained as their closures.

Fig. 2.67

Fig. 2.68

Basic polyhedron 9∗ given by the minimal code |2| 1 2 1 3 2 1 2.

Basic polyhedron 12C given by the code |3| 1 2 1 3 2 1 3 2 3.

For the exhaustive derivation of basic polyhedra with a given number of crossings n there are two possibilities:

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• derive all basic polyhedra from k-tangles with fixed k (k ≥ 3), sufficiently large to obtain all basic polyhedra with n crossings; • for every polyhedron find minimal k such that it can be derived from the basic k-tangle and for this k find its minimal representation. Definition 2.22. Minimal code of basic polyhedron is the code obtained from its minimal representation corresponding to the minimal n. Together with the basic polyhedra which are prime KLs, our algorithm produces non-prime basic polyhedra that should be deleted from the lists of basic polyhedra. The first of them is 6∗ #6∗ with the code |4| 1 3 4 3 2 1 2 4. In the set of different codes describing the same basic polyhedron, the minimal code is taken as the symbol of the basic polyhedron. The list of minimal codes and their corresponding 9 basic polyhedra with n ≤ 11 crossing is given in the following table: |2| 1 2 1 2 |2| 1 2 1 2 1 2 |2| 1 2 1 3 2 1 2 |2| 1 2 1 2 1 2 1 2 |2| 1 2 1 2 3 2 1 2 |3| 1 2 3 2 1 3 2 |2| 1 2 1 2 1 3 2 1 2 |2| 1 2 1 6 1 2 1 2 1 |3| 1 2 3 2 3 2 1 2

6∗ 8∗ 9∗ 10∗ 10∗∗ 10∗∗∗ 11∗ 11∗∗ 11∗∗∗

Among them, composite basic polyhedra 10∗∗∗ and 11∗∗∗ are derived from the elementary 4-tangle |3|. The list of minimal codes and their corresponding 12 basic polyhedra with n = 12 crossings is given in the following table |2| 1 2 1 2 1 2 1 2 1 2 |2| 1 2 1 2 3 2 1 2 1 2 |2| 1 2 1 3 2 1 3 2 1 2 |2| 1 2 1 6 1 2 3 2 1 2 |3| 1 2 1 3 2 1 3 2 3 |3| 1 5 4 3 4 3 2 8 7

12A 12F 12D 12H 12C 12E

|2| 1 2 1 2 1 2 3 2 1 2 |2| 1 2 1 2 3 4 3 2 1 2 |2| 1 2 1 3 2 4 3 2 1 2 |3| 1 2 1 2 3 2 1 2 3 |3| 1 2 3 2 1 3 2 1 2 |3| 1 6 7 6 5 4 3 2 3

12B 12K 12L 12G 12I 12J

The list of 19 basic polyhedra with n = 13 crossings is given in the following table. Every basic polyhedron is represented by its minimal code and its symbol from the LinKnot data base of basic polyhedra.

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|2| 1 2 1 2 1 2 1 3 2 1 2 |2| 1 2 1 2 1 3 2 3 2 1 2 |2| 1 2 1 2 3 2 1 3 2 1 2 |2| 1 2 1 2 3 4 3 2 1 2 1 |2| 1 2 1 6 1 6 1 2 1 2 1 |3| 1 2 1 2 3 2 1 2 3 2 |3| 1 2 3 2 3 2 1 2 1 2 |3| 1 2 6 7 6 5 4 3 2 3 |3| 1 8 1 7 6 5 4 3 2 3 |4| 1 2 1 2 3 4 3 4 3

133∗ 1310∗ 1314∗ 1315∗ 137∗ 134∗ 139∗ 1313∗ 136∗ 1318∗

|2| 1 2 1 2 1 3 2 1 2 1 2 |2| 1 2 1 2 1 6 1 2 1 2 1 |2| 1 2 1 2 3 2 4 3 2 1 2 |2| 1 2 1 6 1 2 1 3 2 1 2 |3| 1 2 1 2 1 3 2 1 2 3 |3| 1 2 3 2 1 3 8 7 6 7 |3| 1 2 3 2 3 6 8 7 6 7 |3| 1 5 4 3 2 1 2 1 2 3 |3| 1 8 1 8 1 2 3 2 1 2

1312∗ 1316∗ 1317∗ 138∗ 132∗ 135∗ 131∗ 1311∗ 1319∗

The following table contains the list of 64 basic polyhedra with n = 14 crossings, given by their minimal codes: |2| 1 2 1 2 1 2 1 2 1 2 1 2 |2| 1 2 1 2 1 2 3 2 1 2 1 2 |2| 1 2 1 2 1 2 3 4 3 2 1 2 |2| 1 2 1 2 1 3 2 4 3 2 1 2 |2| 1 2 1 2 1 6 1 2 3 2 1 2 |2| 1 2 1 2 3 2 1 4 3 2 1 2 |2| 1 2 1 2 3 2 3 4 3 2 1 2 |2| 1 2 1 2 3 2 6 5 4 3 2 3 |2| 1 2 1 2 3 4 5 4 3 2 1 2 |2| 1 2 1 3 2 1 2 1 3 2 1 2 |2| 1 2 1 3 2 1 4 3 2 1 2 1 |2| 1 2 1 3 2 4 3 2 1 2 1 2 |2| 1 2 1 6 1 2 3 2 1 2 1 2 |2| 1 2 1 6 1 6 1 2 3 2 1 2 |3| 1 2 1 2 1 2 3 2 1 3 2 |3| 1 2 1 2 1 3 2 1 3 2 3 |3| 1 2 1 2 3 2 1 2 3 2 3 |3| 1 2 1 2 3 4 3 2 1 2 3 |3| 1 2 3 2 1 3 2 1 2 1 2 |3| 1 2 3 2 1 3 2 8 7 6 7 |3| 1 2 3 2 3 2 1 3 2 1 2 |3| 1 2 3 2 3 2 3 8 7 6 7 |3| 1 2 3 2 3 4 3 2 1 3 2 |3| 1 2 6 7 6 5 4 3 2 3 2 |3| 1 5 4 3 4 3 2 1 2 3 2 |3| 1 6 7 6 5 4 3 2 3 2 3 |3| 1 6 7 6 5 4 5 4 3 2 3 |3| 1 8 1 2 7 6 5 4 3 2 3 |3| 1 8 7 8 7 6 5 4 3 4 3 |4| 1 2 3 4 3 2 1 4 3 2 |4| 1 3 2 4 3 2 1 3 2 4 |4| 1 6 5 4 5 4 3 2 1 2

1449∗ 1410∗ 1450∗ 1437∗ 1421∗ 1418∗ 1444∗ 1412∗ 1427∗ 1443∗ 1454∗ 1416∗ 144∗ 1411∗ 149∗ 145∗ 1417∗ 142∗ 1431∗ 1446∗ 1457∗ 1424∗ 1425∗ 1440∗ 1462∗ 1422∗ 1445∗ 1439∗ 1461∗ 1451∗ 1414∗ 1460∗

|2| 1 2 1 2 1 2 1 2 3 2 1 2 |2| 1 2 1 2 1 2 3 2 3 2 1 2 |2| 1 2 1 2 1 3 2 1 3 2 1 2 |2| 1 2 1 2 1 3 6 5 4 3 2 3 |2| 1 2 1 2 3 2 1 2 3 2 1 2 |2| 1 2 1 2 3 2 3 2 1 2 1 2 |2| 1 2 1 2 3 2 4 3 2 1 2 1 |2| 1 2 1 2 3 4 3 2 1 2 1 2 |2| 1 2 1 2 3 4 6 5 4 3 2 3 |2| 1 2 1 3 2 1 2 4 3 2 1 2 |2| 1 2 1 3 2 1 6 1 2 1 2 1 |2| 1 2 1 6 1 2 1 2 3 2 1 2 |2| 1 2 1 6 1 5 6 5 4 3 2 3 |3| 1 2 1 2 1 2 3 2 1 2 3 |3| 1 2 1 2 1 3 2 1 2 3 2 |3| 1 2 1 2 1 3 4 3 2 3 2 |3| 1 2 1 2 3 2 1 3 2 3 2 |3| 1 2 1 2 3 4 3 2 1 2 5 |3| 1 2 3 2 1 3 2 4 3 2 3 |3| 1 2 3 2 1 3 6 8 7 6 7 |3| 1 2 3 2 3 2 1 8 1 2 1 |3| 1 2 3 2 3 2 6 8 7 6 7 |3| 1 2 6 7 6 5 4 3 2 1 3 |3| 1 5 4 3 2 5 6 8 7 6 7 |3| 1 5 4 3 4 3 2 3 2 3 2 |3| 1 6 7 6 5 4 3 4 3 2 3 |3| 1 6 8 7 6 5 4 3 4 3 2 |3| 1 8 7 6 8 7 6 5 4 3 4 |4| 1 2 1 8 7 6 5 4 3 4 |4| 1 2 10 9 8 7 6 5 4 5 |4| 1 3 4 3 2 3 2 1 2 4 |5| 1 2 1 2 4 3 5 4 5

148∗ 1433∗ 1442∗ 1432∗ 1441∗ 1426∗ 1419∗ 143∗ 1453∗ 1428∗ 1452∗ 1415∗ 141∗ 1447∗ 146∗ 1438∗ 1413∗ 1448∗ 147∗ 1436∗ 1429∗ 1423∗ 1420∗ 1430∗ 1459∗ 1434∗ 1435∗ 1458∗ 1455∗ 1463∗ 1456∗ 1464∗

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A reader familiar with BF Rs can recognize a similarity of coordinate basic polyhedra notation and their corresponding minimal braid words: 6∗ 8∗ 9∗ 10∗ 10∗∗ ...

(Ab)3 (Ab)4 AbACbACbC (Ab)5 AbAbCbACbC ...

The main difference between Crazy Spider Algorithm and BF Rs is that BF Rs can be used only to recognize certain families of basic polyhedra, while Crazy Spider Algorithm is an algorithm for exhaustive derivation of basic polyhedra. Composite basic polyhedra (starting from 10∗∗∗ ) can be represented as the compositions of non-algebraic tangles. The first of them, 10∗∗∗ is 5∗ 5∗ , the product of two hyperbolic tangles |2| 1 2 1. In the same way, 11∗∗∗ is 5∗ 1 5∗ , 12E is 5∗ , 1, 5∗ , 1=5∗ 2 5∗ ∼ 11∗∗∗ 2, 12I is 7∗ 5∗ , 12J is 5∗ 1 1 5∗, etc. Unfortunately, codes denoting basic polyhedra are not unique and their number grows very fast as the number of crossings increases. For the minimization of codes and decreasing their number we use commutativity of vertices belonging to non-adjacent regions, with the requirement that no bigons are created. For example, the code |3| 1 2 3 5 4 5 2 1 2 is minimized to |3| 1 2 3 2 1 2 5 4 5. Instead of minimizing all possible codes according to rules mentioned, it is possible to construct in advance all minimized codes, and then choose the minimal among them. LinKnot function fBasicTan gives all minimized closed n-tangles with k crossings, and the function fBasicPolyTan gives all basic polyhedra with k crossings derived from n-tangles for fixed n. Now we will consider families of minimal representations of basic polyhedra. The first family consists of n-antiprismatic basic polyhedra (2n)∗ (n = 3, 4, . . .), given by the minimal code |2| (1 2)n−1 , where (1 2)n−1 stands for 1 2 . . . 1 2, with 1 2 repeated n − 1 times. Every other minimal code of a basic polyhedron derived from a n-tangle (n ≥ 3) is of the form |n − 1| sk0 s, where sk0 is an alternating sequence 1 2 . . . of the length k (k ≥ 1), and s is a sequence of numbers ri (ri ∈ {1, 2, . . . , 2n}, i = 1, 2, . . .) denoting regions, which do not begin with 1 or 2. Definition 2.23. A family of basic polyhedra derived from s consists of all

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basic polyhedra of the form |n − 1| sk0 s obtained for a fixed s, and can be denoted by (n,s,k). Basic polyhedra belonging to a (n,s,k)-family for fixed n and s are always obtained by the same closure. For example, for s = 3 2 1 2 and k ≥ 3 is obtained the family |2| sk0 s that consists from the basic polyhedra 9∗ , 10∗∗ , 11∗ , 12B,... given by the minimal codes |2| 1 2 1 3 2 1 2, |2| 1 2 1 2 3 2 1 2, |2| 1 2 1 2 1 3 2 1 2, |2| 1 2 1 2 1 2 3 2 1 2,...; for s = 6 1 2 1 2 1 and k ≥ 1 is obtained the family |2| s02k+1 s, beginning with the basic polyhedron 11∗∗ given by the code |2| 1 2 1 6 1 2 1 2 1, etc. The series sk0 is the k-antiprismatic belt of the basic polyhedron (Fig. 2.69), and s is a 3-tangle. Idea of KL families can be extended to families of basic polyhedra and their properties, e.g., in the family |2| sk0 s beginning with 9∗ every basic polyhedron obtained for k = 1 (mod 3) is a two-component link, and knot otherwise (Fig. 2.70).

Fig. 2.69

Even and odd antiprismatic belt of a basic polyhedron.

The proposed (n,s,k)-construction can be extended to the twoparameter families of the form |n − 1| sk0 s sl1 derived from a sequence s, where s is a sequence of the maximal length, beginning and ending by a number greater then 2, sk0 is defined as before, and sl1 is an alternating sequence of the numbers 1 and 2 of the length l, beginning with 1 or 2. To distinguish these two cases, the first sequence will be denoted by a positive, and the other by negative l. Such families can be denoted by (n,s,k,l), and the sequence s is called a generating sequence. In this construction basic polyhedra belonging to the same (n,s,k,l)-family can be obtained by different closures. Using the first, (n,s,k)-construction, all basic polyhedra with at most 16 crossings obtained from an elementary 3-tangle can be derived from the following set of generating s-sequences given in the lexicographic order.

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The family |2| sk0 3 2 1 2 (k ≥ 3) beginning with the basic polyhedron 9∗ .

3 323 3213 32123 34323 61213 321213 321616 324343 343243 365454 612543 3212123 3213243 3216123 3234543 3243543 3432143 3454543 3654543 6156543 32121213 32124543 32145654 61212123 61234543 61654543

6 343 3216 32143 34343 61216 321216 323213 324543 343543 365654 612654 3212143 3213654 3216543 3234654 3243654 3432543 3454654 6121213 6165654 32121243 32126543 32146543 61213213 61546543 65465654

616 3243 32323 34543 61616 321243 323243 324654 343654 612123 615654 3212343 3214323 3232143 3236543 3246543 3432654 3456543 6123213 6546543 32123213 32143213 32161213 61215654 61565654

3654 32343 34654

6123 32654 36543

321323 323654 326543 345654 612323 616123 3212654 3214543 3232343 3243213 3265434 3434543 3465654 6123243

321343 324323 343213 346543 612343 616543 3213213 3214654 3232654 3243243 3432123 3435654 3654354 6123543

32123243 32143243 32165654 61216123 61612123

32123654 32143654 32432123 61216543 61616123

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In the same way, generating s-sequences for n = 4, 5, 6, . . . can be derived. The s-sequence corresponding to the n-antiprismatic basic polyhedra (2n)∗ (n = 3, 4, . . .) is denoted by (0). If we accept the (n,s,k,l)-minimality criterion, this list can be minimized by deleting the s-sequences 3 2 1 3, 3 2 1 4 3, 3 2 6 5 4, 6 1 2 1 3, 3 2 1 2 1 3, 3 2 1 3 4 3, 3 2 3 2 1 3, 3 2 4 6 5 4, 3 4 3 2 1 3, 6 1 2 6 5 4, 3 2 1 2 1 4 3, 3 2 1 2 6 5 4, 3 2 1 3 2 1 3, 3 2 1 4 5 4 3, 3 2 3 2 1 4 3, 3 2 3 2 6 5 4, 3 2 4 3 2 1 3, 3 4 3 2 1 4 3, 3 4 3 2 6 5 4, 6 1 2 1 2 1 3, 6 1 2 3 2 1 3, 3 2 1 2 1 2 1 3, 3 2 1 2 3 2 1 3, 3 2 1 4 3 2 1 3, 3 2 1 6 1 2 1 3, 6 1 2 1 3 2 1 3, and adding the s-sequences 6 1 6 5 4, 3 2 1 6 5 4, 3 2 1 2 3 2 3, 6 1 2 1 3 2 3. In this case the minimality criterion is the minimal length of the generating sequence s and the lexicographic order. The use of (n,s,k,l)-construction instead of (n,s,k)-construction results in certain differences with regard to the preceding tables. For the generating sequences s = 3 and s = 6 the results are the same as before, but the basic polyhedron 12D=(3,(3, 2, 3),3,4) will be obtained from a shorter sequence s = 3 2 3 as |2| 1 2 1 3 2 3 1 2 1 2, and not from the sequence s = 3 2 1 3 as |2| 1 2 1 3 2 1 3 2 1 2. In the same way, from s = 3 4 3 we obtain two basic polyhedra with 12 crossings 12K=(3,(3, 4, 3),4,−3), 12H=(3,(3, 4, 3),2,−5), etc., so (n,s,k,l)-minimality criterion is more economical for the notation of basic polyhedra. Using (n,s,k,l)-minimality criterion, the tables of the basic polyhedra remain the same for n = 6, 8, 9, 10, 11 crossings, but for the basic polyhedra with n = 12 crossings we have a new table (3,(0),12,0) (3,(3),6,−3) (3,(3,4,3),2,−5) (3,(3,2,4,3),3,−3) (4,(3,2,4,3),4,1) (4,(3,2,1,3,2,3),3,0)

12A 12B 12H 12L 12G 12C

(3,(3),4,−5) (3,(3,2,3),3,4) (3,(3,4,3),4,−3) (4,(3,2,3),2,4) (4,(5,4,3,4,3),1,−3) (4,(8,7,6,5,6,5),1,2)

12F 12D 12K 12I 12E 12J

and for n = 13 crossings the table (3,(3),5,−5) (3,(6),5,5) (3,(3,2,3),5,−3) (3,(6,1,6),3,5) (3,(6,1,2,3),3,4) (4,(3,2,3),5,2) (4,(3,2,4,3,2,3),2,2) (4,(8,1,8,1,2,3),1,−3) (4,(8,7,6,5,4,3,4),1,−2) (5,(3,2,4,3,4),1,−3)

1312∗ 1316∗ 1310∗ 137∗ 138∗ 132∗ 131∗ 1319∗ 1313∗ 1318∗

(3,(3),7,−3) (3,(3,2,3),4,4) (3,(3,4,3),4,−4) (3,(3,2,1,6),3,4) (4,(3,2,3),2,−5) (4,(3,2,1,2,3),4,−1) (4,(6,5,4,3,2,3),2,−2) (4,(5,4,3,2,1,2,5),1,2) (4,(8,7,6,5,4,3,4),1,2)

133∗ 1314∗ 1315∗ 1317∗ 139∗ 134∗ 135∗ 1311∗ 136∗

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Analogous (n,s,k,l)-tables can be obtained for n = 14, 15, . . . The members of two-parameter family of the basic polyhedra obtained from the s-sequence s = 3 as the closures of 3-tangles, are given in the following table: (3,(3),3,−3) (3,(3),5,−3) (3,(3),6,−3) (3,(3),7,−3) (3,(3),8,−3)

= = = = =

9∗ 111∗ 12B 133∗ 148∗

(3,(3),4,−3) (3,(3),4,−5) (3,(3),5,−5) (3,(3),6,−5) etc.;

= = = =

10∗∗ 12F 1312∗ 1410∗

the following family is obtained from s = 6 (3,(6),3,5)= 112∗ etc.;

(3,(6),5,5) = 1316∗

and the following family is obtained from s = (3, 2, 3) (3,(3,2,3),3,4) = 12D (3,(3,2,3),5,−3) = 1310∗ (3,(3,2,3),5,4) = 1442∗ etc.

2.7.3

(3,(3,2,3),4,4) = 1314∗ (3,(3,2,3),4,−5) = 1426∗ (3,(3,2,3),6,−3) = 1433∗

Non-algebraic tangle compositions and component algebra

In the set of n-tangles we introduce several operations under the common name– compositions. Definition 2.24. A composition of two n-tangles is a n-tangle obtained by joining in pairs n adjacent arcs emerging from the first tangle with n adjacent arcs emerging from the other. Note that the set of all n-tangles (n ≥ 2) is closed under compositions. For example, in the set of 2-tangles we have three operations: sum, product, and ramification resulting in a new 2-tangle. The concept of tangle composition can be extended to the set of tangles that consists from n1 -, n2 -,...,nm -tangles (ni ≥ 2, i ∈ {1, 2, . . . m}), where the number of joined arcs is chosen in such a way that every tangle obtained by a composition has 2ni free arcs (i.e., so that a set of ni -tangles is closed under tangle composition). We will consider only 2-tangles and 3-tangles.

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The graph consisting of a regular 2n-gon and n chords joining its distinct vertices is called a chord diagram of order n, or shortly n-diagram. Let the symmetry group G act on a chord diagram. Definition 2.25. Two n-diagrams are equivalent iff there exists an element of the group G that transforms one to another. Definition 2.26. The set of n-diagrams quotient by the equivalence relation coming from the action of identity group G is called the complete set of n-diagrams, or the set of positions of n-diagrams. If G is the dihedral group G = Dn of the order 2n we get the set of basic n-diagrams. Main goal of this section is to determine number of components of KLs obtained as a closure of a composition of n-diagrams (Fig. 2.71).

Fig. 2.71

The composition of two 3-diagrams.

First introduce n-tangle types. Vertices in every n-tangle can be substituted by algebraic tangles. There are three basic types of algebraic tangles, [1]k , [0]k , and [∞]k , where k is the number of internal closed components. From every n-tangle we obtain its corresponding n-diagrams (or Gauss n-diagrams). The number of chord diagrams can be computed combinatorially. The number of the basic chord diagrams for n = 3, . . . 11 is given in the following table (Khruzin, 2000). n

3 5

4 17

5 79

6 554

7 5283

8 65346

9 966156

10 16411700

11 3127002217

The five diagrams obtained for n=3 and 17 diagrams obtained for n = 4 are illustrated in Fig. 2.72 and denoted, respectively, by 3.1-3.5 and 4.14.17.

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In order to make compositions of n-diagrams and count the number of components we use the complete set of chord diagrams and closures of n-tangles. For example, the basic chord diagrams 3.1-3.5 have, respectively, 1, 3, 2, 3, 6 possible positions (Fig. 2.73). Among all basic chord diagrams for n = 3, 4, only one diagram, 4.17, has the left and right form and the maximal number of 16 possible positions.

Fig. 2.72

The basic 3- and 4-diagrams.

The set of n-diagrams is closed with regard to n-tangle compositions modulo internal closed components. For example, the composition of the chord diagram 3.3 with itself gives the same chord diagram with one additional closed internal component (Fig. 2.74), denoted by 3.31 . Hence, we can work in the complete set of n-diagrams and their compositions, and keep the record of the internal closed components by adding subscripts.

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Fig. 2.73

Fig. 2.74

The set of 3-diagrams.

The occurrence of an internal closed component.

The complete set of n-diagrams with the operation of n-tangle composition is the non-commutative monoid – a non-commutative semigroup with the neutral element, known as Brauer semigroup (Wilcox, 2006). The neutral element is the n-diagram with horizontal parallel chords (e.g., 3.4, 4.9). This set has (2n − 1)!! elements, where (2n − 1)!! is the odd factorial number (2n − 1)!! = 1 · 3 · . . . (2n − 1). The number of n-diagrams is given by the sequence A001147 from the Encyclopedia of Integer Sequences by N. Sloane: 1, 3, 15, 105, 945, 10395, . . . and can be easily computed from the general formula. For n = 3 the minimal set of generators of the complete

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set of 3-diagrams consists from three diagrams without connected adjacent vertices (e.g., the diagram 3.1 from Fig. 2.72, and two positions of the diagram 3.2 from Fig. 2.73). For n > 3 we start from the basic n-diagrams without connected adjacent vertices (e.g., the diagrams 4.1-4.7 for n = 4, etc.) From this set we can choose different minimal sets of generators, each consisting from appropriately chosen positions of three different basic diagrams. For example, we can use the diagrams from Fig. 2.75a for n = 4, or the diagrams from Fig. 2.75b for n = 5. Theorem 2.14. For every n (n ≥ 2) the minimal set of generators consists from three diagrams (Radovi´c, 2006). For n = 2, 3, 4, 5, 6, 7, 8, 9 . . . the number of basic n-diagrams is 1, 2, 7, 36, 300, 3218, 42335, 644808, . . ., given by the sequence A007474 from the Encyclopedia of Integer Sequences by N. Sloane (Bar Natan, 1995). The LinKnot function ListOfOneFactors, written by T. Bertok and corrected by the authors, gives as the result all basic n-diagrams without connected adjacent vertices. The function MultTan multiplies n-diagrams. The function fGenSet checks if a given set of diagrams is the generator set of the complete set of n-diagrams.

Fig. 2.75

Minimal sets of generators for (a) 4-diagrams; (b) 5-diagrams.

If the elements of the complete set of 3-diagrams are denoted by 1-15 (Fig. 2.73), the following multiplication table is obtained (subscripts denote internal closed components):

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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1 7 2 13 10 9 8 1 6 5 4 11 12 3 15 14

2 2 21 12 11 12 11 2 11 12 11 111 121 12 2 2

3 10 15 4 7 12 8 3 11 5 13 6 9 1 2 14

4 13 14 7 3 9 11 4 6 12 1 8 5 10 15 2

5 8 14 8 14 51 8 5 81 5 5 8 5 14 141 14

6 9 15 15 9 9 61 6 6 91 15 6 9 6 15 151

7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

8 5 14 14 5 5 81 8 8 51 14 8 5 8 14 141

9 6 15 6 15 91 6 9 61 9 9 6 9 15 151 15

10 3 15 1 13 5 11 10 8 12 7 6 9 4 14 2

11 12 2 2 12 12 111 11 11 121 2 11 12 11 2 21

12 11 2 11 2 121 11 12 111 12 12 11 12 2 21 2

13 4 14 10 1 12 6 13 11 9 3 8 5 7 2 15

14 14 141 5 8 5 8 14 8 5 8 81 51 5 14 14

15 15 151 9 6 9 6 15 6 9 6 61 91 9 15 15

Since every basic polyhedron or polyhedral KL is the composition of ntangles, previous results can be used for finding the number of components of any polyhedral KL. By substituting vertices in n-tangle by algebraic tangle types we obtain n-tangle types. Unfortunately, the correspondence between n-diagrams and n-tangle types is not one-to-one: different n-tangle types can give the same n-diagram. For n = 3, the set of different tangle types can be easily described: in every vertex of the 3-tangle (Fig. 2.76) a “vertical” or “horizontal” mirror can be placed. As the result, following ten 3-tangle types are obtained: (1, 1, 1), (1, ∞, 1), (∞, ∞, ∞), (0, 1, 0), (0, ∞, 0), (1, 0, 1), (∞, 0, ∞), (∞, 1, ∞), (0, 1, ∞), (0, 0, 0). Furthermore, they will give five distinct basic 3-diagrams without internal closed components: (1, 1, 1), (1, ∞, 1), (∞, ∞, ∞), (∞, 1, ∞), (1, 0, 1), and (0, 0, 0) with an internal closed component. This means that the basic 3-diagram 3.3 can be obtained from three tangle types: (∞, ∞, ∞), (0, 1, 0) and (0, ∞, 0), the diagram 3.4 can be obtained from (1, 0, 1) and (∞, 0, ∞), and the diagram 3.5 can be obtained from (∞, 1, ∞) and (0, 1, ∞). For n > 3 things become more complicated. From the basic n-diagram that consists of main diagonal chords (the diagrams 3.1, 4.1, etc.), considering the chords as strands, we obtain n-tangle without multiple crossings, where every strand (chord) intersects each other exactly once. This tangle has n2 vertices. All basic n-diagrams can be obtained from this one, by substituting vertices with the elementary algebraic tangles of the types [1], [0], [∞], i.e., by placing two-sided mirrors in the vertices, and deleting repeated diagrams. Again, different n-tangle types can give the same basic n-diagram, so the correspondence between n-tangle types and basic n-diagrams is many-to-one. For computing the number of components we will use basic n-diagrams, their different positions and compositions.

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Fig. 2.76

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3-tangle and different chord diagrams obtained from it.

Let us give a simple example how one can see that Borromean rings, i.e., the basic polyhedron 6∗ is a three-component link. We take its representation as the composition of two 3-tangles, compute the type of the tangle composition, make its closure, and the conclusion is obvious: Borromean rings are the three-component link (Fig. 2.77). Moreover, we get that every vertex substitution in the basic polyhedron 6∗ , where all substitutes are algebraic tangles of the type [1] gives a three-component link. This method– tangle-type computation, enables us to determine the number of components of any polyhedral KL. The number of components of a basic polyhedron (i.e., its corresponding alternating KL) is not the property of a single basic polyhedron, but of the family of basic polyhedra and depends from family parameters. The same holds for some other KL invariants, like signature or BJunknotting (unlinking) number. This can be illustrated by the example of the one-parameter family of basic polyhedra |2| sk0 s (k ≥ 3, s = 3 2 1 2), starting with the basic polyhedron 9∗ . For k = 3, . . . , 21 we obtained the

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Fig. 2.77

Composition of 3-tangles and Borromean rings obtained as its closure.

following table, where the second column gives the ordering number of the basic polyhedron from the LinKnot data base (for k ≤ 14), the third column the number of components, the fourth column the signature, and the fifth column BJ-unknotting (unlinking) number of the basic polyhedron (see page 83). k 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

BP 9∗ 10∗∗ 11∗ 12B 133∗ 148∗ 1510∗ 1625∗ 17455∗ 182675∗ 195031∗ 2031002∗

σ 2 2 1 1 3 2 3 3 2 2 3 3 3 4 3 3 4 4 4

u 2 1 0 0 1 0 2 1 0 0 1 0 2 1 0 0 1 0 2

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In the table above one can recognize the following patterns: • the number of components is 2 for k = 1 (mod 3), and 1 otherwise; • the signature is 1 for k = 1 (mod 3), 0 for k = 2 (mod 3) or k = 0 (mod 6), and 2 otherwise; • BJ-unknotting (unlinking) number is k for k = 0 (mod 6), k + 1 for k = 2 (mod 6) or k = 5 (mod 6), and k + 2 otherwise. Connections between the families of basic polyhedra and KL invariants (e.g., polynomial) are an open field for research. Another interesting question is the connection between braid family representatives (Gittings, 2004; Jablan and Sazdanovi´c, 2005b) and new basic polyhedra notation introduced. With the new notation of basic polyhedra and canonical orientation of algebraic tangles substituting their vertices (Fig. 2.65), instead of the tables (or data base) of basic polyhedra, we hope to be able to establish hierarchical order in the world of polyhedral KLs, in the same way as it is established for algebraic KLs (rational, stellar, arborescent, etc.)

2.8

KL tables

The first (and still the best program) for knot theory is Knotscape with the tables of knots with n ≤ 16 crossings, giving possibility to compute their various invariants (Alexander, Jones, HOMFLYPT and Kauffman polynomials, hyperbolic invariants, signature, symmetries, etc.) There are several sources on the Internet providing knot data bases, containing polynomials of knots and some other knot invariants. Almost all of them are based on Hoste-Thistlethwaite Knotscape tables of knots (using only a part of them: knots with n ≤ 11 or n ≤ 12 crossings). The best available source of that kind is the Table of Knot Invariants by C. Livingston and J.C. Cha (http://www.indiana.edu/ knotinfo/) which contains knots with n ≤ 12 crossings (and not links), providing a reader the most complete computational results for various knot invariants and some possibilities for interactive computation. More possibilities for interactive use provides Knot Atlas by Dror Bar Natan (http://katlas.math.toronto.edu/wiki/). Knot Atlas contains the tables of knots with n ≤ 11 crossings and links with n ≤ 11 crossings, which can be used for further computations in the Mathematicabased program Knot Theory. Excellent program is R. Scharein’s KnotPlot (http://www.pims.math.ca/knotplot/), which has a superb graph-

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ics. The program Knotilus by S. Rankin (http://srankin.math.uwo.ca/cgibin/retrieve.cgi/html/start.html) contains the largest tables of alternating knots with n ≤ 22 crossings and provides wide possibilities for manipulating with Gauss codes and virtual knots. All the programs and data bases mentioned are based on the classical notation of KLs and their Dowker and Gauss codes. LinKnot uses Conway notation, and the idea of families, classes and worlds of KLs (rational, stellar,...), which makes its data bases (e.g., basic polyhedra) easy to manipulate and extend. For a KL given in Conway notation, e.g., 6∗ 2.3, we can recognize the family 6∗ (2k).(2l + 1) it belongs to. We can take a basic polyhedron or source link from a data base and derive polyhedral KLs. Hence, LinKnot data bases are the source of infinite classes of KLs. The program LinKnot (K2KC) includes the data base KnotLinkBase.m of KLs given in Conway notation as Mathematica strings. This data base contains the complete lists of alternating KLs with n ≤ 12 crossings, and lists of non-alternating KLs with n ≤ 10 crossings, as well as the list of nonalternating knots with n = 11 crossings. There are two functions working with the lists and particular elements of these data bases: NumberOfKL and GetKnotLink. As an input, the function NumberOfKL uses the Mathematica string. To choose a list of alternating KLs with k crossings, you need to write the string ”ak” where a stands for alternating KLs, and k is a number of crossings. For a list of non-alternating KLs with k crossings you need to write ”nk”, where the letter ”n” stands for “nonalternating”, and k is the number of crossings. The list ”n11” contains only non-alternating knots with n = 11 crossings. The output of the function NumberOfKL is the number of alternating (non-alternating) KLs with a specified number of crossings. Input of the function GetKnotLink is a Mathematica string ”aN” or ”nN”, where N is an integer that represents the number of the desired KL in the list. As the output, the function Get KnotLink returns the corresponding Conway symbol that can be used for further calculations. The structure of the database KnotLinkBase.m corresponds to the classification of KLs proposed by A. Caudron (1982): the “worlds” or their “subworlds” (e.g., rational, stellar, arborescent, polyhedral, etc.) in the file KnotLinkBase.nb are denoted by different colors. Generating KLs (whose Conway symbols contain only single vertices and chains of 2 or 3 bigons), are given in the second part of a particular list of alternating aN or non-alternating nN (N = 1, 2, . . . , 12) KLs in the file KnotLinkBase.m and emphasized with different colorings. A family of

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KLs can be obtained from each generating link by adding 2 to the chains of bigons. For example, from the generating rational link 3 1 2 with n = 6 crossings we obtain 5 1 2 and 3 1 4 for n = 8, then 7 1 2, 5 1 4, and 3 1 6 for n = 10, etc. As we pointed out many times, all important properties of KLs (including their Dowker codes, polynomial invariants, minimum braids, signatures, unlinking numbers, linking numbers, symmetry properties, etc.) are well ordered according to families. Unfortunately, the data base KnotLinkBase.nb is perhaps the weakest part of the program LinKnot, since it is created manually and probably contains some errors or misprints. Recall that the most of knot tables that are currently in use are based on the more then 100 year old results of T.P. Kirkman (1885a,b), P.G. Tait (1876/77a,b,c, 1883/84, 1884/85), and C.N. Little (1885, 1890, 1892, 1900), and have been corrected several times. Computer derivations of KLs have only appeared in the last few decades and are mostly restricted to knots (Thistlethwaite, 1999), and links with a low number of crossings given in Dowker notation (Doll and Hoste, 1991; Cerf, 1998). Based on recent results obtained by the program LinKnot, it is reasonable to expect that soon it will be possible to derive KLs in Conway notation by computer, or at least alternating KLs. We are able to generate all rational and stellar KLs and all alternating KLs obtained from some source link– the only restriction on the number of crossings is coming from computer limitations. As it was already explained, derivation of alternating KLs is basically a series of tangle substitutions made in previously generated source KLs, mostly based on specific partitions or compositions of numbers. Symmetry plays an important role in reducing the number of possibilities and recognizing in advance possible repetitions and duplicates of KLs. Since we have no general algorithmic solution for implementation of symmetry and its numerous particular cases in a computer program for KL derivation, sometimes is necessary to create all possible Conway symbols, and then select those with different minimal Dowker codes computed by the LinKnot function MinDowAltKL. Non-alternating KLs of the polyhedral world represent a bigger problem, because the same non-alternating KL can be generated from different basic polyhedra. For example, the Conway symbols 8∗ 2 : −2 0 and 9∗ − 2 0 represent the same non-alternating knot (Fig. 2.78). We can derive nonalternating KLs given by Conway symbols, using the following algorithm: (1) in a Conway symbol of a KL make all combinations of bigon chains

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(including single vertices) with different signs; (2) reduce every KL with the K2K function Reduction KnotLink; (3) calculate different polynomial invariants of reduced KLs and delete repeated ones.

Fig. 2.78 9∗ − 2 0.

Two different representations of the same non-alternating knot 8∗ 2 : −2 0 =

In practice, the K2K function Reduction KnotLink fails for some classes of KLs, so we need to use polynomial invariants to ensure that we have obtained a new non-alternating KL with n crossings (not the already derived one, or an alternating or non-alternating KL with a lower number of crossings). Unfortunately, even polynomials fail to distinguish some pairs (classes) of KLs, so we use all available KL invariants to ensure that we have obtained correct results. In the computer program for the derivation of non-alternating KLs in Dowker notation, M. Thistlethwaite used the following criteria: (1) First filter contains some reduction procedures used in the initial generation of KLs; (2) Polynomials are used at the next stage – they can be quickly computed, although they are not very powerful at distinguishing KLs. They divide KLs into relatively small equivalence classes; (3) Now we introduce more powerful invariants, like homomorphisms of the KL group or those arising from hyperbolic structure (if it exists). Finally, one has to work hard to show that the groups of diagrams that haven’t been distinguished so far, actually do represent the same KL.

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Following the work of J. Conway (1970) and A. Caudron (1982) we propose another approach, using symmetry, graph theory and combinatorics. We try to establish a hierarchy of different representations of nonalternating KLs, criteria for getting one-to-one correspondence between KLs and Conway symbols, and find an algorithm for exhaustive derivation of non-alternating KLs. 2.8.1

Non-alternating and almost alternating KLs

The list of non-alternating KLs with n ≤ 11 crossings is given by J.H. Conway (1970) and A. Caudron (1982). In The Knot Book (1994) C.C. Adams introduced the idea of almost alternating KLs. Definition 2.27. A projection of a KL is called almost alternating if one crossing change can make it alternating. A non-alternating KL is called almost alternating if it has an almost alternating projection. For all KLs with n ≤ 11 crossings, with the exception of four of them, we succeeded in finding their minimal almost alternating representations. In their corresponding Conway symbols the sign +− denotes the crossing change from a1 , ..., an + to a1 , ..., an −, and 1− denotes the crossing change from +1 to −1. Generally, the problem of finding minimal almost alternating representations is a difficult problem with a lot of open questions. The minimality of almost alternating KL-representations can be proven using graph-theoretical argumentation (Caudron, 1982), but for most of them we have only empirical tools: construct all almost alternating representations with a fixed number of crossings and choose the first corresponding to a given non-alternating KL. Whenever it was possible, we requested that the minimal almost alternating representation has to belong to the same family (class, subworld,. . .) of the non-alternating KL. For n ≤ 9 all the sources (Conway, 1970; Caudron, 1982; Adams, 1994; Rolfsen, 1976) agree on the number of non-alternating knots: there are 3 non-alternating knots for n = 8, and 8 for n = 9. For n = 10 in Conway (1970) and Rolfsen (1976) one knot was repeated (Perko pair, 10162 = 10161 ), i.e., 2 1 : −2 0 : −2 0 = 3 : −2 0 : −2 0 After this correction, for n = 10 there are 42 non-alternating knots. For n = 11 crossings, 182 non-alternating knots are given in Conway’s paper, where

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the knots −2 1 0 : 3 : 2, 8∗ − 2 1 0 : .2 and 8∗ − 3 0 :: 2 0 are missing. The complete list, containing 185 knots, is given by A. Caudron and confirmed by computer derivation. For non-alternating links with n ≤ 9 crossings all the sources agree: there is one 3-component link for n = 6, two 2-component links for n = 7, 8 non-alternating links for n = 8 (two 2-component, four 3-component, and two 4-component links), and 28 non-alternating links for n = 9 (nineteen 2-component, and nine 3-component links) (Conway, 1970; Caudron, 1982; Rolfsen, 1976; Doll and Hoste, 1991; Cerf, 1998). These results are confirmed by the computer derivation (Doll and Hoste, 1991; Cerf, 1998). For n = 10 the only sources are works by J.H. Conway (1970) and A. Caudron (1982). They need to be corrected according to the computer derived list of non-alternating links with n = 10 crossings, recently completed by M. Thistlethwaite, who obtained 113 non-alternating links. Derivation of links with n ≤ 10 crossings represented by braids with at most four strands (Kawauchi and Tayama, 2006) confirmed Caudron’s corrections (Caudron, 1982, page 114) of Conway’s results (Conway, 1970). Since an algebraic link that has exactly one negative sign in its Conway notation has an almost alternating projection (Adams, 1994, pp. 140, Exercise 5.32), we can directly conclude that all but 18 of the non-alternating knots in the list of 11-crossing prime knots given in the Conway notation are almost alternating, and for those 18 we will try to find their almost alternating minimal representations. The general derivation rules exist for stellar and arborescent nonalternating KLs, and the same holds for almost alternating representations of stellar and arborescent non-alternating KLs. All stellar non-alternating KLs of the form a1 , . . . , ai −, i = 3, 4, . . . can be directly derived from the KLs of the form a1 , . . . , ai + by replacing + by −. Non-alternating arborescent KLs of the form (a1 , a2 ) (a3 , a4 −) are given by the almost alternating minimal representation (a1 , a2 ) (a3 , a4 +− ). Almost alternating representations of some non-alternating polyhedral KLs can also be obtained in a direct way. Remaining non-alternating KLs that can not be represented directly are given in the following tables. For n = 8 we have the following results: 8310

(2, 2) − (2, 2)

(2, 2) (3 1− , 2)

843 2, 2, 2, 2 − −

2, 2, 2, 3 1−

Two links from this table are given by minimal 10-crossing almost alternating representations, and all the other non-alternating KLs with n = 8 crossings are given by their minimal 9-crossing almost alternating representations.

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For n = 9 we have the following results: 947 9259 9261 9317 9321

8∗ − 2 0 (3, 2) − (2, 2) 2 : −2 0 : −2 0 3, 2, 2, 2 − − . − (2, 2)

8∗ 2 1− (3, 2)(3 1− , 2) 2 0.1− .2.2 : 2 0 3, 2, 2, 3 1− .(2, 3 1− )

2 1− : 2 : 2 (2 1, 2)(3 1− , 2) .(2, 2+− ) (2, 2+)(3 1− , 2)

949 −2 0 : 2 : 2 9260 (2 1, 2) − (2, 2) 9320 .(2, 2−) 9319 (2, 2+) − (2, 2)

From 42 non-alternating knots for n = 10, thirty nine of them can be given by an almost alternating minimal 11-crossing representations, and only three of them, 10152 , 10153 and 10154 , have 12-crossing minimal almost alternating representations. They are given in the following table, which does not include non-alternating knots represented in general form as a1 , a2 , a3 − = a1 , a2 , a3 +− or (a1 , a2 ) (a3 , a4 −) = (a1 , a2 ) (a3 , a4 +− ). 10152 10154 10156 10158 10160 10162 10164

(3, 2) − (3, 2) (21, 2) − (21, 2) −3 : 2 : 2 0 −3 0 : 2 : 2 −3 0 : 2 0 : 2 0 −3 0 : −2 0 : −2 0 8∗ 2 : −2 0

(3, 2)(3 1− , 2 1) (2 1, 2)(3 1− , 3) 2 1 1− 0 : 2 : 2 0 2 1 1− : 2 : 2 2 1 1− : 2 0 : 2 0 3 : 2 1− : 2 8∗ 2 : 2 1−

10153 (3, 2) − (21, 2) 10155 −3 : 2 : 2 10157 −3 : 2 0 : 2 0 10159 −3 0 : 2 : 2 0 10161 3 : −2 0 : −2 0 10163 8∗ − 3 0 10165 8∗ 2 : . − 2 0

(3, 2)(3 1− , 3) 2 1 1− 0 : 2 : 2 2 1 1− 0 : 2 0 : 2 0 2 1 1− : 2 : 2 0 2 0.1− .2.2 : 2 1 0 8∗ 2 1 1− 8∗ 2 : .2 1−

For n = 11 we have 64 non-alternating stellar knots of the form a1 , a2 , a3 − or a1 , a2 , a3 , a4 −, with a minimal 12-crossing almost alternating representation of the form a1 , a2 , a3 +− or a1 , a2 , a3 , a4 +− . Then we have 18 non-alternating arborescent knots of the form (a1 , a2 )(a3 , a4 −) or (a1 , a2 +)(a3 , a4 −), with a minimal 12-crossing almost alternating representation of the form (a1 , a2 )(a3 , a4 +− ) or (a1 , a2 +)(a3 , a4 +− )3 . In the same way, the following eight 11-crossing non-alternating knots are obtained from their 12-crossing minimal almost alternating representations. .(3, 2−).2 .2.(3, 2−) .(3, 2−).20 .2 0.(3, 2−)

.(3, 2+− ).2 .2.(3, 2+− ) .(3, 2+− ).20 .2 0.(3, 2+− )

.(2 1, 2−).2 .2.(2 1, 2−) .(2 1, 2−).20 .2 0.(2 1, 2−)

.(2 1, 2+− ).2 .2.(2 1, 2+− ) .(2 1, 2+− ).20 .2 0.(2 1, 2+− )

Among 11-crossing non-alternating knots, three stellar and ten arborescent knots have 13-crossing minimal almost alternating representations: 3, 3, 3, 2 − − 3, 2 1, 3, 2 − − (2 2, 2) − (2 1, 2) (2 1 1, 2) − (2 1, 2) (3, 2 1) − (2 1, 2) (3, 2+) − (2 1, 2) (2 1, 2+) − (2 1, 2) 3 In

3, 3, 3, 3 1− 3, 2 1, 3, 3 1− (2 2, 2)(3 1− , 3) (2 1 1, 2)(3 1− , 3) (3, 2 1)(3 1− , 3) (3, 2+)(3 1− , 3) (2 1, 2+)(3 1− , 3)

3, 3, 2 1, 2 − − (2 2, 2) − (3, 2) (2 1 1, 2) − (3, 2) (3, 2 1) − (3, 2) (3, 2+) − (3, 2) (2 1, 2+) − (3, 2)

all following tables, such trivial derivations are omitted.

3, 3, 2 1, 3 1− (2 2, 2)(3 1− , 2 1) (2 1 1, 2)(3 1− , 2 1) (3, 2 1)(3 1− , 2 1) (3, 2+)(3 1− , 2 1) (2 1, 2+)(3 1− , 2 1)

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In the same way, from the following 13-crossing minimal almost alternating representations we obtain the next six non-alternating knots: . − (3, 2).2 .2. − (3, 2) . − (3, 2).2 0

.(3 1− , 2 1).2 .2.(3 1− , 2 1) .(3 1− , 2 1).2 0

. − (2 1, 2).2 .2. − (2 1, 2) .2 0. − (3, 2)

.(3 1− , 3).2 .2.(3 1− , 3) .2 0.(3 1− , 2 1)

Finally, from 76 non-alternating polyhedral knots, 74 of them are given by almost alternating minimal 12-crossing representations, but we did not manage to obtain almost alternating representations for the remaining two knots, 2 0. − 3. − 2 0.2 and 2 0. − 2 1. − 2 0.2 −2 2 : 2 : 2 −2 2 : 2 0 : 2 0 2 2 : −2 0 : −2 0 −2 1 1 0 : 2 : 2 0 −2 1 1 : 2 0 : 2 0 −4 0 : 2 : 2 −4 0 : 2 0 : 2 0 −3 1 0 : 2 : 2 −3 1 0 : −2 0 : −2 0 −2 1 1 : 2 : 2 0 −2 1 1 0 : −2 0 : −2 0 −3 0 : 2 1 : −2 0 −2 1 0 : 3 : 2 −2 1 0 : −3 0 : −2 0 −2 1 0 : −2 1 0 : −2 0 2.2 1. − 2.2 2.3. − 2.2 0 2. − 3. − 2 0.2 0 2.2. − 2.2.2 0 2.2 0. − 2.2.2 0 8∗ − 3 1 0 8∗ − 3 0.2 0 8∗ − 2 1 0 : 2 8∗ 3 0 : −2 0 8∗ 2 1 0 : −2 0 8∗ − 2 1 0 : .2 0 8∗ 3 :: −2 0 8∗ 2. − 2 0.2 8∗ 2.2 0. − 2 0 8∗ 2 : 2 0 : −2 0 8∗ 2 0 : −2 0 : 2 0 8∗ 2 : . − 2 0 : .2 8∗ − 2 1 0 : .2 9∗ . − 3 9∗ 2. − 2 9∗ .2 : . − 2 9∗ .2 0 : . − 2 10∗∗ − 2 0 2 0. − 3. − 2 0.2

2 1 1 1− 0 : 2 : 2 2 1 1 1− 0 : 2 0 : 2 0 2 1 1 0.2 0.1− .2.2 2 2 1− : 2 : 2 0 2 2 1− 0 : 2 0 : 2 0 3 1 1− : 2 : 2 3 1 1− : 2 0 : 2 0 4 1− : 2 : 2 3 1 : 2 1− : 2 2 2 1− 0 : 2 : 2 0 2 1 1 : 2 1− : 2 2 0 : 2.2 1.1− .2 1 0 3 1− : 3 : 2 2 1 : 3 : 2 1− 2 1 : 2 1 : 2 1− 2.2 1.2 1− 0.2 2.3.2 1− 0.2 0 8∗ 2 0.2 0.1− .3 2.2.2 1− 0.2.2 0 2.2 0.2 1− 0.2.2 0 8∗ 4 1− 8∗ 2 1 1− .2 0 8∗ 3 1− : 2 8∗ 3 0 : 2 1− 8∗ 2 1 0 : 2 1− 8∗ 3 1− : .2 0 8∗ 3 :: 2 1− 8∗ 2.2 1− .2 8∗ 2.2 0.2 1− 8∗ 2 : 2 0 : 2 1− 8 ∗ 2 0 : 2 1− : 2 0 8∗ 2 : .2 1− : .2 8∗ 3 1− : .2 9∗ .2 1 1− 0 9∗ 2.2 1− 0 9∗ .2 : .2 1− 0 9∗ .2 0 : .2 1− 0 10∗∗ 2 1− ?

−2 2 0 : 2 : 2 0 −2 2 : −2 0 : −2 0 −2 1 1 : 2 : 2

2 1 1 1− : 2 : 2 0 2 2 0 : 2 1− : 2 2 2 1− 0 : 2 : 2

−2 1 1 : −2 0 : −2 0 −4 : 2 : 2 0 −4 0 : −2 0 : −2 0 −3 1 0 : 2 0 : 2 0 −2 1 1 0 : 2 : 2 −2 1 1 0 : 2 0 : 2 0 −3 0 : 2 1 : 2 −3 0 : 2 1 0 : 2 −2 1 0 : 3 0 : 2 −2 1 0 : 2 1 : 2 2. − 2 1.2.2 2. − 3.2.2 0 20.3. − 2.2 2. − 2 1. − 2 0.2 0 2.2. − 2.2 0.2 0 8∗ − 4 0 8∗ − 2 1 1 0 8∗ 3 : −2 0 8∗ − 3 0 : 2 0 8∗ − 2 1 0 : 2 0 8∗ − 3 0 : .2 0 8∗ 3 0 :: −2 0 8∗ 2 1 :: −2 0 8∗ 2. − 2 0.2 0 8∗ 2 : 2 : −2 0 8∗ 2 : −2 0 : 2 0 8∗ 2 0 : 2 0 : −2 0 8∗ 2 : .2 : . − 2 0 8∗ − 3 0 :: 2 0 9∗ . − 2 1 9∗ 2 0. − 2 9∗ . − 2 : . − 2 10∗ − 2 0

2 1 1 0 : 2 1− : 2 3 1 1− 0 : 2 : 2 0 4 : 2 1− : 2 4 1− : 2 0 : 2 0 2 2 1− : 2 : 2 2 2 1− : 2 0 : 2 0 2 1 1− : 2 1 : 2 2 1 1− : 2 1 0 : 2 3 1− : 3 0 : 2 3 1− : 2 1 : 2 2.3 1− 0.2.2 2.2 1 1− 0.2.2 0 2 0.3.2 1− 0.2 8∗ 2 0.2 0.1− .2 1 2.2.2 1− 0.2 0.2 0 8∗ 3 1 1− 8∗ 2 2 1− 8∗ 3 : 2 1− 8∗ 2 1 1− : 2 0 8∗ 3 1− : 2 0 8∗ 2 1 1− : .2 0 8∗ 3 0 :: 2 1− 8∗ 2 1 :: 2 1− 8∗ 2.2 1− .2 0 8∗ 2 : 2 : 2 1− 8 ∗ 2 : 2 1− : 2 0 8∗ 2 0 : 2 0 : 2 1− 8∗ 2 : .2 : .2 1− 8∗ 2 1 1− :: 2 0 9∗ .3 1− 0 9∗ 2 0.2 1− 0 8∗ 2 0.2 0.1− .2 0.2 0 10∗ 2 1−

2 0. − 2 1. − 2 0.2

?

In conclusion, for n ≤ 11 there are only two non-alternating 11-crossing knots, 2 0. − 3. − 2 0.2 and 2 0. − 2 1. − 2 0.2, whose almost alternating representations we did not determined (see Adams, 1994, Unsolved question, page 140, Fig. 5.54) (Fig. 2.79), but we are sure that, if they ex-

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ist, they must have more then 16 crossings. Knowing one of them, the other follows immediately. For one of the knots from Adams’ question, 9∗ . − 2 : . − 2, we obtained 12-crossing minimal non-alternating representation 8∗ 2 0.2 0.1−.2 0.2 0.

Fig. 2.79

The 11-crossing knots 2 0. − 3. − 2 0.2 and 2 0. − 2 1. − 2 0.2.

Completing the tables derived by J. Conway (1970) and A. Caudron (1982), one can search for almost alternating representations of nonalternating links given in the Conway notation for n ≥ 10. 2.9

Projections of KLs and chirality

A knot or link can have more then one minimal diagram. Every minimal diagram of an alternating KL can be obtained from any other minimal diagram of the same KL by a finite series of flypes (Theorem 1.11). If a bigon is denoted by a bold line, an elementary flype can be illustrated as in Fig. 2.80b. Flype diagrams (or vertex-bicolored diagrams) of KLs are obtained in the following way: after collapsing every chain of bigons with both ends incident to the same vertex into a black point, new bigons obtained by the collapse are denoted by bold lines. In vertex-bicolored diagrams, transition from one projection to another by a flype is represented by a mutual place exchange of differently colored vertices connected by a bold edge. For example, the transition from one minimal projection of the link 2 2 2 to the other projection ((1, 2, 1), 1, 1) of the same link, expressed in the language of flype diagrams, is illustrated in Fig. 2.80b. This basic flyping algorithm can be applied only to algebraic tangles. Non-isomorphic projections of KLs with n ≤ 9 crossings and their corresponding flype diagrams are given in Figs. 2.81 and 2.82.

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Families of knots obtained by addition/collapse of bigon chains (i.e., n-moves), are considered by M. Thistlethwaite (1985) in his survey article, which gives a brief account of the work of P.G. Tait, T.P. Kirkman, and C.N. Little.

Fig. 2.80 (a) A flype; (b) an elementary flype and its vertex-colored interpretation; (c) flype diagrams of the link 2 2 2.

A KL is achiral iff it is ambient isotopic to its mirror image (Definition 1.26). For every achiral diagram D of an oriented KL, the writhe w(D) is equal to 0 (Theorem 1.8). Hence, w(D) = 0 is a necessary (but not sufficient, and very weak) condition for the achirality of oriented KLs. First chiral alternating knot with the writhe equal to 0 is 4 1 3 with n = 8 crossings. From 26 alternating knots with n = 10 crossings and with zero writhe, 14 are achiral, and 12 are chiral: 2 1 0 : 2 : 2 0, 2 2 1 1 1 1 2, 2 2, 2 1, 2+, 2 3, 3, 2, 3 0 : 2 0 : 2 0, 3 1 1, 3, 2, 3 1 1 3 2, .3.2 0.2, 3 : 2 0 : 2 0, .3.2.2 0, 4 1 1 1 3, and 4 1, 3, 2. Next we introduce two polynomial invariants of alternating KL projections, in order to distinguish alternating KL projections and determine their chirality. Let us consider an oriented alternating KL diagram D with generators g1 , . . . , gn . In every vertex of D there are three generators: passing generator gi , and incoming and outgoing generators gj , gk , respectively. If ǫ(V ) is the sign of the crossing V , then aii = ǫ(V )t, aij = 1, aik = −1, and dD (t) = det(aij ). For example, for the achiral figure-eight knot 2 2 (or 41 ):

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Fig. 2.81 Projections of algebraic alternating source KLs with n ≤ 8 crossings and their corresponding vertex-bicolored flype diagrams.

t 1 −1 0 0 −t 1 −1 = t4 − 2t2 . dD (t) = −1 0 t 1 1 −1 0 −t

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Fig. 2.82 Projections of polyhedral source KLs with n ≤ 9 crossings and their corresponding vertex-bicolored flype diagrams.

In order to prove next theorem we need to define a permutation matrix: Definition 2.28. A permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column, and zeros elsewhere.

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Theorem 2.15. If dD′ (t) 6= dD′′ (t), two oriented alternating knot diagrams D′ and D′′ are non-isomorphic (Jablan, 1995). Proof. Let D′ and D′′ denote two isomorphic alternating knot diagrams, and A′ and A′′ their corresponding matrices. If D′ and D′′ are isomorphic, there exists a permutation matrix P such that P A′ P −1 = A′′ . From the relations: |P A′ P −1 | = |A′′ |, |P A′ P −1 | = |P ||A′ ||P −1 | = |P ||A′ | |P1 | = |A′ | = |A′′ |, |A′ | = |A′′ |, it follows that dD (t) = det(aij ) is the invariant of alternating knot diagrams. The projection invariant obtained is not a complete invariant of alternating knot diagrams, meaning that two non-isomorphic diagrams can give the same polynomial. The polynomial dD (t) = cn tn + · · · + c1 t has the following properties: (1) for every alternating knot projection D, the degree of dD (t) is n, |cn | = 1 and |c1 | = |w(D)|, where w(D) is the writhe of D; (2) dD (t) and dD (−t) correspond to obverse (mirror-symmetric) knot diagrams; (3) for n = 0 (mod 2), a change of the orientation results in a change of dD (t) to dD (−t), and for n = 1 (mod 2) in a change of dD (t) to −dD (−t). Notice that the matrix defined above is a signed KL generator adjacency matrix, so its determinant remains invariant (up to the sign) with regards to any permutation of columns or rows. This means that we can always transform A′ into the matrix with all diagonal entries equal to t or −t. Hence, |cn | = 1 and the coefficient c1 is the sum of diagonal entries, so |c1 | = |w(D)|. The proof of the second and third property is trivial. According to (2) and (3), in the set of all polynomials dD (t) we may distinguish even polynomials (dD (t) = dD (−t)), containing only even degrees of t, corresponding to achiral knot projections, and odd polynomials (dD (t) = −dD (−t)), containing only odd degrees of t, which are invariant under the change of orientation. As with every polynomial invariant, the projection polynomial dD (t) sometimes fails to detect isomorphism of knot projections or achirality. For example, for n = 10 from 364 non-isomorphic projections of alternating knots it recognizes 363 of them as different, and sometimes fails to detect achirality (this means, sometimes yields an even polynomial d(t) for a chiral knot projection). The following table contains polynomials dD (t) for all different projec-

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tions of knots with n ≤ 7 crossings. A symbol of a knot in the classic notation is in the first column, followed by the Conway symbol of the corresponding projection, minimal Dowker code and the projection polynomial dD (t) given by the sequence of its coefficients in descending order. 31 41 51 52 61 62 63 71 72 73 74 75 7′5 76 7′6 77 7′7

3 22 5 32 42 312 2112 7 52 43 313 322 ((1, 3, 1), 1, 1) 2212 (((1, 2, 1), 1), 1, 1) 21112 ((((1, 2), 1), 1), 1, 1)

462 4682 6 8 10 2 4 4 8 10 2 6 4 8 12 10 2 6 4 8 10 12 2 6 4 8 10 2 12 6 8 10 12 14 2 4 6 4 10 14 12 2 8 6 6 10 12 14 2 4 8 6 10 12 14 4 2 8 4 10 14 12 2 6 8 4 10 12 14 2 8 6 4 8 12 2 14 6 10 4 8 12 10 2 14 6 4 8 10 12 2 14 6 4 8 12 14 2 6 10

1 1 1 1 1 1 1 1 1 −1 1 −1 −1 1 −1 1 −1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 −2 5 1 −4 1 2 7 −3 −1 −4 −2 −3 −1 0 −1 0

0 0 0 −2 −1 0 0 0 0 0 1 0 −1 −1 −1 −1

5 5 3 −3 3 14 −2 8 2 4 4 −1 3 −1 3

2 2 0 0 0 0 0 0 0 −2 −1 −2 −1

7 7 7 7 7 7 3 3 3 3

The LinKnot function fDiffProjectionsAltKL calculates all nonisomorphic projections of an alternating KL given by its Conway symbol. Flypes (Theorem 1.11) make the computation of all minimal diagrams of an alternating KL significantly easier than for non-alternating KLs. One minimal non-alternating KL diagram can originate from very different sources. A minimal non-alternating algebraic KL diagram can be obtained from different polyhedral alternating KL diagrams by an appropriate choice of crossing changes. For example, non-alternating knot 2 1, 2 1, −2 can be obtained from different projections of 2 1, 2 1, 2, but also from minimal diagrams of .2.2 or .2.2 0. Hence, three non-alternating knots with n = 8 crossings have the following minimal diagrams, given by the Knotscape Dowker codes: 2 1, 2 1, −2 {{8}, {4, 8, −12, 2, 14, −6, 16, 10}} {{8}, {4, 8, −14, 2, 12, 16, −6, 10}} {{8}, {4, 8, −14, 2, 12, 16, −6, 10}} {{8}, {4, 8, 12, 2, −14, −16, 6, −10}} {{8}, {6, 8, 12, 14, 4, −16, 2, −10}} {{8}, {−6, 8, 14, −12, 4, 16, −2, 10}}

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3, 2 1, −2 {{8}, {6, 8, 12, 2, −14, −16, 4, −10}} {{8}, {4, 8, 12, 2, −14, 6, −16, −10}} {{8}, {4, 8, 14, 2, −12, −16, 6, −10}} {{8}, {4, 8, −12, 2, 14, 16, −6, 10}} {{8}, {−6, 8, 12, −14, 4, 16, −2, 10}} {{8}, {6, 8, 14, 12, 4, −16, 2, −10}} {{8}, {6, 8, −10, 12, 14, −16, 2, −4}} 3, 3, −2 {{8}, {6, 8, −12, 2, 14, 16, −4, 10}} {{8}, {−4, −8, 12, −2, 14, 16, 6, 10}} {{8}, {6, −8, −12, 14, −4, 16, −2, 10}} {{8}, {−6, 8, −14, 12, 4, 16, −2, 10}} {{8}, {6, −8, 10, −12, 14, −16, 2, −4}}

For n = 9 crossings we have the following non-alternating knots: 2 2, 3, −2 with seventeen, 2 1 1, 3, −2 with fourteen, 2 2, 2 1, −2 with seventeen, 2 1 1, 2 1, −2 with fourteen, 3, 3, −3 with eight, 2 1, 2 1, −3 with four, −2 0 : −2 0 : −2 0 with two, and 8∗ − 2 0 with two minimal diagrams. The projection polynomial dD (t) can be also computed for link projections. In this case, the result is a polynomial of the form: dD (t) = cn tn + · · · + ck tk , where n is the number of crossing points, and k is the number of link components. For every link, |cn | = 1. If ai are link components, aii = w(ai ), and |ck | = |det(aij )|, where aij =lk(ai , aj ) is the linking number of the components ai , aj . In order to increase the selectivity of this polynomial, one can use different variables for generators belonging to different components. For example, Borromean rings have projection polynomial dD (x, y, z) = x2 y 2 z 2 , which implies that their components are interchangeable, and that Borromean rings are (probably) achiral. A similar polynomial AD (t), introduced somewhat earlier by C. Liang and Y. Jiang (1982), is effectively used by C. Liang and K. Mislow (1994a) for recognition of achiral knots. In that polynomial, tǫ(V ) stands instead of ǫ(V )t from the preceding polynomial, and aij = s if the vertices i, j are connected with multiplicity s (s = 0, 1, 2). For achiral knot projections AD (t) = AD (t−1 ). For example, the achiral knot 2 2 (or 41 ) has AD (t) = AD (t−1 ) = −4t2 + 8t − 3 + 8t−1 − 4t−2 . As C. Liang and K. Mislow pointed out, achirality is the result of antisymmetry (vertex sign-changing symmetry), where a rotational antireflection produces invertible achiral knots, and a single 2-antirotation produces noninvertible achiral knots. Hence, achiral knots will be the members of KL families with preserved antisymmetry.

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The LinKnot functions JablanPoly and LiangPoly calculate above mentioned polynomial invariants for a KL projection given by its Conway symbol, Dowker code, or P -data.

Fig. 2.83

Achiral knots with n = 12 crossings from the work of M.G. Haseman.

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Achiral knots with n = 12 crossings were derived by M.G. Haseman (1918). In her original work, all of them except one (H59 = H60 ) are shown by their centro-antisymmetric presentations (Fig. 2.83). After correction for the knot 10. 2 2 2 2 2 2 (shortly written as 26 ), and identification of 7 duplicates (Thistlethwaite, 1985), the complete list contains 54 knots. They are given by Conway symbols showing their (anti)symmetry: 01. 02. 03. 04. 05. 06. 07. 08. 09. 10. 11. 12. 13. 14. 15.

62 5 12 5 (3, 3)2 3 2 12 2 3 34 (2 1, 2 1)2 3 1 22 1 3 22 14 22 2 18 2 26 4 22 4 2 1 2 12 2 1 2 (2 1 1, 2)2 (3, 2+)2 (2 1, 2+)2

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

= 13. (2 2, 2)2 2 42 2 2.2.2.2.2 0.2 0 2 0.2 1.2 1.2 0 2.2.2 0.2.2.2 0 12L 10∗∗ : 2 0 :: .2 0 10∗ 2 0 :: .2 0 8∗ 2 : 2 0.2 0 : 2 8∗ 2.2.2.2 2.2 1.2 1.2 .2 1.2 1.2.2 3.2.2.3 .3.3.2.2

31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.

12K 8∗ 2.2.2 0 :: .2 0 8∗ .2 1.2 1 8∗ 2.2 : .2 0.2 0 10∗∗ 2 0 :: .2 0 = 20. 8∗ 2 1 0 : .2 1 0 .4.4 .3 1.3 1 8∗ 2.2 : .2.2 8∗ 3.3 8∗ 3 0.3 0 10∗ 2 ::: .2 10∗∗ : 2 :: .2 10∗∗ 2 :: .2

46. 8∗ 2 0.2 0.2 0.2 0 47. 3 0.2.2.3 0 48. 8∗ 3 : .3 49. 8∗ 3 0 : .3 0 50. .2 2.2 2 51. = 50. 52. .3.3.2 0.2 0 53 10∗ 2.2 54. = 40. 55. 10∗∗ : 2 0.2 0 56. 10∗∗ : 2.2 57. = 45. 58. 12B 59. .2 1.2.2 1 0.2 0 60. = 59. 61. = 6.

In the case of the P -world for n ≤ 12, we will restrict the discussion of achirality to the basic polyhedra and alternating KLs generated from them as families. We conjecture that alternating achiral KLs can only be derived from achiral basic polyhedra or achiral source KLs by an arrangement of tangles preserving achirality of a generating KL. We already mentioned that some more sensitive polynomial invariants (e.g., Jones, Kauffman or HOMFLYPT polynomials) are able to recognize chiral KLs, but not always. The origin of polyhedral achiral knots will be achiral basic polyhedra with a symmetry group G which contains rotational antireflection, antirotation of order 2, or anti-inversion. The first achiral basic polyhedron is 6∗ (Borromean rings). For n ≤ 12 we obtain from it the following achiral knots: n=8 .2.2 n = 10 .2 1.2 1

.2.2.2 0.2 0

2.2.2.2

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n = 12 .4.4 .3.3.2 0.2 0 2 0.2 1.2 1.2 0

.3 1.3 1 .2 1.2.2 1 0.2 0 2.2.2.2.2 0.2 0

.2 2.2 2 3.2.2.3 2.2.2 0.2.2.2 0

.3.3.2.2 2.2 1.2 1.2

.2 1.2 1.2.2 3 0.2.2.3 0

For 6∗ , G = [3, 4] and G′ = [3+ , 4], so the achiral knots derived from it will contain antirotation or rotational antireflection. In this way, from 6∗ we derive non-invertible achiral knots (e.g., .2.2, .2.2.2.2, etc.), as well as invertible ones (e.g., .2.2.2 0.2 0). The invertible achiral knot 8∗ with the antireflection corresponds to the basic polyhedron 8∗ . The following achiral knots are derived from the basic polyhedron 8∗ : n = 10 8∗ 2 0.2 0

8∗ 2 : .2

8∗ 3.3 8∗ 2 1 0 : .2 1 0 8∗ 2.2 : .2 0.2 0

8∗ 2 1.2 1 8∗ .20 : 2.2 : 20 8∗ 2 0.2 0 : .2 0.2 0

n = 12 8∗ 3 0.3 0 8∗ 2.2.2.2

8∗ 3 : .3 8∗ 20.20.20.20

8∗ 3 0 : .3 0 8∗ 2.20.20.2

Some of the obtained achirals are invertible (e.g. 8∗ , etc.), and others are not (e.g., 8∗ 2 0.2 0, 8∗ 2 : .2, etc.). The achiral invertible knot 10∗ corresponds to the basic polyhedron 10∗ . The following achiral knots with n = 12 crossings are derived from 10∗ : 10∗ 2 ::: .2, 10∗ 2 0 ::: 2 0, 10∗ 2 :::: .2 From the basic polyhedron 10∗∗ we derive achiral knots: n = 12 10∗∗ 2 :: .2 10∗∗ : 2 0 :: .2 0

10∗∗ 2 0 :: .2 0

10∗∗ : 2.2

10∗∗ : 2 0.2 0

10∗∗ : 2 :: .2

Finally, for n = 12 we have three achiral knots corresponding to the basic polyhedra 12B (122∗), 12K (1211∗), and 12L (1212∗). The axis of 2-antirotation is projected in the perpendicular projection plane into the center of antisymmetry. Each of the achiral knots mentioned, except the knot .2 1.2.2 1 0.2 0, contains 2-antirotation and admits one or several centro-antisymmetric projections, i.e., has a discernible antisymmetry (Liang and Mislow, 1994a). For the exceptional knot .2 1.2.2 1 0.2 0 = H59 = H60 , where the second and third symbol corresponds to the notation from M. Haseman tables of achiral knots, the achirality is the result

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of anti-inversion, and none of its projections is centro-antisymmetric. The corresponding projection polynomials dH59 (t) = t12 − 4t8 + 2t6 − t4 − 6t2 , dH60 (t) = t12 − 4t8 − 2t6 + 3t4 + 2t2 , with the property dD (t) = dD (−t), reveal this concealed antisymmetry. The same holds for symmetric projection polynomials: A(H59 (t)) = −4t−6 − 6t−5 + 49t−4 + 18t−3 − 113t−2 + 68t−1 − 19 + 68t − 113t2 + 18t3 + 49t4 − 6t5 − 4t6 , A(H60 )(t) = 108t−4 −204t−3−340t−2 +594t−1−19+594t−340t2−204t3+108t4, with the property AD (t) = AD (t−1 ). This concealed antisymmetry is visible on the corresponding vertex-bicolored graphs on a sphere (Fig. 2.84).

Fig. 2.84 Achiral knot H59 = H60 and two its vertex-antisymmetric 3D-presentations, based on the anti-inversion.

Still, there is no simple criterion for the recognition of achiral knots, even for alternating ones. One of the attempts was Kauffman Conjecture (Kauffman, 1990a; van Mill and Reed, 1991): Conjecture 2.3. Let K be an achiral alternating knot. Then there exists a reduced alternating diagram D of K, such that G(D) is isomorphic to G∗ (D), where G(D) is a checkerboard-graph of D and G∗ (D) its dual. The counterexample to the preceding conjecture, knot (2 1, 3) 1 1 (3, 2 1), was found by Dasbach and Hougardy (1996). They proved that none of the eight graphs corresponding to different embeddings of this knot is isomorphic to its dual.

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The same knot is an example of how difficult it can be to find an antisymmetric drawing that shows concealed antisymmetry, e.g., the achirality of the projection (2 1, 3) 1 1 (3, 2 1). Achirality of this projection can be explained by the presence of the rotational antireflection, i.e., by purely geometrical antisymmetry arguments (Fig. 2.85). As we concluded many times before for different KL properties, achirality is a property of families. For example, from the knot (2 1, 3) 1 1 (3, 2 1) we derive family of achiral knots ((2m) 1, (2n + 1)) 1 1 ((2n + 1), (2m) 1), (m, n ≥ 1).

Fig. 2.85

The antisymmetric presentation of the achiral knot (2 1, 3) 1 1 (3, 2 1).

The LinKnot function AmphiProjAltKL tests the achirality of a projection of an alternating oriented KL given by its Conway symbol, Dowker code, or P -data, and the function AmphiAltKL tests the achirality of an alternating oriented KL given by its Conway symbol. As in all cases mentioned before, this applies only to minimal projections. As the best tool for recognizing chirality you can use the program SnapPea by J. Weeks. In the same way as with achiral alternating knots, we can work with achiral alternating oriented links. The first of them is the basic polyhedron 6∗ , famous Borromean rings. For n = 8 crossings we have stellar achiral 4-component link 2, 2, 2, 2 that generates two families of achiral links: (2m), (2m), (2n), (2n) and (2m), (2n), (2m), (2n), (m, n ≥ 1); arborescent achiral 3-component link (2, 2)(2, 2) that generates the family of achiral links ((2m), (2n)) ((2m), (2n)), (m, n ≥ 1); and polyhedral 3-component link .2 : 2 0 that generates the family of achiral links .(2m) : (2m) 0 (m ≥ 1). For n = 10 we have the beginnings of four new families of achiral links: 3component achiral link (2, 2+) (2, 2+) that generates the family of achiral

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links ((2m), (2n) + (2p)) ((2m), (2n) + 2p), (m, n, p ≥ 1); 3-component achiral link .3.3 that generates the family of achiral links .(2m + 1).(2m + 1), (m ≥ 1); 3-component achiral link .3 : 30 that generates the family of achiral links .(2m + 1) : (2m + 1) 0; and 3-component achiral link 8∗ 2.2 that generates the family of achiral links 8∗ (2m + 1).(2m + 1), (m ≥ 1). For n = 12, together with the links belonging to the families already mentioned (4, 4, 2, 2; 4, 2, 4, 2; (4, 2) (4, 2); (2, 2 + +) (2, 2 + +), and 4. : 4 0), we have the first elements of new families of achiral links: Generating link

Family

Comp. No. 3 1, 2, 3 1, 2 (2m + 1) 1, (2n), (2m + 1) 1, (2n) (m, n ≥ 1) 4 (3 1, 2) (3 1, 2) ((2m + 1) 1, (2n)) ((2m + 1) 1, (2n)) (m, n ≥ 1) 3 (2, 2, 2) (2, 2, 2) ((2m), (2n), (2p)) ((2m), (2n), (2p)) (m, n, p ≥ 1) 5 ((2, 2), 2) ((2, 2), 2) (((2m), (2n), (2p)) (((2n), (2m)), (2p)) (m, n, p ≥ 1) 3 (2, 2), (2, 2), (2, 2) ((2m), (2m)), ((2m), (2m)), ((2m), (2m)) (m ≥ 1) 5 .2 1 1.2 1 1 .(2m) 1 1.(2m) 1 1 (m ≥ 1) 3 .2 1 : 2 1 1 0 .(2m) 1 : (2m) 1 1 0 (m ≥ 1) 3 .2 2 : 2 2 0 .(2m) (2n) : (2m) (2n) 0 (m ≥ 1) 3 2.3.3.2 (2m).(2n + 1).(2n + 1).(2m) (m, n ≥ 1) 3 .3.2.3 0.2 0 .(2m + 1).(2n).(2m + 1) 0.(2n) 0 (m, n ≥ 1) 3 2.2.2.2.2.2 (2m).(2n).(2p).(2p).(2n).(2m) (m, n, p ≥ 1) 3 8∗ 2 1 0.2 1 0 8∗ (2m) 1 0.(2m) 1 0 (m ≥ 1) 3 8∗ .2 : 2.2 : 2 8∗ .(2m) : (2n).(2n) : (2m) (m, n ≥ 1) 3

and two achiral basic polyhedra 121∗ (12A) and 1210∗ (12J). In the same way, we can try to find achiral non-alternating KLs. It is interesting that in the existing knot tables for n ≤ 10 there are no nonalternating achiral knot. The first non-alternating achiral knot 1211∗ − 1. − 1. − 1 :: −1. − 1. − 1 has n = 12 crossings (Fig. 2.86). We give the list of achiral non-alternating knots with n = 14 crossings and their corresponding families, and the analogous lists of achiral nonalternating links with n = 10 crossings, but we can not guarantee that our lists are complete. For n = 14 we have the following non-alternating achiral knots and the families of achiral knots generated from them4 : Generating link ((−2 − 1, 2), 2) (2, (−2 − 1, 2)) ((2 1, 2), −2) (−2, (2 1, 2)) (2, −2 − 1) 1 1 1 1 (−2 − 1, 2) (−2, 2 1) 1 1 1 1 (2 1, −2) 4 All

Family ((−(2m) − 1, (2n)), (2p)) ((2p), (−(2m) − 1, (2n))) (((2m) 1, (2n)), −(2p)) (−(2p), ((2m) 1, (2n))) ((2m), −(2n) − 1) 1 1 1 1 (−(2n) − 1, (2m)) (−(2m), (2n) 1) 1 1 1 1 ((2n) 1, −(2m))

parameters are greater or equal to 1.

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Fig. 2.86 Non-alternating achiral knot 1211∗ − 1. − 1. − 1 :: −1. − 1. − 1 with n = 12 crossings. (−2, −2 − 1) 1 1 1 1 (−2 − 1, −2) (−2 − 1, 3) 1 1 (−2 − 1, 3) .(−2 − 1, 2).(2, −2 − 1) .(2 1, −2).(−2, 2 1) .(−2 − 1, 2).(2, −2 − 1) 8∗ 2 0. − 2 − 1 0. − 2 − 1 0.2 0 8∗ − 2 0. − 2. − 1. − 2 0.2 0.2 : 2 0 10∗∗ : −2 − 10 :: . − 2 − 1 0 10∗∗ − 2 − 1 0 :: . − 2 − 1 0 10∗∗ − 2 0.2 0 :: −2 0.2 0 10∗∗ 2 0. − 2. − 1. − 1. − 2.2 0 10∗∗ − 2 0.2 : .2. − 2 0 10∗∗∗ . − 2 0. − 2 : −2. − 2 0 128∗ . − 2 : . − 1. − 1. − 1. − 2 0. − 1 128∗ . − 2 ::: −2 0 128∗ . − 2 0 : . − 1. − 1. − 1. − 2. − 1 1211∗ − 2 0 ::: −2 0 1211∗ . − 2 0 ::: −2 0 1212∗ . − 2 ::::: −2 0

(−(2m), −(2n) − 1) 1 1 1 1 (−(2n) − 1, −(2m)) (−(2m) − 1, (2n + 1)) 1 1 (−(2m) − 1, (2n + 1)) .(−(2m) − 1, (2n)).((2n), −(2m) − 1) .((2m) 1, −(2n)).(−(2n), (2m) 1) .(−(2m) − 1, (2n)).((2n), −(2m) − 1) 8∗ (2m) 0. − (2n) − 1 0. − (2n) − 1 0.(2m) 0 8∗ − (2m) 0. − (2n). − 1. − (2p) 0.(2m) 0.(2n) : (2p) 0 10∗∗ : −(2m) − 10 :: . − (2m) − 1 0 10∗∗ − (2m) − 1 0 :: . − (2m) − 1 0 10∗∗ − (2m) 0.(2n) 0 :: −(2m) 0.(2n) 0 102∗ (2m) 0. − (2n). − 1. − 1. − (2n).(2m) 0 102∗∗ − (2m)0.(2n) : .(2n). − (2m)0 10∗∗∗ . − (2m) 0. − (2n) : −(2n). − (2m) 0 128∗ . − (2m) : . − 1. − 1. − 1. − (2m) 0. − 1 128∗ . − (2m) ::: −(2n) 0 128∗ . − (2m) 0 : . − 1. − 1. − 1. − (2m). − 1 1211∗ − (2m) 0 ::: −(2n) 0 1211∗ . − (2m) 0 ::: −(2m) 0 1212∗ . − (2m) ::::: −(2m) 0

and the non-alternating achiral knot 1428∗ : . − 1. − 1. − 1. − 1. − 1. − 1. − 1 : . − 1. Families of achiral knots from the preceding table are derived from their generating knots. If we permit a substitution of elementary tangles 1 and −1, we expect that from every achiral KL can be derived an infinite class of achiral KLs. For example, from the non-alternating knot 1428∗ : . − 1. − 1. − 1. − 1. − 1. − 1. − 1 : . − 1 can be obtained the class of achiral KLs of the form 1428∗t1 .t2 .t3 . − t4 . − t5 . − t6 . − t6 . − t7 . − t7 . − t4 .t3 .t2 . − t5 .t1 ,

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where each tangle ti can also be replaced by ti 0 (i = 1, 2, . . . , 7). Unfortunately, obtained KLs are too large for a systematic experimental check. The first non-alternating achiral link is the 4-component link 10−∗∗∗ , or more precisely 10∗∗∗ − 1. − 1. − 1. − 1 :: . − 1, i.e., the basic polyhedron 10∗∗∗ turned by suitable crossing changes into the non-alternating link (Fig. 2.87). It generates the following families: • the non-alternating 4-component links 10∗∗∗ − (2m + 1). − (2n + 1). − (2p + 1). − (2q + 1).(2n + 1).(2m + 1).(2q + 1).(2p + 1). − (2r + 1).(2r + 1), (m, n, p, q, r ≥ 1); • the non-alternating achiral 2-component links 10∗∗∗ − (2m). − (2n). − (2p). − (2q).(2n).(2m).(2q).(2p). − (2r).(2r), (m, n, p, q, r ≥ 1); • the non-alternating achiral 2-component links 10∗∗∗ −(2m) −1.−(2n) − 1. − (2p) − 1. − (2q) − 1.(2n) 1.(2m) 1.(2q) 1.(2p) 1. − (2r) − 1.(2r) 1, (m, n, p, q, r ≥ 1), etc.. An infinite collection of achiral KLs can be derived from non-alternating achiral link 10−∗∗∗ by any tangle substitution of the form 10∗∗∗ − t1 . − t2 . − t3 . − t4 .t2 .t1 .t4 .t3 . − t5 .t5 , where ti are arbitrary tangles (i = 1, 2, . . . , 5), and each of them can be also substituted by ti 0.

Fig. 2.87 Non-alternating achiral 4-component link 10∗∗∗ − 1. − 1. − 1. − 1 :: . − 1 with n = 10 crossings.

Since every (anti)symmetric structure can be obtained from another (anti)symmetric structure by a series of symmetric replacements, the origins of achiral KLs are lower level achirals. For example, polyhedral achiral

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alternating KLs originate from achiral basic polyhedra and can be obtained by symmetric tangle substitutions. Every palindromic rational knot with an even number of crossings is achiral, and the same statement holds for rational links (Theorem 1.20, Theorem 1.22). We hope that the more general statement holds: every algebraic alternating KL of the symmetric (palindromic) form p p is achiral, where p is any algebraic tangle, and we suppose that all achiral alternating algebraic KLs can be described in this way. For achiral alternating polyhedral KLs, origins are achiral basic polyhedra (Fig. 2.88). Their list for n ≤ 16 is: n=6 6∗ No. of basic polyhedra: 1 n=8 8∗ No. of basic polyhedra: 1 n = 10 10∗ 10∗∗ 10∗∗∗ No. of basic polyhedra: 3 n = 12 121∗

122∗

127∗ 128∗ 1210∗ 1211∗ No. of basic polyhedra: 7

1212∗

n = 14 144∗ 1419∗ 1453∗

147∗ 1420∗ 1455∗

148∗ 149∗ 1411∗ 1412∗ ∗ ∗ ∗ 1428 1433 1444 1447∗ 1456∗ 1458∗ 1460∗ No. of basic polyhedra: 21

1413∗ 1449∗

1417∗ 1450∗

n = 16 162∗ 1643∗ 1685∗ 16132∗ 16206∗ 16242∗ 16285∗ 16361∗ 16391∗ 16431∗

167∗ 1646∗ 1686∗ 16133∗ 16223∗ 16256∗ 16286∗ 16367∗ 16393∗ 16437∗

1617∗ 1647∗ 1689∗ 16142∗ 16226∗ 16263∗ 16293∗ 16369∗ 16402∗ 16442∗ No.

1621∗ 1623∗ 1648∗ 1649∗ ∗ 1692 1697∗ 16150∗ 16156∗ 16227∗ 16230∗ 16270∗ 16273∗ 16327∗ 16346∗ 16374∗ 16377∗ 16412∗ 16416∗ ∗ 16451 16461∗ of basic polyhedra:

1625∗ 1651∗ 16110∗ 16160∗ 16234∗ 16280∗ 16347∗ 16380∗ 16419∗ 16494∗ 78

1632∗ 1674∗ 16113∗ 16175∗ 16235∗ 16282∗ 16351∗ 16384∗ 16428∗

1639∗ 1680∗ 16128∗ 16204∗ 16239∗ 16284∗ 16360∗ 16388∗ 16429∗

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Fig. 2.88

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The achiral basic polyhedron 162∗ .

Instead of selecting achiral KLs from complete lists of KLs with a given number of crossings, we can derive them from achiral basic polyhedra. First, we select alternating achiral source links derived from achiral basic polyhedra, and then make tangle substitutions preserving achirality. As the first filter, we used the equality of Kauffman polynomials computed for each source KL and its mirror image, and for final selection the program SnapPea by J. Weeks. There are 9 achiral source KLs derived from the basic polyhedron 6∗ : ∗ 6 2 : .2, 6∗ 2.2, 6∗ 2.2.2 0 : 2, 6∗ 2.2 : 2 0.2 0, 6∗ 2.2.2 0.2.2.2 0, 6∗ 2.2 : 2.2, 6∗ 2.2.2.2, 6∗ 2.2.2.2.2 0.2 0, 6∗ 2.2.2.2.2.2. Two classes of achiral KLs, 6∗ p.p : q.q and 6∗ p.q : p.q, are derived from 6∗ 2.2 : 2.2; 6∗ 2.2.2 0.2.2.2 0 generates 6∗ p.p.q 0.r.r.q 0 and 6∗ p.q.r 0.p.q.r 0; 6∗ 2.2.2.2.2.2 generates 6∗ p.p.q.r.r.q and 6∗ p.q.r.p.q.r, and all other source KLs generate one class each. Instead of the term “family” here we use more general term “class”, because p, q, r, . . ., are arbitrary tangles, and not only chains of bigons. Hence, from the basic polyhedron 6∗ the following classes of achiral alternating KLs are obtained: 6∗ p.p 6∗ p.q.p 0

:q 6∗ p.q : p.q

6∗ p.p.q 0.r.r.q 0 6∗ p.q.r.p.r 0.q 0

6∗ p : .p 6∗ p.p

: q 0.q 0 6∗ p.q.q.p

6∗ p.q.r 0.p.q.r 0 6∗ p.p.q.r.r.q

6∗ p.p : q.q 6∗ p.q.q.p.r 0.r 0 6∗ p.q.r.p.q.r

From the basic polyhedron 8∗ the following classes of achiral alternating KLs are derived:

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8∗ p : .p 8∗ p.p

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8∗ p 0.p 0

8∗ p.p : .q 0.q 0 8∗ p.q.q.p 8∗ p : q 0.q 0 : p 8∗ p.q 0 : .q 0.p 8∗ p 0.q 0 : .q 0.p 0

8∗ p 0 : .p 0

8∗ p.p.q 0 :: .q 0 8∗ p.p : .q.q 8∗ p 0.p 0 : q 0 : .q 0 8∗ p.q 0.q 0.p 8∗ p.p : q 0 : .q 0

8∗ p.q 0.q 0.p : r 0.r 0 8∗ p.p : q.r 0.r 0.q 8∗ p.q.r.r.q.p 8∗ p 0.q 0.r 0.r 0.q 0.p 0 8∗ p.q.r 0.r 0.q.p 8∗ p.q 0.q 0.p.r 0 : .r 0

8∗ p 0.q 0.q 0.p 0 8∗ p.q : .q.p 8∗ p 0.p 0 : .q 0.q 0 8∗ p.p : q : .q

8∗ p.p.q 0 : r 0.r 0 : q 0 8∗ p.p.q 0.r : .r.q 0 8∗ p.q.q.p : r.r 8∗ p.q.q.p.r 0 : .r 0 8∗ p.p.q 0 : r.r : q 0

8∗ p.p.q 0.r 0.s 0.s 0.r 0.q 0 8∗ p.q.q.p.r 0.s 0.s 0.r 0 8∗ p 0.p 0.q 0.r 0.s 0.s 0.r 0.q 0 8∗ p.p.q.r.s.s.r.q

8∗ p 0.q 0.q 0.p 0 : r 0.r 0 8∗ p.q 0.r 0.r 0.q 0.p 8∗ p.p.q 0.r 0 : .r 0.q 0 8∗ p.p : q 0.r 0.r 0.q 0 8∗ p.q.q.p : r 0.r 0

8∗ p.p.q 0.r 0.s.s.r 0.q 0 8∗ p.q.r.r.q.p.s 0.s 0 8∗ p.q 0.q 0.p.r 0.s 0.s 0.r 0

8∗ p.q.r 0.r 0.q.p.s 0.s 0 8∗ p.q.q.p.r 0.s.s.r 0 8∗ p.p.q 0.r.s 0.s 0.r.q 0

From the basic polyhedron 10∗ the following classes of achiral alternating KLs are derived:

10∗ p 0 :: .p 0 10∗ p ::: .p

10∗ p 0 : .p 0 10∗ p : .p

10∗ p 0 : q 0 : .p 0 : q 0 10∗ p 0.q 0 :: p 0.q 0 10∗ p 0.p 0 : q 0 :: .q 0 10∗ p : q 0.q 0 : p 10∗ p.q 0.q 0.p 10∗ p : .p : .q 0.q 0 10∗ p.p : q 0 :: .q 0 10∗ p : q : .q : p 10∗ p.p : .q : .q 10∗ p.q.q.p

10∗ p 0.p 0 10∗ p.p

10∗ p 0 : q 0 : .q 0 : p 0 10∗ p 0.p 0 : .q 0 : .q 0 10∗ p 0.q 0.q 0.p 0 10∗ p.q 0 :: .q 0.p 10∗ p : q 0 : .p : q 0 10∗ p.p :: q 0.q 0 10∗ p.p.q 0 ::: .q 0 10∗ p.p :: q.q 10∗ p.q : .q.p

10∗ p 0.p 0 :: q 0.q 0 10∗ p 0.q 0 : .q 0.p 0 10∗ p : q 0 : .q 0 : p 10∗ p.q 0 : .q 0.p 10∗ p.q 0 :: p.q 0 10∗ p.p : .q 0 : .q 0 10∗ p : q : .p : q 10∗ p.q :: p.q 10∗ p.p : q :: .q

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10∗ p 0.p 0 : q 0 : r 0.r 0 : q 0 10∗ p 0.q 0.r 0 : .p 0.q 0.r 0 10∗ p 0.q 0.q 0.p 0 : r 0 : .r 0 10∗ p.q 0 : r 0.r 0 : q 0.p 10∗ p : q 0.q 0 : p : r 0.r 0 10∗ p.q 0 : .q 0.p.r 0 : .r 0 10∗ p.q 0.r 0 : .p.q 0.r 0 10∗ p.q 0.q 0.p.r 0 :: .r 0 10∗ p.q 0.r : .p.q 0.r 10∗ p.p : q 0 : r 0.r 0 : q 0 10∗ p.p.q 0 : r 0 : .r 0 : q 0 10∗ p.q.r 0 : .r 0.q.p 10∗ p.q : r 0 : p.q : r 0 10∗ p.q.r 0 : .p.q.r 0 10∗ p.p : .q.r 0.r 0.q 10∗ p.q.q.p : .r 0.r 0 10∗ p.p : q : r.r : q 10∗ p.q.r : .p.q.r 10∗ p.q.q.p : r : .r

10∗ p 0.q 0 : r 0 : p 0.q 0 : r 0 10∗ p 0.q 0.r 0 : .r 0.q 0.p 0 10∗ p 0.q 0.r 0.r 0.q 0.p 0 10∗ p.q 0.r 0 : .r 0.q 0.p 10∗ p : q 0.r 0 : p : q 0.r 0 10∗ p.q 0 : .r 0.p.q 0 : .r 0 10∗ p.q 0.q 0.p : .r 0.r 0 10∗ p.q 0.q 0.p : r : .r 10∗ p.q 0.r : .r.q 0.p 10∗ p.p : q 0.r 0 : .r 0.q 0 10∗ p.p.q 0.r 0 :: .r 0.q 0 10∗ p.q.r 0.r 0.q.p 10∗ p.p : q 0.r : .r.q 0 10∗ p.p.q 0 : r : .r : q 0 10∗ p.p : q : r 0.r 0 : q 10∗ p.q.q.p : r 0 : .r 0 10∗ p.q : r : p.q : r 10∗ p.q.r : .r.q.p 10∗ p.q.r.r.q.p

10∗ p 0.q 0.q 0.p 0 : r 0.s 0.s 0.r 0 10∗ p 0.q 0.r 0.r 0.q 0.p 0 : s 0.s 0 10∗ p.q 0.r 0.s 0.s 0.r 0.q 0.p 10∗ p.q 0.r 0.r 0.q 0.p.s 0 : .s 0 10∗ p.q 0.r 0.s 0 : p.q 0.r 0.s 0 10∗ p.q 0.q 0.p.r 0 : s 0.s 0 : r 0 10∗ p.q 0.q 0.p : r.s 0.s 0.r 10∗ p.q 0.r.s 0.s 0.r.q 0.p 10∗ p.p : q 0.r 0.s 0.s 0.r 0.q 0 10∗ p.p.q 0.r 0 : s 0.s 0 : r 0.q 0 10∗ p.q.r 0.s 0.s 0.r 0.q.p 10∗ p.q.r 0 : s 0.p.q.r 0 : s 0 10∗ p.q.r 0.s 0 : p.q.r 0.s 0 10∗ p.q.r 0.r 0.q.p : s 0.s 0 10∗ p.q.r 0.r 0.q.p : s.s 10∗ p.p.q 0 : r.s 0.s 0.r : q 0 10∗ p.p.q 0.r.s 0 : .s 0.r.q 0 10∗ p.q.r 0.s : p.q.r 0.s 10∗ p.p : q.r 0.s 0.s 0.r 0.q 10∗ p.q.r.s 0 : p.q.r.s 0 10∗ p.q.q.p.r 0 : s 0.s 0 : r 0 10∗ p.q.q.p : r 0.s.s.r 0 10∗ p.q.q.p.r 0.s : .s.r 0 10∗ p.q.r.r.q.p : s 0.s 0 10∗ p.q.q.p : r.s.s.r 10∗ p.q.q.p.r.s : s.r

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10∗ p 0.q 0 : r 0.r 0 : q 0.p 0 10∗ p 0.q 0.q 0.p 0 : .r 0.r 0 10∗ p : q 0.r 0.r 0.q 0 : p 10∗ p.q 0.r 0.r 0.q 0.p 10∗ p.q 0 : .q 0.p : r 0.r 0 10∗ p.q 0 : r 0 : p.q 0 : r 0 10∗ p.q 0.q 0.p : r 0 : .r 0 10∗ p.q 0 : r : p.q 0 : r 10∗ p.p : .q 0.r 0.r 0.q 0 10∗ p.p.q 0 : .r 0.r 0 : .q 0 10∗ p.q : r 0.r 0 : q.p 10∗ p.p : q 0 : r.r : q 0 10∗ p.p.q 0 : .r.r : .q 0 10∗ p.p.q 0.r :: .r.q 0 10∗ p.p : q.r 0 : .r 0.q 10∗ p.q.q.p.r 0 :: .r 0 10∗ p.q : r.r : q.p 10∗ p.q.q.p : .r.r

10∗ p 0.q 0.r 0.s 0 : p 0.q 0.r 0.s 0 10∗ p 0.q 0.r 0.s 0.s 0.r 0.q 0.p 0 10∗ p.q 0.r 0.r 0.q 0.p : s 0.s 0 10∗ p.q 0.r 0 : s 0.p.q 0.r 0 : s 0 10∗ p.q 0.q 0.p : r 0.s 0.s 0.r 0 10∗ p.q 0.q 0.p.r 0.s 0 : .s 0.r 0 10∗ p.q 0.r 0.s : p.q 0.r 0.s 10∗ p.q 0.r.s 0 : p.q 0.r.s 0 10∗ p.p.q 0 : r 0.s 0.s 0.r 0 : q 0 10∗ p.p.q 0.r 0.s 0 : .s 0.r 0.q 0 10∗ p.p.q 0 : r 0.s.s.r 0 : q 0 10∗ p.p.q 0.r 0 : s.s : r 0.q 0 10∗ p.p.q 0.r 0.s : .s.r 0.q 0 10∗ p.q.r 0.r 0.q.p.s 0 : .s 0 10∗ p.p : q 0.r.s 0.s 0.r.q 0 10∗ p.p.q 0.r : s 0.s 0 : r.q 0 10∗ p.q.r 0.s.s.r 0.q.p 10∗ p.p.q 0.r : s.s : r.q 0 10∗ p.q.r.s 0.s 0.r.q.p ∗ 10 p.q.q.p : r 0.s 0.r 0.r 0 10∗ p.q.q.p.r 0.s 0 : .s 0.r 0 10∗ p.q.q.p.r 0 : s.s : r 0 10∗ p.q.q.p : r.s 0.s 0.r 10∗ p.q.r.r.q.p.s 0 : .s 0 10∗ p.q.r.s : p.q.r.s 10∗ p.q.r.s.s.r.q.p

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10∗ p 0.p 0.q 0.r 0.s 0.t 0.t 0.s 0.r 0.q 0 10∗ p 0.q 0.r 0.r 0.q 0.p 0.s 0.t 0.t 0.s 0 10∗ p.q 0.r 0.r 0.q 0.p.s 0.t 0.t 0.s 0 10∗ p.q 0.q 0.p.r 0.s 0.t 0.t 0.s 0.r 0 10∗ p.p.q 0.r 0.s 0.t 0.t 0.s 0.r 0.q 0 10∗ p.q.r 0.s 0.t 0.p.q.r 0.s 0.t 0 10∗ p.q.r 0.r 0.q.p.s 0.t 0.t 0.s 0 10∗ p.p.q 0.r.s 0.t.t.s 0.r.q 0 10∗ p.q.r 0.s.s.r 0.q.p.t 0.t 0 10∗ p.q.q.p.r 0.s 0.t 0.t 0.s 0.r 0 10∗ p.q.q.p.r 0.s.t 0.t 0.s.r 0 10∗ p.q.r.s.t 0.p.q.r.s.t 0 10∗ p.q.r.r.q.p.s 0.t.t.s 0 10∗ p.p.q.r.s.t.t.s.r.q 10∗ p.q.r.r.q.p.s.t.t.s

10∗ p 0.q 0.q 0.p 0.r 0.s 0.t 0.t 0.s 0.r 0 10∗ p 0.q 0.r 0.s 0.t 0.p 0.q 0.r 0.s 0.t 0 10∗ p.q 0.r 0.s 0.t 0.p.q 0.r 0.s 0.t 0 10∗ p.q 0.r.s 0.t 0.p.q 0.r.s 0.t 0 10∗ p.p.q 0.r 0.s 0.t.t.s 0.r 0.q 0 10∗ p.p.q 0.r 0.s.t 0.t 0.s.r 0.q 0 10∗ p.p.q 0.r.s 0.t 0.t 0.s 0.r.q 0 10∗ p.q.r 0.s.t 0.p.q.r 0.s.t 0 10∗ p.q.r.s 0.t 0.p.q.r.s 0.t 0 10∗ p.q.q.p.r 0.s 0.t.t.s 0.r 0 10∗ p.q.q.p.r 0.s.t.t.s.r 0 10∗ p.q.r.r.q.p.s 0.t 0.t 0.s 0 10∗ p.q.r.s.s.r.q.p.t 0.t 0 10∗ p.q.q.p.r.s.t.t.s.r 10∗ p.q.r.s.t.p.q.r.s.t

From the basic polyhedron 10∗∗ the following classes of achiral alternating KLs are derived: 10∗∗ : p 0 :: .p 0 10∗∗ .p 0 : .p 0 10∗∗ : p.p 10∗∗ p :: .p

10∗∗ : p 0.p 0 10∗∗ p 0 :: .p 0 10∗∗ .p :: .p

10∗∗ : p 0.p 0 :: q 0.q 0 10∗∗ .p 0 : .p 0 : q 0 : .q 0 10∗∗ .p 0.q 0 :: p 0.q 0 10∗∗ p 0 : q 0.q 0 : p 0 10∗∗ .p 0 : .p 0 : .q.q 10∗∗ p 0 : q : .p 0 : q 10∗∗ p 0 : q.q : p 0 10∗∗ p : q 0.q 0 : p 10∗∗ .p : .q 0 : p : .q 0 10∗∗ p : q 0 : .p : q 0 10∗∗ .p : .p : .q 0.q 0 10∗∗ : p.q :: p.q 10∗∗ .p : .q : p : .q 10∗∗ .p.q.q.p 10∗∗ p.q :: p.q 10∗∗ .p 0.q 0 : r 0 : p 0.q 0 : r 0 10∗∗ .p 0.q 0.q 0.p 0 : r 0 : .r 0 10∗∗ p 0.q 0 : .q 0.p 0 : r 0.r 0 10∗∗ p 0.q 0 : r 0 : p 0.q 0 : r 0 10∗∗ .p 0.q 0.q 0.p 0 : .r.r

10∗∗ .p 0 :: .p 0 10∗∗ : p :: .p 10∗∗ .p : .p

10∗∗ : p 0.q 0 :: p 0.q 0 10∗∗ .p 0 : .q 0 : p 0 : .q 0 10∗∗ .p 0.q 0.q 0.p 0 10∗∗ p 0.q 0 :: p 0.q 0 10∗∗ .p 0 : q : .p 0 : q 10∗∗ p 0.q :: p 0.q 10∗∗ p 0.q : .q.p 0 10∗∗ p.q 0 : .q 0.p 10∗∗ .p : q 0 : .p : q 0 10∗∗ p.q 0 :: p.q 0 10∗∗ .p : .p : q 0 : .q 0 10∗∗ .p : .p : .q.q 10∗∗ .p : q : .p : q 10∗∗ p : q : .p : q 10∗∗ p.q : .q.p 10∗∗ .p 0.q 0.r 0 : .p 0.q 0.r 0 10∗∗ p 0 : q 0.q 0 : p 0 : r 0.r 0 10∗∗ p 0.q 0 : .q 0.p 0.r 0 : .r 0 10∗∗ p 0.q 0.r 0 : .p 0.q 0.r 0 10∗∗ p 0.q 0 : .q 0.p 0 : r.r

10∗∗ .p 0 : .p 0 : .q 0.q 0 10∗∗ .p 0 : q 0 : .p 0 : q 0 10∗∗ p 0 : q 0 : .p 0 : q 0 10∗∗ p 0.q 0 : .q 0.p 0 10∗∗ .p 0.q :: p 0.q 10∗∗ .p 0.q.q.p 0 10∗∗ .p.q 0.q 0.p 10∗∗ : p.q 0 :: p.q 0 10∗∗ .p.q 0 :: p.q 0 10∗∗ : p.p :: q 0.q 0 10∗∗ : p.p :: q.q 10∗∗ .p : .p : q : .q 10∗∗ .p.q :: p.q 10∗∗ p : q.q : p 10∗∗ .p 0.q 0.q 0.p 0 : .r 0.r 0 10∗∗ p 0 : q 0.r 0 : p 0 : q 0.r 0 10∗∗ p 0.q 0 : .r 0.p 0.q 0 : .r 0 10∗∗ p 0.q 0.r 0.r 0.q 0.p 0 10∗∗ .p 0.q 0.r : .p 0.q 0.r

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10∗∗ p 0.q 0 : r : p 0.q 0 : r 10∗∗ .p 0.q : r 0 : p 0.q : r 0 10∗∗ p 0.q : .r 0.p 0.q : .r 0 10∗∗ p 0.q.r 0.r 0.q.p 0 10∗∗ p 0 : q.q : p 0 : r 0.r 0 10∗∗ .p 0.q.r : .p 0.q.r 10∗∗ p 0 : q.r : p 0 : q.r 10∗∗ p 0.q : .r.p 0.q : .r 10∗∗ p 0.q.r.r.q.p 0 10∗∗ .p.q 0 : r 0 : p.q 0 : r 0 10∗∗ .p.q 0.q 0.p : r 0 : .r 0 10∗∗ p.q 0 : .q 0.p : r 0.r 0 10∗∗ p.q 0 : r 0 : p.q 0 : r 0 10∗∗ .p.q 0.q 0.p : r : .r 10∗∗ .p.q 0.r : .p.q 0.r 10∗∗ p.q 0.r.r.q 0.p 10∗∗ .p : .p : q 0.r.r.q 0 10∗∗ .p.q.r 0 : .p.q.r 0 10∗∗ p.q : r 0 : p.q : r 0 10∗∗ .p.q.q.p : r 0 : .r 0 10∗∗ p.q : .q.p.r 0 : .r 0 10∗∗ .p.q.q.p : .r.r 10∗∗ p : q.r : p : q.r 10∗∗ p.q : .r.p.q : .r 10∗∗ p.q.r.r.q.p

10∗∗ p 0.q 0.r : .p 0.q 0.r 10∗∗ .p 0.q.r 0 : .p 0.q.r 0 10∗∗ p 0.q : r 0 : p 0.q : r 0 10∗∗ .p 0.q.q.p 0 : .r 0.r 0 10∗∗ p 0.q : .q.p 0 : r 0.r 0 10∗∗ .p 0.q.q.p 0 : .r.r 10∗∗ p 0.q : .q.p 0 : r.r 10∗∗ p 0.q : r : p 0.q : r 10∗∗ p.q 0.r 0.r 0.q 0.p 10∗∗ .p.q 0.r 0 : .p.q 0.r 0 10∗∗ p : q 0.q 0 : p : r 0.r 0 10∗∗ p.q 0 : .q 0.p.r 0 : .r 0 10∗∗ p.q 0.r 0 : .p.q 0.r 0 10∗∗ p.q 0 : .q 0.p : r.r 10∗∗ p.q 0 : r : p.q 0 : r 10∗∗ .p : .p : q 0.r 0.r 0.q 0 10∗∗ .p : q.r 0 : p : q.r 0 10∗∗ p : q.r 0 : p : q.r 0 10∗∗ p.q.r 0 : .p.q.r 0 10∗∗ p : q.q : p : r 0.r 0 10∗∗ .p.q : r : p.q : r 10∗∗ .p.q.q.p : r : .r 10∗∗ p.q : .q.p : r.r 10∗∗ p.q : r : p.q : r

10∗∗ .p 0.q 0.q 0.p 0 : r 0.s 0.s 0.r 0 10∗∗ p 0.q 0.r 0 : s 0.p 0.q 0.r 0 : s 0 10∗∗ p 0.q 0.r 0.r 0.q 0.p 0 : s 0.s 0 10∗∗ p 0.q 0.r 0.r 0.q 0.p 0 : s.s 10∗∗ p 0.q 0.r : s 0.p 0.q 0.r : s 0 10∗∗ p 0.q 0.r.r.q 0.p 0 : s 0.s 0 10∗∗ p 0.q 0.r.s : p 0.q 0.r.s 10∗∗ .p 0.q.r 0.s 0 : p 0.q.r 0.s 0 10∗∗ p 0.q.r 0 : s 0.p 0.q.r 0 : s 0 10∗∗ p 0.q.r 0.r 0.q.p 0 : s 0.s 0 10∗∗ p 0.q.r 0.r 0.q.p 0 : s.s 10∗∗ p 0.q.r 0 : s.p 0.q.r 0 : s 10∗∗ .p 0.q.q.p 0 : r 0.s 0.s 0.r 0 10∗∗ .p 0.q.q.p 0 : r 0.s.s.r 0 10∗∗ p 0.q : .q.p 0.r 0.s.s.r 0 10∗∗ p 0.q.r : s 0.p 0.q.r : s 0 10∗∗ p 0.q.r : s.p 0.q.r : s 10∗∗ p 0.q.r.r.q.p 0 : s 0.s 0 10∗∗ p 0.q.r.r.q.p 0 : s.s 10∗∗ p.q 0.r 0.r 0.q 0.p : s 0.s 0

10∗∗ p 0.q 0.r.r.q 0.p 0 10∗∗ p 0 : q.r 0 : p 0 : q.r 0 10∗∗ p 0.q.r 0 : .p 0.q.r 0 10∗∗ .p 0.q.q.p 0 : r 0 : .r 0 10∗∗ p 0.q : .q.p 0.r 0 : .r 0 10∗∗ p 0 : q.q : p 0 : r.r 10∗∗ p 0.q : .q.p 0.r : .r 10∗∗ p 0.q.r : .p 0.q.r ∗∗ 10 .p : q 0.r 0 : p : q 0.r 0 10∗∗ .p.q 0.q 0.p : .r 0.r 0 10∗∗ p : q 0.r 0 : p : q 0.r 0 10∗∗ p.q 0 : .r 0.p.q 0 : .r 0 10∗∗ .p.q 0.q 0.p : .r.r 10∗∗ .p.q 0 : r : p.q 0 : r 10∗∗ p.q 0.r : .p.q 0.r 10∗∗ p.q.r 0.r 0.q.p 10∗∗ .p.q : r 0 : p.q : r 0 10∗∗ p.q : .r 0.p.q : .r 0 10∗∗ .p.q.q.p : .r 0.r 0 10∗∗ p.q : .q.p : r 0.r 0 10∗∗ .p.q.r : .p.q.r 10∗∗ p : q.q : p : r.r 10∗∗ p.q : .q.p.q : .r 10∗∗ p.q.r : .p.q.r

10∗∗ .p 0.q 0.r 0.s 0 : p 0.q 0.r 0.s 0 10∗∗ p 0.q 0.r 0.s 0 : p 0.q 0.r 0.s 0 10∗∗ p 0.q 0.r 0.r 0.q 0.p 0.s 0 : .s 0 10∗∗ p 0.q 0.r 0.s : p 0.q 0.r 0.s 10∗∗ p 0.q 0.r.s 0 : p 0.q 0.r.s 0 10∗∗ p 0.q 0.r.r.q 0.p 0.s 0 : .s 0 10∗∗ p 0.q 0.r.r.q 0.p 0 : s.s 10∗∗ p 0.q : r 0.s 0.p 0.q : r 0.s 0 10∗∗ p 0.q.r 0.s 0 : p 0.q.r 0.s 0 10∗∗ p 0.q.r 0.r 0.q.p 0.s 0 : .s 0 10∗∗ p 0.q.r 0.r 0.q.p 0.s : .s 10∗∗ p 0.q.r 0.s : p 0.q.r 0.s 10∗∗ p 0.q : .q.p 0.r 0.s 0.s 0.r 0 10∗∗ .p 0.q.r.s 0 : p 0.q.r.s 0 10∗∗ p 0.q : r.s 0.p 0.q : r.s 0 10∗∗ p 0.q.r.s 0 : p 0.q.r.s 0 10∗∗ p 0.q.r.s : p 0.q.r.s 10∗∗ p 0.q.r.r.q.p 0.s 0 : .s 0 10∗∗ p 0.q.r.r.q.p 0.s : .s 10∗∗ p.q 0.r 0.r 0.q 0.p.s 0 : .s 0

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10∗∗ p.q 0.r 0.r 0.q 0.p : s.s 10∗∗ p.q 0.r 0 : s 0.p.q 0.r 0 : s 0 10∗∗ .p.q 0.q 0.p : r 0.s 0.s 0.r 0 10∗∗ .p.q 0.q 0.p : r.s 0.s 0.r 10∗∗ p.q 0.r 0.s : p.q 0.r 0.s 10∗∗ p.q 0.r : s 0.p.q 0.r : s 0 10∗∗ p.q 0.r.r.q 0.p : s 0.s 0 10∗∗ p.q 0.r.r.q 0.p : s.s 10∗∗ .p.q.r 0.s 0 : p.q.r 0.s 0 10∗∗ p.q.r 0 : s 0.p.q.r 0 : s 0 10∗∗ p.q.r 0.r 0.q.p : s 0.s 0 10∗∗ p.q.r 0.r 0.q.p : s.s 10∗∗ .p.q.r 0.s : p.q.r 0.s 10∗∗ p.q.r 0.s : p.q.r 0.s 10∗∗ p.q : r.s 0.p.q : r.s 0 10∗∗ p.q.r.s 0 : p.q.r.s 0 10∗∗ q.p : .p.q.r 0.s 0.s 0.r 0 10∗∗ p.q : .q.p.r 0.s.s.r 0 10∗∗ p.q.r.r.q.p : s 0.s 0 10∗∗ .p.q.q.p : r.s.s.r 10∗∗ p.q.r.s : p.q.r.s 10∗∗ p.q.r.r.q.p.s : .s

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10∗∗ .p.q 0.r 0.s 0 : p.q 0.r 0.s 0 10∗∗ p.q 0.r 0.s 0 : p.q 0.r 0.s 0 10∗∗ .p.q 0.q 0.p : r 0.s.s.r 0 10∗∗ .p.q 0.r 0.s : p.q 0.r 0.s 10∗∗ .p.q 0.r.s 0 : p.q 0.r.s 0 10∗∗ p.q 0.r.s 0 : p.q 0.r.s 0 10∗∗ p.q 0.r.r.q 0.p.s 0 : .s 0 10∗∗ p.q 0.r.s : p.q 0.r.s ∗∗ 10 p.q : r 0.s 0.p.q : r 0.s 0 10∗∗ p.q.r 0.s 0 : p.q.r 0.s 0 10∗∗ p.q.r 0.r 0.q.p.s 0 : .s 0 10∗∗ p.q.r 0.r 0.q.p.s : .s 10∗∗ p.q.r 0 : s.p.q.r 0 : s 10∗∗ .p.q.r.s 0 : p.q.r.s 0 10∗∗ p.q.r : s 0.p.q.r : s 0 10∗∗ .p.q.q.p : r 0.s 0.s 0.r 0 10∗∗ .p.q.q.p : r 0.s.s.r 0 10∗∗ .p.q.q.p : r.s 0.s 0.r 10∗∗ q.r.s.s.r.q.p 0 : .p 0 10∗∗ p.q.r : s.p.q.r : s 10∗∗ p.q.q.p.r.s : s.r

10∗∗ p 0.q 0.r 0.r 0.q 0.p 0.s 0.t 0.t 0.s 0 10∗∗ p 0.q 0.r.s 0.t 0.p 0.q 0.r.s 0.t 0 10∗∗ p 0.q 0.r.r.q 0.p 0.s 0.t.t.s 0 10∗∗ p 0.q.r 0.s 0.t 0.p 0.q.r 0.s 0.t 0 10∗∗ p 0.q.r 0.q 0.q.p 0.s 0.t.t.s 0 10∗∗ p 0.q.r 0.s 0.t.p 0.q.r 0.s 0.t 10∗∗ p 0.q.r.s 0.t 0.p 0.q.r.s 0.t 0 10∗∗ p 0.q.r.s.t 0.p 0.q.r.s.t 0 10∗∗ p 0.q.r.r.q.p 0.s 0.t.t.s 0 10∗∗ p 0.q.r.r.q.p 0.s.t.t.s 10∗∗ p.q 0.r 0.r 0.q 0.p.s 0.t 0.t 0.s 0 10∗∗ p.q 0.r.s 0.t 0.p.q 0.r.s 0.t 0 10∗∗ p.q 0.r.r.q 0.p.s 0.t.t.s 0 10∗∗ p.q.r 0.s 0.t 0.p.q.r 0.s 0.t 0 10∗∗ p.q.r 0.r 0.q.p.s 0.t.t.s 0 10∗∗ p.q.r 0.s 0.t.p.q.r 0.s 0.t 10∗∗ p.q.r.s 0.t 0.p.q.r.s 0.t 0 10∗∗ p.q.r.s.t 0.p.q.r.s.t 0 10∗∗ p.q.r.r.q.p.s 0.t.t.s 0 10∗∗ p.q.r.r.q.p.s.t.t.s

10∗∗ p 0.q 0.r 0.s 0.t 0.p 0.q 0.r 0.s 0.t 0 10∗∗ p 0.q 0.r.r.q 0.p 0.s 0.t 0.t 0.s 0 10∗∗ p 0.q 0.r.s.t 0.p 0.q 0.r.s.t 0 10∗∗ p 0.q.r 0.r 0.q.p 0.s 0.t 0.t 0.s 0 10∗∗ p 0.q.r 0.r 0.q.p 0.s.t 0.t 0.s 10∗∗ p 0.q.r 0.s.t 0.p 0.q.r 0.s.t 0 10∗∗ p 0.q.r.s 0.t.p 0.q.r.s 0.t 10∗∗ p 0.q.r.r.q.p 0.s 0.t 0.t 0.s 0 10∗∗ p 0.q.r.r.q.p 0.s.t 0.t 0.s 10∗∗ p 0.q.r.s.t.p 0.q.r.s.t 10∗∗ p.q 0.r 0.s 0.t 0.p.q 0.r 0.s 0.t 0 10∗∗ p.q 0.r.r.q 0.p.s 0.t 0.t 0.s 0 10∗∗ p.q 0.r.s.t 0.p.q 0.r.s.t 0 10∗∗ p.q.r 0.r 0.q.p.s 0.t 0.t 0.s 0 10∗∗ p.q.r 0.r 0.q.p.s.t 0.t 0.s 10∗∗ p.q.r 0.s.t 0.p.q.r 0.s.t 0 10∗∗ p.q.r.s 0.t.p.q.r.s 0.t 10∗∗ p.q.r.r.q.p.s 0.t 0.t 0.s 0 10∗∗ p.q.r.r.q.p.s.t 0.t 0.s 10∗∗ p.q.r.s.t.p.q.r.s.t

In the above lists containing classes of achiral alternating KLs derived from the basic polyhedra 6∗ , 8∗ , 10∗ , and 10∗∗ it is possible to recognize symmetry patterns and regular distribution of the classes.

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Achiral basic polyhedron 10∗∗∗ is atypical and more difficult to handle: it is the first basic polyhedron which can be obtained as a product of two non-algebraic tangles (10∗∗∗ = 5⋆ 5⋆ ) (Fig. 2.89) and P ET (Polya Enumeration Theorem) can not be used for calculating the number of source KLs derived from it.

Fig. 2.89 The achiral basic polyhedron 10∗∗∗ expressed as the product of two nonalgebraic tangles 5⋆ .

First surprise comes as the result of of the chirality tests using Kauffman polynomial and the program SnapPea: 2-component link 10∗∗∗ 2 :: .2 0 recognized by Kauffman polynomials as achiral is chiral. All other polynomials presumably able to detect chirality, Jones and HOMFLYPT polynomial, fail as well. All chiral KLs that can not be recognized as chiral by any of polynomials mentioned will be called non-detectable chiral KLs. Moreover, chiral link 10∗∗∗ 2 :: .2 0 generates an infinite class of non-detectable chiral KLs of the form 10∗∗∗ p :: .p 0. The same property holds for chiral source KLs 10∗∗∗ 2 :: .2 0 : .2 0.2, 10∗∗∗ 2 :: .2 0 : .2.2 0, 10∗∗∗ 2 : 2 0 : .2 : 2 0, 10∗∗∗ 2 : 2 : .2 0 : 2 0, 10∗∗∗ 2.2 : .2 0 : .2 0, 10∗∗∗ 2 : 2 0 : .2 : 2 0.2.2 0, etc., and infinite classes of chiral KLs 10∗∗∗ p :: .p 0 : .q 0.q, 10∗∗∗ p :: .p 0 : .q.q 0, 10∗∗∗ p : p 0 : .q : q 0, 10∗∗∗ p : q 0 : .q : p 0, 10∗∗∗ p.q 0 : .q : .p 0, 10∗∗∗ p : q : .p 0 : q 0, 10∗∗∗ p : q : .q 0 : p 0, 10∗∗∗ p.q : .q 0 : .p 0, 10∗∗∗ p : q 0 : .q : p 0.r.r 0, etc. By different substitutions in the same achiral source KL we can obtain classes of achiral KLs and non-detectable chiral KLs. For example, source knot 10∗∗∗ 2 : .2 0.2.2 0 generates an infinite class of achiral KLs 10∗∗∗ p : .q 0.p.q 0 and an infinite class of non-detectable chiral KLs 10∗∗∗ p : .q 0.q.p 0; source knot 10∗∗∗ 2.2 0 : .2 : .2 0 generates an infinite class of achiral KLs 10∗∗∗ p.q 0 : .p : .q 0 and an infinite class of non-detectable chiral KLs 10∗∗∗ p.q 0 : .q : .p 0, etc.

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However, there are non-detectable source KLs that do not generate infinite classes of non-detectable chiral KLs, for example 10∗∗∗ 2.2 ::: 2 0.2, 10∗∗∗ 2.2 :: 2 0 : .2, 10∗∗∗ 2.2 :: .2 : 2, 10∗∗∗ 2.2 : .2 :: 2, etc. Next tables contain classes of achiral KLs with at most two parameters obtained from the basic polyhedron 10∗∗∗ : 10∗∗∗ p ::: p 10∗∗∗ :::: p.p 0

10∗∗∗ p :: p 10∗∗∗ p : p 0

10∗∗∗ p : p 0 : q 0 : q 10∗∗∗ p : p 0 : q : q 0 10∗∗∗ p ::: p : q 0.q 10∗∗∗ p : q : q : p 10∗∗∗ p.p.p.p

10∗∗∗ p : q 0 : q 0 : p 10∗∗∗ p : q 0 : p : q 0 10∗∗∗ p :: p :: q 0.q 10∗∗∗ p.q :: q.p 10∗∗∗ p ::: p : q.q 0

10∗∗∗ p : .q 0.p.q 0 10∗∗∗ p.q 0 : .p : .q 0 10∗∗∗ p : q : p : q 10∗∗∗ p.q : .p : .q 10∗∗∗ p :: p :: q.q 0

Fig. 2.90 (a) Diagram of the achiral source knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2.2 0; (b) the achiral knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0 without antisymmetric minimal projection.

Among classes of achiral KLs derived from source KLs with four bigons there is one exceptional one-parameter class: the class of links 10∗∗∗ p.p.p.p. A 2-parameter class 10∗∗∗ p : p : .p 0 : p 0.q.q 0 is the origin of a very interesting subclass: class of knots with no minimal antisymmetric (i.e., achiral) diagram. For n ≤ 16 all achiral alternating knots have an achiral minimal diagram. The first knot without it is 18-crossing achiral knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0. A simple explanation of this phenomenon is the following: source knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2.2 0 has an antisymmetric minimal diagram (Figs. 2.90a, 2.91a). Its achirality is the result of the rotational antireflection of order 4 (Fig. 2.91b). The achiral knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0 has only one minimal diagram, 10∗∗∗ 2 : 2 : .2 0 : 2 0.(2, 1).(2, 1) 0 which is chiral (Fig. 2.90b).

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Fig. 2.91 (a) Achiral source knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2.2 0; (b) its 3-D antisymmetric representation with the visible rotational antireflection of order 4.

However, since tangles 2 1 and 1 2 are flype-equivalent, knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0 can be represented simultaneously by four diagram states: 10∗∗∗ 2 : 2 : .2 0 : 2 0.(2, 1).(2, 1) 0, 10∗∗∗ 2 : 2 : .2 0 : 2 0.(2, 1).(1, 2) 0, 10∗∗∗ 2 : 2 : .2 0 : 2 0.(1, 2).(2, 1) 0, and 10∗∗∗ 2 : 2 : .2 0 : 2 0.(1, 2).(1, 2) 0. So, this “dynamic” flyping antisymmetry is compatible with the rotational antireflection of order 4 and results in the achirality of the knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0 (Fig. 2.92). However, its “static”, fixed diagram is chiral. The same property holds for an infinite class of KLs 10∗∗∗ p : p : .p 0 : p 0.q.q 0, where q is a symmetric tangle (e.g., (2, 2)), or a tangle that is flype-equivalent to its reverse (e.g., 2 1 = 1 2). Otherwise, KLs obtained are chiral (e.g., the knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.(3, 2).(3, 2) 0 is chiral). The following table contains the remaining classes of achiral KLs derived from the basic polyhedron 10∗∗∗ : 10∗∗∗ p : q 0 : q 0 : p : r.r 0 10∗∗∗ p : .q 0.p.q 0 : .r.r 0 10∗∗∗ p.q 0.r : r.q 0.p 10∗∗∗ p.q.r 0 : r 0.q.p 10∗∗∗ p.q.r 0 : p : r 0.q 10∗∗∗ p : q : q : p : r 0.r 10∗∗∗ p.q.r : p : r.q 10∗∗∗ p : q : q : p : r.r 0

10∗∗∗ p : .q 0.p.q 0 : .r 0.r 10∗∗∗ p : q 0 : p : q 0 : r.r 0 10∗∗∗ p.q 0.r : p : r.q 0 10∗∗∗ p : p : .p 0 : p 0.p.p 0 10∗∗∗ p.q : r 0 : q.p.r 0 10∗∗∗ p.q :: q.p : r 0.r 10∗∗∗ p.q.r : r.q.p 10∗∗∗ p.q :: q.p : r.r 0

10∗∗∗ p.q 0 : .p : .q 0.r 0.r 10∗∗∗ p.q 0 : .p : .q 0.r.r 0 10∗∗∗ p : p : .p 0 : p 0.p 0.p 10∗∗∗ p.q : r 0.p.r 0 : q 10∗∗∗ p : q : p : q : r 0.r 10∗∗∗ p.q : .p : .q.r 0.r 10∗∗∗ p : q : p : q : r.r 0 10∗∗∗ p.q : .p : .q.r.r 0

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Fig. 2.92 Dynamic antisymmetric 3-D representation of the achiral knot 10∗∗∗ 2 : 2 : .2 0 : 2 0.2 1.2 1 0 with the rotational antireflection of order 4 preserved since tangles 2 1 and 1 2 are flype-equivalent. 10∗∗∗ p.q 0.r.s 0.p.s 0.r.q 0 10∗∗∗ p.q 0.r : r.q 0.p : s.s 0 10∗∗∗ p.q 0.r : p : r.q 0.s.s 0 10∗∗∗ p.q.r 0 : r 0.q.p : s 0.s 10∗∗∗ p.q : r 0.p.r 0 : q.s 0.s 10∗∗∗ p.q : r 0.p.r 0 : q.s.s 0 10∗∗∗ p.q : r 0 : q.p.r 0.s 0.s 10∗∗∗ p.q.r.s 0.p.s 0.r.q 10∗∗∗ p.q.r : p : r.q.s 0.s 10∗∗∗ p.q.r.s.p.s.r.q 10∗∗∗ p.q.r : r.q.p : s.s 0

10∗∗∗ p.q 0.r : r.q 0.p : s 0.s 10∗∗∗ p.q 0.r : p : r.q 0.s 0.s 10∗∗∗ p.q.r 0.s 0.p.s 0.r 0.q 10∗∗∗ p.q.r 0 : r 0.q.p : s.s 0 10∗∗∗ p.q.r 0 : p : r 0.q.s 0.s 10∗∗∗ p.q.r 0 : p : r 0.q.s.s 0 10∗∗∗ p.q : r 0 : q.p.r 0.s.s 0 10∗∗∗ p.q.r.s 0.r.q.p.s 0 10∗∗∗ p.q.r : r.q.p : s 0.s 10∗∗∗ p.q.r : p : r.q.s.s 0

10∗∗∗ p.q 0.r.s 0.p.s 0.r.q 0.t 0.t 10∗∗∗ p.q.r 0.s 0.p.s 0.r 0.q.t 0.t 10∗∗∗ p.q.r.s 0.p.s 0.r.q.t 0.t 10∗∗∗ p.q.r.s 0.r.q.p.s 0.t 0.t 10∗∗∗ p.p.p.p.p 0.p 0.p 0.p 0.q.q 0 10∗∗∗ p.q.r.s.r.q.p.s.t.t 0

10∗∗∗ p.q 0.r.s 0.p.s 0.r.q 0.t.t 0 10∗∗∗ p.q.r 0.s 0.p.s 0.r 0.q.t.t 0 10∗∗∗ p.q.r.s 0.p.s 0.r.q.t.t 0 10∗∗∗ p.q.r.s 0.r.q.p.s 0.t.t 0 10∗∗∗ p.q.r.s.p.s.r.q.t.t 0

In the preceding table there are three 2-parameter classes of achiral KLs: 10∗∗∗ p : p : .p 0 : p 0.q 0.q, 10∗∗∗ p : p : .p 0 : p 0.q.q 0, and 10∗∗∗ p.p.p.p.p 0.p 0.p 0.p 0.q.q 0. Also, there are non-detectable links of the same kind like 10∗∗∗ p.p.p.p.q : q 0.

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The following tables contain KLs up to n = 12 crossings obtained from the families of KLs mentioned before. Our results agree with the known results for achiral alternating knots with n ≤ 12 crossing, and achiral alternating links with n ≤ 8 crossings. Moreover, we hope that this symmetry approach can be extended to all achiral KLs. n=2 2 n=4 22 n=6 33

2112

6∗

44

2222

3113

n=8 211112

(2, 2) (2, 2)

6∗ 2.2

6∗ 2 : .2

8∗

n = 10 55 221122 (2 1, 2) (2 1, 2) 6∗ 2 1 : .2 1 8∗ 2.2 10∗∗

2332 311113 (2, 2) 1 1 (2, 2) 6∗ 2.2.2.2 8∗ 2 : .2 10∗∗∗

3223 21111112 6∗ 3.3 6∗ 2.2 : 2.2 8∗ 2 0.2 0

4114 (2, 2+) (2, 2+) 6∗ 2 1.2 1 6∗ 2.2.2 0 : 2 8∗ 2 0 : .2 0

212212 (3, 2) (3, 2) 6∗ 3 : .3 6∗ 2.2 : 2 0.2 0 10∗

n = 12 66 5115 312213 21211212 (4, 2) (4, 2) (3, 2) 1 1 (2, 3) (2, 2 + +) (2, 2 + +) (2, 2) 1 1 1 1 (2, 2) (2 1 1, 2) (2 1 1, 2) 6∗ 2 1 1.2 1 1 6∗ 2 2 : .2 2 6∗ 2.3.3.2 6∗ 3.2 : 3.2 6∗ 2.3.2 0 : 3 6∗ 2 1.2 1 : 2 0.2 0 8∗ 3.3 8∗ 3 0.3 0 8∗ 2.2.2.2 8∗ 2.2 : 2 0 : .2 0 8∗ 2 : 2 0.2 0 : 2 10∗ 2.2

2442 213312 321123 22111122 (3, 3) (3, 3) (3 1, 2) (3 1, 2) (2, 2) 2 2 (2, 2) (2 1, 2 1) (2 1, 2 1) 6∗ 4.4 ∗ 6 (2, 2).(2, 2) 6∗ 2 1 1 : .2 1 1 6∗ 2 1.2.2.2 1 6∗ 2 1.2 1 : 2.2 6∗ 2 1.2.2 1 0 : 2 6∗ 2.2.2.2.2.2 8∗ 2 1.2 1 8∗ 2 1 0.2 1 0 8∗ 2.2 : 2 : .2 8∗ 2.2 : .2 0.2 0 8∗ 2 0.2 0.2 0.2 0 10∗ 2 : .2

3333 222222 411114 31111113 (3, 2 1) (3, 2 1) (2 2, 2) (2 2, 2) ((2, 2), 2) ((2, 2), 2) (2 1, 2+) (2 1, 2+) 6∗ 3 1.3 1 6∗ 4 : .4 6∗ (2, 2) : .(2, 2) 6∗ 2.2 1.2 1.2 6∗ 2 1.2 : 2 1.2 6∗ 2.2 1.2 0 : 2 1 6∗ 2.2.2.2.2 0.2 0 8∗ 3 : .3 8∗ 3 0 : .3 0 8∗ 2.2 : .2.2 8∗ 2.2 0.2 0.2 8∗ 2 0.2 0 : 2 0 : .2 0 10∗ 2 ::: .2

4224 231132 21122112 2111111112 (3, 2+) (3, 2+) (2, 2, 2) (2, 2, 2) (2, 2+) 1 1 (2, 2+) (2 1, 2) 1 1 (2, 2 1) 6∗ 2 2.2 2 6∗ 3 1 : .3 1 6∗ 3.2.2.3 6∗ 3.3 : 2.2 6∗ 3.2.3 0 : 2 6∗ 3.3 : 2 0.2 0 6∗ 2.2.2 0.2.2.2 0 8∗ 2 1 : .2 1 8∗ 2 1 0 : .2 1 0 8∗ 2.2.2 0 :: .2 0 8∗ 2.2 0 : .2 0.2 ∗ 8 2 0.2 0 : .2 0.2 0 10∗ 2 0.2 0

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10∗ 2 0 : .2 0 10∗∗ .2 0 :: .2 0 10∗∗ : 2.2 10∗∗∗ 2 : 2 0 121∗ 1210∗

10∗ 2 0 :: .2 0 10∗∗ .2 0 : .2 0 10∗∗ .2 :: .2 10∗∗∗ :::: 2.2 0 122∗ 1211∗

10∗∗ : 2 0 :: .2 0 10∗∗ 2 0 :: .2 0 10∗∗ .2 : .2 10∗∗∗ 2 :: 2 127∗ 1212∗

10∗∗ : 2 0.2 0 10∗∗ : 2 :: .2 10∗∗ 2 :: .2 10∗∗∗ 2 ::: 2 128∗

For future derivation, one may check achirality of all rational, stellar, arborescent and polyhedral alternating source KLs using Kauffman polynomials computed for a KL and its mirror image, the LinKnot functions AmphiProjAltKL and AmphiAltKL and the program SnapPea by J. Weeks, and then systematically derive classes of achiral KLs.

Fig. 2.93

Achiral oriented link 8∗ . − 2 0.2 0. − 2 0 with n = 11 crossings.

Fig. 2.94 Antisymmetric presentation 123∗ − 2 0. − 1. − 1.2 0 : . − 2 0. − 1. − 1.2 0 with n = 16 crossings of the achiral oriented link 8∗ . − 2 0.2 0. − 2 0 with n = 11 crossings.

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Fig. 2.95

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Achiral knot 10∗∗ 2 0 : 2 0. − 2. − 1.2 0 : 2 0. − 2. − 1 with n = 15 crossings.

Fig. 2.96 Antisymmetric representation 10∗∗ − 2 0. − 1. − 2 0.2 : −2 0. − 1. − 2 0.2 with n = 16 crossings of the achiral knot 10∗∗ 2 0.2 : −2 0 : 2 0. − 1. − 1. − 1. − 2 0 with n = 15 crossings.

P.G. Tait conjectured that every achiral KL must have an even number of crossings, so neither P.G. Tait nor M.G. Haseman considered the possibility of the existence of achiral knots with an odd crossing number. However, this Tait’s Conjecture was disproved (see Hoste, Thistlethwaite, and Weeks, 1998). The first oriented achiral non-alternating link 8∗ . − 2 0.2 0. − 2 0 with n = 11 crossings was discovered in 1998 (Liang, Mislow and Flapan, 1998). The achiral non-alternating knot 10∗∗ 2 0.2 : −2 0 : 2 0.−1.−1.−1.−2 0 with n = 15 crossings was found by M. Thistlethwaite, who also recognized few repetitions in Haseman’s tables. However, Tait’s Conjecture about achiral KLs holds for alternating KLs: there is no alternating achiral KL with an odd number of crossings (Corollary 1.1).

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The non-alternating achiral oriented link 8∗ . − 2 0.2 0. − 2 0 with n = 11 crossings (Fig. 2.93) has a non-minimal antisymmetric representation 123∗ − 2 0. − 1. − 1.2 0 : . − 2 0. − 1. − 1.2 0 with n = 16 crossings (Fig. 2.94), that shows its achirality. The nonalternating achiral knot 10∗∗ 2 0.2 : −2 0 : 2 0. − 1. − 1. − 1. − 2 0 with n = 15 crossings (Fig. 2.95) has a non-minimal antisymmetric representation 10∗∗ − 2 0. − 1. − 2 0.2 : −2 0. − 1. − 2 0.2 with n = 16 crossings (Fig. 2.96). This property can be extended to a family of achiral knots with an odd number of crossings. From the antisymmetric representation 10∗∗ − 2 0. − 1. − 2 0.2 : −2 0. − 1. − 2 0.2 of the knot 10∗∗ 2 0 : 2 0. − 2. − 1.2 0 : 2 0. − 2. − 1 we derive the three-parameter family 10∗∗ (−2p) 0. − 1.(−2q) 0.(2r) : (−2p) 0. − 1.(−2q) 0.(2r) of achiral knots with n = 2p + 2q + 2r + 9 crossings. For example, the non-minimal chiral antisymmetric representation 10∗∗ − 4 0. − 1. − 6 0.8 : −4 0. − 1. − 6 0.8 with n = 40 crossings can be reduced to the achiral knot with n = 39 crossings given by the Dowker code {{39}, {8, −32, −34, 64, −62, −60, −66, 42, 40, 38, −68, −70, 44, 46, 48, 50, −4, −6, 24, 18, 16, 142, 6, 28, 30, 36, −72, −74, −76, −12, −10, −78}}.

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Families of undetectable KLs

A conversion of Conway symbols, Dowker codes, and P -data to P D (planar diagrams), which are the main input for the program KnotTheory is provided by LinKnot functions fConwayToPD, fKnotscapeDow ToPD, fDowkerToPD, and fPdataToPD. The minimum braid words corresponding to KLs can be computed by the function BR. The function fBraidW gives the corresponding braid word, and the function fPDfromBW computes P D from a given braid word. We will consider the set of polynomial invariants P , where P is Alexander, Conway, Jones, Khovanov5, A2, Links-Gould, HOMFLYPT, Kauffman, and colored Jones polynomial. In our research, Khovanov, A2, and colored Jones polynomials are computed using the functions Kh, A2, and ColouredJones from the program KnotTheory (http://katlas.math. toronto.edu/wiki/The Mathematica Package KnotTheory), and LinksGould invariants are computed using the functions LinksGould and LinksGouldInv from the program Links-Gould Explorer written by David de Wit (http://www.maths.uq.edu.au/∼ ddw/Links–Gould Explorer/Links–GouldExplorer.htm), included in LinKnot. One of the main questions about every polynomial invariant is whether or not there exists a P -unknot (unlink), i.e., a nontrivial knot (link) L with the trivial polynomial P (L). For example, Alexander and Conway unknots and Jones unlinks exist (Eliahou, Kauffman and Thistlethwaite, 2003). There are entire families of KLs with these properties. The existence problem for Jones unknots is open; it has been shown that Jones unknots must be non-alternating (Murasugi, 1987a,b) with at least n = 18 crossings (Dasbach and Hougardy, 1997). For the remaining P -invariants the question about P -unknot is open as well. Definition 2.29. Two non-isotopic knots or links L1 and L2 are called P -undetectable if P (L1 ) = P (L2 ) for some polynomial invariant P . There are infinitely many pairs of P -undetectable KLs with the same or different number of crossings, and there is an infinite number of undetectable KLs for any polynomial invariant P (see, e.g., Kanenobu, 1986; Przytycki, 1995; Watson, 2004). L. Watson showed that an arbitrary tangle T can be extended to produce diagrams of two distinct knots that can not be distinguished by the Jones polynomial. For a prime tangle T , the 5 The function Kh computes Khovanov polynomial (Shumakovitch, 2004). In the same sense, we will use the term “Links-Gould polynomial” (De Wit, 2000).

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resulting knots are prime non-mutant. (Watson, 2006a). Moreover, according to experimental results, there are families of KLs that can not be detected by certain polynomial invariant P : P (L) is the same for all KLs from such a family F . For example, for every knot from the family (2k + 1), 3, −3 the Alexander polynomial is 2-5t+2t2 and the Conway polynomial is 1 − 2x. Notice that families of KLs need to be disjoint. For example, 2p 2p will be considered as one-parameter family, and 2p 2q as two-parameter family with p 6= q. For families of alternating KLs, we propose the following conjecture: Conjecture 2.4. For every two alternating non-isotopic KLs L1 and L2 belonging to the same family F , P (L1 ) 6= P (L2 ) for every polynomial invariant P . With non-alternating KLs, the situation is different. Definition 2.30. A family F of non-alternating KLs will be called P undetectable if P (L) is the same for any link L from F . If a polynomial from P can recognize as different any two non-isotopic KLs from F , we will call this family P -detectable. Since Alexander polynomial is obtained from the Conway polynomial from the same skein relation by a change of variable, the two polynomials carry identical information. Hence, every Alexander-undetectable family is Conway-undetectable and vice versa. K. Kanenobu (1986) found an infinite class of knots with the same Jonesand/or HOMFLYPT-polynomial (Fig. 2.97). All KLs of the form K(p, q) = 6∗ − 2. − 2.p 0.2.2.q 0 are Jonesundetectable iff |p − q| = const, where const is an arbitrary constant. K(p, q) and K(p′ , q ′ ) are Jones-, HOMFLYPT- (and Khovanovundetectable) iff |p − q| = |p′ − q ′ |, p = p′ (mod 2), and q = q ′ (mod 2). Kanenobu’s examples can be reconstructed in the framework of S. Eliahou, L. Kauffman, and M. Thistlethwaite (2003), and this was done by L. Watson (2005). Watson also gave a construction of infinite families of distinct knots with identical Khovanov homology (2006). K. Luse and Y. Rong (2006) constructed a new class of Jones- and HOMFLYPT-undetectable knots (Fig. 2.98). Kanenobu’s knots can be obtained from this class for a = 1, t1 = 2p, t2 = 2q. We extended their construction to the family F of the form L(x, y) = 6∗ − a. − b.t1 x 0.b.a.t2 y 0, where are a, b are fixed positive integers, t1 and

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Fig. 2.97 Distinct non-mutant knots 41 #41 and 3 1 1 3, obtained from 6∗ − 2. − 2.p 0.2.2.q 0 for p = q = 0 and p = q = 1, with identical Jones, but different HOMFLYPT polynomials.

Fig. 2.98

The family 6∗ t1 . − 2.2 0. − t2 .(2a). − (2a) 0.

t2 are arbitrary fixed tangles, and x, y are integers (Fig. 2.99). This family has the following properties: (1) all KLs from F are Alexander- and Conway-undetectable for every x, y with x = y (mod 2); (2) all KLs from F are Jones-undetectable for every x, y satisfying the condition |x − y| = const, for arbitrary constant const; (3) the links L(x, y) and L(x′ , y ′ ) from F are Jones-, Khovanov-, HOMFLYPT-, A2-, and Links-Gould-undetectable iff |x−y| = |x′ −y ′ |, x = x′ (mod 2), and y = y ′ (mod 2).

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Fig. 2.99

Fig. 2.100

The family 6∗ − a. − b.t1 x 0.b.a.t2 y 0.

The family 8∗ − a. − b. − c. − t1 x 0.c.b.a.t2 y 0.

All undetectable KLs obtained are Kauffman- and colored Jonesdetectable. KLs obtained are achiral for t1 = t2 , and chiral otherwise. The following table contains a summary of the previous results where plus or minus means that given polynomial invariant does or does not distinguish KLs from a given family:

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Invariant Alexander Conway Jones Khovanov HOMFLYPT A2 Links-Gould Kauffman Colored Jones

(1) − − + + + + + + +

(2) − − − + + + + + +

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(3) − − − − − − − + +

The following knots are obtained for a = b = 2, (x, y) ∈ {(2, 6), (2, 2), (3, 3), (4, 4), (5, 5)} : K1 = 6∗ − 2. − 2.2 0.2.2.6 0 K3 = 6∗ − 2. − 2.3 0.2.2.3 0 K5 = 6∗ − 2. − 2.5 0.2.2.5 0

K2 = 6∗ − 2. − 2.2 0.2.2.2 0 K4 = 6∗ − 2. − 2.4 0.2.2.4 0

All of them are Alexander and Conway-undetectable; K1 can be recognized as different from the other four knots by all remaining polynomial invariants from P ; K2 and K3 , or K4 and K5 can be distinguished by HOMFLYPT, A2 or Links-Gould invariant, but they do not distinguish K2 from K4 , or K3 from K5 . The five of them can be distinguished by the Kauffman polynomial or by the colored Jones polynomial.

Fig. 2.101 Undetectable knots 10∗ − 2. − 2. − 2. − 2.3 2 0.2.2.2.2.3 2 0 and 10∗ − 2. − 2. − 2. − 2.3 4 0.2.2.2.2.3 4 0 from the family 10∗ − a. − b. − c. − d.t1 x 0.d.c.b.a.t2 y 0.

Let us consider the family of antiprismatic basic polyhedra 6∗ , 8∗ , 10∗ , 12A (121∗ ), 1449∗, . . . Several P -undetectable families can be derived from each of these basic polyhedra.

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Here is the list of families we have obtained, where x and y are parameters, and all other tangles are fixed (Figs. 2.99-2.100):

(1) the families 6∗ − a. − b.t1 x 0.b.a.t2 y 0; 6∗ − a.b.t1 x 0. − b.a.t2 y 0; (2) the families 8∗ − a. − b. − c. − t1 − x 0.c.b.a.t2 y 0, 8∗ − a. − b.c. − t1 − x 0. − c.b.a.t2 y 0, 8∗ − a.b. − c. − t1 − x 0.c. − b.a.t2 y 0; (3) the families 10∗ − a. − b. − c. − d.t1 x 0.d.c.b.a.t2 y 0, 10∗ − a. − b.c.d.t1 x 0. − d. − c.b.a.t2 y 0; (4) the families 121∗ − a. − b. − c. − d. − e. − t1 − x 0.e.d.c.b.a.t2 y 0, 121∗ − a. − b. − c. − d.e. − t1 − x 0. − e.d.c.b.a.t2 y 0, 121∗ − a. − b. − c.d. − e. − t1 − x 0.e. − d.c.b.a.t2 y 0, 121∗ − a. − b. − c.d.e. − t1 − x 0. − e. − d.c.b.a.t2 y 0, 121∗ − a. − b.c. − d.e. − t1 − x 0. − e.d. − c.b.a.t2 y 0, 121∗ − a. − b.c.d. − e. − t1 − x 0.e. − d. − c.b.a.t2 y 0, 121∗ − a.b. − c.d. − e. − t1 − x 0.e. − d.c. − b.a.t1 y 0. (5) the families 1449∗ − a. − b. − c. − d. − e. − f.t1 x 0.c.t2 y 0.f.b.a.d.e, 1449∗ − a. − b. − c. − d.e. − f.t1 x 0.c.t2 y 0.f.b.a.d. − e, 1449∗ − a. − b.c.d. − e. − f.t1 x 0. − c.t2 y 0.f.b.a. − d.e, 1449∗ − a. − b.c.d.e. − f.t1 x 0. − c.t2 y 0.f.b.a. − d. − e, 1449∗ − a.b. − c. − d.e.f.t1 x 0.c.t2 y 0. − f. − b.a.d. − e, 1449∗ − a.b.c.d. − e.f.t1 x 0. − c.t2 y 0. − f. − b.a. − d.e, Notice that all the families of P -undetectable KLs obtained from the antiprismatic basic polyhedra (2k)∗ (k = 3, 4, 5, . . .) are of the form (2k)∗ w.t1 x 0.w′ .t2 y 0, where w and w′ are palindromic words of the form a.b . . . with the antisymmetric distribution of signs6 . The following P -undetectable families are obtained from the basic polyhedron 10∗∗∗ (Fig. 2.102): 6 According to the Conway notation, the basic polyhedron 1449∗ from LinKnot data base is 14∗ , i.e., (2n)∗ for n = 7.

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Fig. 2.102

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The family 10∗∗∗ − a. − b.b 0.a 0. − c 0. − d 0.d.c. − t1 − x 0.t2 y 0.

10∗∗∗ − a. − b.b 0.a 0. − c 0. − d 0.d.c. − t1 10∗∗∗ − a.b. − b 0.a 0. − c 0.d 0. − d.c. − t1 10∗∗∗ − a. − b.b 0.a 0. − c. − d.d 0.c 0. − t1 10∗∗∗ − a.b. − b 0.a 0. − c.d. − d 0.c 0. − t1

− x 0.t2 y 0, − x 0.t2 y 0, − x 0.t2 y 0, − x 0.t2 y 0

with the same properties. In particular, for Jones unknot we need to search only in Jonesundetectable families with t1 = t2 . Conjecture 2.5. Families of P -undetectable KLs can be derived from every achiral basic polyhedron. Achiral basic polyhedra with n = 12 crossings are: 12A (121∗ ), 12B (122∗ ), 12J (1210∗), 12K (1211∗ ), and 12L (1212∗). Conjecture 2.6. All algebraic alternating KL families are detectable by any polynomial invariant P . Algebraic non-alternating KL families can be only Alexander- and Conway-undetectable. Another type of undetectable knots was discovered by Dunfield, Garoufalidis, Shumakovitch and Thistlethwaite (2006). In their paper Behavior of knot invariants under genus 2 mutation knots with the same colored Jones, HOMFLYPT, and Kauffman polynomials, signature and hyperbolic volume are considered. These knots also have equal Alexander polynomial and the A2 invariant.

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The authors proved the following proposition: Proposition 2.2. There exist knots with the same colored Jones polynomials (for all colors), HOMFLYPT and Kauffman polynomials, volume and signature, but different Khovanov (and reduced Khovanov) homology. Definition 2.31. Two knots are almost mutant or algebraically mutant if they have the same HOMFLYPT and Kauffman polynomials, signature, and hyperbolic volume.

Fig. 2.103

Cabled tangle T (1, n).

Among pairs almost mutant non-alternating knots with n ≤ 16 crossings the authors found pairs with different Khovanov homology. There is one pair of such non-alternating 14- crossing knots, four pairs for n = 15, and 27 pairs for n = 16 crossings, in total 32 pairs of knots with at most n ≤ 16 crossings. For n ≤ 15 the authors recognized a pattern: all five pairs of knots are cabled mutant knots with the same closure: tangle T (1, 2) (Fig. 2.103). Authors noticed that many pairs for n = 16 consist of cabled mutant knots, but have not verified this pattern for all pairs. In the following table those 32 pairs of knots are given in Knotscape notation and denoted by (1)-(32), n (1) 14n 22185 ,1422589 n (2) 15n ,15 57606 57436 n (3) 15n ,15 115375 51748 n (4) 15133697 ,14n 135711 n (5) 15n 148673 ,15151500 n (6) 16n 257474 ,16293658 n (7) 16n ,16 258027 380926 n n (8) 16258035 ,16359938 n (9) 16n 261803 ,16300395 n (10) 16262535 ,16n 300387 n (11) 16n 306846 ,16307597

(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

n 16n 332130 ,16707045 n 16n ,16 337388 697474 n 16n ,16 472161 635329 n 16564024 ,14n 564036 n 16n 564059 ,16564068 n 16n ,16 789164 797712 n 16n ,16 789206 797688 n n 16809314 ,16850490 n n 16809334 ,16850512 n 16n 812818 ,16850972 n 16n ,16 820956 820968

(23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

n

16n 822219 ,16822229 n 16n 878609 ,16944604 n n 16884231 ,14884268 n 16n 885298 ,16885312 n n 16885305 ,16885319 n 16n 885467 ,16885968 n 16n ,16 890470 944600 n n 16937845 ,16947557 n n 16939163 ,16945493 n 16n 943082 ,14943119

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where the bar above the number of crossings means the mirror image of the corresponding knot. In order to generalize their results, we described ten classes of KLs, given in Conway notation and enumerated by I-X, containing 32 mentioned pairs of Khovanov-detectable almost mutant knots. Most of them are cabled mutants, but few of them (3,9,10,21,24,29) are obtained by a more general method: inserting tangles in the top crossings of the tangle T (1, n). A non-alternating representation of KL is called non-reducible if it can not be reduced to a KL with a lower number of crossings. Conjecture 2.7. All non-reducible KLs from the classes I-X with arbitrary tangles a,b,..., x,y are almost mutant KLs with different Khovanov polynomials. In other words, the pair-classes are equivalence classes of almost mutant KLs detectable by Khovanov homology. Before describing particular classes, we will explain common construction principle: the tangle T = T (1, n) is rotated around vertical 2-axis, giving the tangle T ′ . By composing T and T ′ with an arbitrary (n + 1)tangle T1 , we obtain two KLs (T, T1 ) and (T ′ , T1 ). If the tangle T1 consists from 2-tangles a,b,..., and x and y are tangles placed in the top crossings of T and T ′ , by appropriate choice of signs and positions of the constituting tangles a, b,...,x,y we obtain a class-pair of almost mutant KLs with distinct Khovanov homology. All other “non-appropriate” choices giving non-reducible KLs result in completely undetectable almost mutant KLs. The pair-classes I-X are derived from four basic polyhedra, 112∗ , 127∗ , 132∗ , and 137∗ . 112∗ − 1. − 1.b 0.2 0. − a. − c. − 2 0.1.y.1. − x 0 112∗ 1.1.b 0. − 2 0. − a. − c.2 0. − 1. − x 0. − 1.y 112∗ − 1. − 1.b 0.2 0. − a 0. − c. − 2 0.1.y.1. − x 0 112∗ 1.1.b 0. − 2 0. − a 0. − c.2 0. − 1. − x 0. − 1.y 112∗ − 1. − 1.b 0.2 0. − a. − c 0. − 2 0.1.y.1. − x 0 112∗ 1.1.b 0. − 2 0. − a. − c 0.2 0. − 1. − x 0. − 1.y 112∗ − 1. − 1.b.2 0.a 0. − c 0. − 2 0.1.y.1. − x 0 112∗ 1.1.b. − 2 0.a 0. − c 0.2 0. − 1. − x 0. − 1.y 112∗ − 1. − 1.b 0.2 0. − a. − c. − 2 0.1.y 0.1. − x 112∗ 1.1.b 0. − 2 0. − a. − c.2 0. − 1. − x. − 1.y 0 127∗ c.a. − b.d. − 1.2. − 2 0. − 1.1.1.y. − x 0 127∗ c.a. − b.d.1. − 2.2 0.1. − 1. − 1. − x.y 0 127∗ − c.a.b. − d. − 1.2. − 2 0. − 1.1.1.y. − x 0 127∗ − c.a.b. − d.1. − 2.2 0.1. − 1. − 1. − x.y 0

I II III IV V VI VII

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128∗ − 1. − x 0. − 1.2 0.a. − b 0.c. − 2 0.d 0.1.y 0.1 128∗ 1. − y 0.1. − 2 0.a. − b 0.c.2 0.d 0. − 1.x 0. − 1 132∗ d.1.e.b.c. − 2.1.x 0. − a 0.2 0. − 1. − y. − 1 132∗ d. − 1.e.b.c.2. − 1. − y 0. − a 0. − 2 0.1. − x.1 137∗ − x 0.y 0. − 1. − 1. − a.d.1.1.2.b.e.c. − 2 137∗ y 0. − x 0.1.1. − a.d. − 1. − 1. − 2.b.e.c. − 2

VIII IX X

The above-mentioned 32 knots are the members of infinite pair-classes of almost mutant KLs detectable by Khovanov polynomial. In the following table there are given values of tangles a,b,c,...,x,y for these 32 pairs.

I I VI II I VI I I I II II VI VI VI III III I I I II IV VIII IX I I III III I V X II VII

a 1 1 1 2 1 1 1 21 1 2 2 2 1 1 1 1 1 1 3 3 2 1 2 1 1 1 1 1 1 2 2 1

b 2 21 2 2 2 21 22 2 21 2 21 2 2 2 21 2 31 4 2 2 1 2 1 2 3 3 2 2 2 1 2 2

c 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 21 1 1 1 1 2 1 1 2 2 2 3 3 2 1 2 2

d

e

1

1

1 2 1

2 1

1

1

1

1

x 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1

y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

The pair-classes are not disjoint. For example, the knot (1) can also be obtained in the class II, or III for b = 2.

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All classes I-X are constructed in the same way, as a closure of tangle T (1, 2) by an antiprismatic k-belt consisting from k triangles (k = 1, 2, ...) (Fig. 2.69). However, instead of antiprismatic belt, arbitrary (n + 1)-tangle T1 can be used (see, e.g., the pair-class derived from the basic polyhedron 171312∗, Fig. 2.106). The same tangle T (1, n) and (n + 1)-tangle T1 used as its closure can result in different basic polyhedra, according to different pairing of free strands (e.g., the classes VI and VIII). Representing the pairclasses I-X by KL diagrams with labelled tangles a,b,c,...,x,y, we obtain five diagrams, corresponding to the class-pairs I-V,VI-VII,VIII,IX, and X, respectively (Fig. 2.104).

Fig. 2.104

Diagrams of classes I-X with labelled tangles a,b,c,...,x,y.

Figure 2.105 shows the knot pair (9) with x = 2, that can not be derived as a pair of cabled mutant knots, without introducing tangle x in the top crossing of the generating tangle T (1, 2). Following the same pattern, we can derive almost mutant KLs detectable by Khovanov polynomial from other basic polyhedra with a higher number of crossings. For example, we derived the following pair-classes:

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Fig. 2.105

Knot pair (9).

1411∗f.d 0. − y. − 1.e. − b. − c.x. − 1.2 0. − a.1.1. − 2 0 1411∗f.d 0.x.1.e. − b. − c. − y. − 1. − 2 0. − a. − 1. − 1.2 0

from the basic polyhedron 1411∗,

156∗ 1.y 0.1. − 2. − 1. − x 0. − f. − g 0.2. − 1. − c 0.d.e. − a. − b 156∗ − 1. − x 0. − 1.2.1.y 0. − f. − g 0. − 2.1. − c 0.d.e. − a. − b

from 156∗ ,

1680∗f.h. − g.e.d. − 2.1.y 0.c.b.2 0.1. − x.a 0. − 1. − 1 1680∗f.h. − g.e.d.2. − 1. − x 0.c.b. − 2 0. − 1.y.a 0.1.1

from 1680∗, etc. From the basic polyhedron 171312∗ we derived the pairclasses 171312∗ − 1. − y 0. − 1.1. − e.x.1.1. − 2.d. − c 0.1. − 1.2 0. − b 0.a. − 1 171312∗1.x 0.1. − 1. − e. − y. − 1. − 1.2.d. − c 0. − 1.1. − 2 0. − b0.a.1 that contain tangle T (1, 3) (Fig. 2.106). The construction of almost mutant Khovanov-detectable pair-classes can be extended using tangles T (k, n) (k > 1) (Fig. 2.107). For example, the following pair-classes are obtained from the basic polyhedron 1912169∗ and tangle T (2, 2) 1912169∗ − 1. − 1. − y. − 1.2 0.c 0. − v 0. − 1.2. − a. − b 0.u.1. − 2 0. − 2 0.1.x.1.1

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Fig. 2.106

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Diagram of the pair-class derived from the basic polyhedron 171312∗ .

1912169∗1.1.x.1. − 2 0.c 0.u.1. − 2. − a. − b 0. − v. − 1.2 0.2 0. − 1. − y. − 1. − 1 giving, e.g., for b = 2, c = 2 and a = d = x = y = u = v = 1 Khovanovdetectable almost mutant knots {{25}, {6, 12, −22, 20, −30, 2, 46, −40, −34, 10, 50, −4, −48, 42, 36, −8, 28, −18, 32, 26, −16, 38, −24, 14, −44}} and {{25}, {6, −12, 24, 22, −32, 20, −4, −48, 42, 36, 8, 2, 46, −40, −34, −10, −38, 18, −30, −44, 16, −28, 50, −14, 26}}. The open question is: are there some other constructions resulting in Khovanov-detectable almost mutant KLs. Polynomials are determined via LinKnot and the function Kh from KnotTheory program. Hyperbolic volumes of knots are computed via Knotscape, and those of links via SnapPea (possibly with an insufficient precision). For example, Khovanov-detectable almost mutant links obtained in the class I for b = 3, a = c = x = y = 1 have equal multi-variable Alexander polynomials and, conjecturally, the same volume (approximately 12.6684303...) (Fig. 2.108).

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Fig. 2.107 (a) Diagram of the pair-class derived from the basic polyhedron 1912169∗ ; (b) Khovanov-detectable almost mutant knots obtained from it for for b = 2, c = 2 and a = d = x = y = u = v = 1.

Fig. 2.108 Khovanov-detectable almost mutant links obtained from pair-class I for b = 3, a = c = x = y = 1.

We expect that two Khovanov-detectable almost mutant KLs can have different unknotting numbers. The best potential candidate to prove this is the pair (9). The unknotting number of the first knot from this pair, 16n261803 , is 2, and BJ-unknotting number of the other knot, 16n300395 , is 3 (Fig. 2.109). The same conclusion can be obtained if one succeeds to prove that the unknotting number of any of the following knots: 16n258027 (the fist from the pair (7)), 16n635329 (the second from (14)), 16n789206 (the first from (18)), or 16n812818 (the first from (21)) is 3. For n ≤ 16 almost mutant knots always have the same number of crossings (Dunfield, Garoufalidis, Shumakovitch and Thistlethwaite, 2006), but for larger values of n is possible to obtain almost mutant knots with dif-

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Fig. 2.109

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Try to prove that the unknotting number of this knot, 16n 300395 , is 3.

ferent number of crossings. For example, the following pair of 19-crossing knots is obtained from the pair-class I, for a = 3, b = 1, c = 2, x = 2, y = 3 {{19}, {−18, 24, −36, 14, 38, −28, 6, 8, −22, 30, −2, −16, −32, −34, −10, 20, −26, −4, 12}}

{{19}, {20, 24, −36, −12, 28, −38, −6, 22, 2, −30, 14, 16, −32, −34, 10, −18, −26, −4, −8}},

Knotscape reduces the first knot to 18 crossings: {{18}, {−6, −10, −24, −32, −2, −18, 20, 36, 34, −26, 28, 14, −4, −12, 22, 16, −8, 30}},

and the second to 17 crossings: {{17}, {6, 10, −20, 14, 2, −30, 8, 28, 24, 32, −4, 16, 18, 34, −12, 22, 26}}.

All polynomial invariants (Alexander, Conway, Jones, HOMFLYPT, colored Jones (for all colors), Kauffman, A2, and Khovanov polynomial), signature, hyperbolic volume (23.4165685828), and even BJ-unknotting numbers (3) are equal for these two knots. The question is: what will happen with reducible KLs from the pair-classes I-X, or other pair-classes constructed in the analogous manner? In certain cases, both almost mutant knots can be reduced to the same number of crossings and recognized by Khovanov polynomial (e.g., 18-crossing knots from pair-class VI obtained for a = 3, b = 2, y = 2 that reduce to 17-crossing knots). Is it possible to find a pair of almost mutant KLs with different crossing numbers, detectable by Khovanov polynomial?

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Detecting chirality of KLs by polynomial invariants

A KL is achiral (or amphicheiral) if it is ambient isotopic to its mirror image. Otherwise, it is chiral (Definition 1.26). An oriented link L is achiral if there is an ambient isotopy transforming it into its mirror image L∗ and preserving the orientation of components (Definition 1.53). We will consider n-colored Jones polynomial (n ≥ 3) and the set of polynomial invariants P : Jones, Khovanov, HOMFLYPT, and Kauffman polynomial, and compare their ability to detect chiral KLs. HOMFLYPT polynomial is used only for detecting chiral knots and links with an odd number of components. Definition 2.32. A chiral link is called chiral P -undetectable if P (L) = P (L∗ ), where L∗ is the mirror image of the link L. Otherwise, it is called chiral P -detectable. For chiral P -undetectable links, instead of terms: Jones-, Khovanov-, HOMFLYPT-, and Kauffman-undetectable KLs, we will use concise terms: J-, Kh-, H-, and K-undetectable, respectively. Examples of chiral undetectable knots can be found in different sources, e.g., a traditional example of chiral knot 2 2, 3, 2− (942 ) which is Hundetectable (Adams, 1994, page 179; Flapan, 2000, page 48). The best source for K- and H-undetectable chiral knots is the paper by C. Liang and K. Mislow (1994a). Their list contains the following chiral undetectable knots: 2 2, 3, 2− (942 ), 2 2, 2 1, 2+ (1071 ), .2.(2 2, 2) 0 (12126 ), .2.(2, 2 2) 0 (12132 ), .2 1 1 1.2 (12214 ), 8∗ 2 1 1 0 : .2 0 (12222 ), and 9∗ 2 1 0.2 (12697 ), that are undetectable by all mentioned polynomial invariants, except 2 2, 3, 2− (942 ) which is Kh-detectable. Their chirality can be detected by n-colored Jones polynomial (n ≥ 3). Definition 2.33. A mutation which transforms achiral knot or link L into a chiral knot or link L′ is called chiral mutation. Since all polynomial invariants are unaffected by mutation, obtained link L′ can not be recognized as chiral by any polynomial invariant. At the beginning we need to underline that all P -undetectable KLs that we considered are 3-colored Jones-detectable, so we propose the following main conjecture: Conjecture 2.8. All chiral KLs are 3-colored Jones-detectable (up to a chiral mutation).

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Chirality problem is completely solved for rational KLs: a rational knot is achiral iff its Conway symbol is mirror-symmetric (palindromic) and has an even number of crossings (Siebenmann, 1975; Caudron, 1982) (Theorem 1.20). All rational links are 2-component. Since all oriented alternating links with an even number of components are chiral (Cerf, 1997), there are no oriented rational achiral links. A rational non-oriented link is achiral iff its Conway symbol is mirrorsymmetric (palindromic) and has an even number of crossings (Kauffman and Lambropoulou, 2002) (Theorem 1.22). Chiral rational KLs are the only class of chiral KLs which are P detectable by all mentioned polynomial invariants. Theorem 2.16. Every alternating pretzel (Montesinos) KL is chiral. First surprise occurs for alternating pretzel KLs: for n = 8 we have J-undetectable 3-component link 3 1, 2, 2, and for n = 10 first J-, Kh-, and H-undetectable knot 4 1, 3, 2, and the J- and Kh-undetectable 2-component link 2 1 1, 3 1, 2. Knot 4 1, 3, 2 is the beginning of the family of J-, Kh-, and H-undetectable pretzel knots (k + 2) 1, 3, k (k ≥ 2). For an odd n there are no chiral P -undetectable alternating pretzel KLs. Notice that for H-polynomial only KLs with an odd number of components are taken into account. Number of P -undetectable alternating chiral pretzel KLs for 8 ≤ n ≤ 16 is the following :

n=8 n = 10 n = 12 n = 14 n = 16

J1 2 11 30 65

Kh2 4 20 32

H1 1 5 10 23

It is well known that writhe and signature of every achiral alternating knot is equal to 0, so this can be used as the first (but very weak) achirality criterion. Most of P -undetectable knots satisfy this property. However, for n = 12 there is a J- and H-undetectable knot 2 5, 3, 2 with writhe and signature equal 2. Alternating chiral pretzel KLs are completely detectable by Kauffman polynomial.

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Theorem 2.17. Every alternating pretzel (Montesinos) KL with pluses is chiral. First alternating chiral K-undetectable knot is 2 2, 2 1, 2+ (also undetectable by all other polynomials). For pretzel KLs with pluses (n ≤ 16), we have the following results:

n = 10 n = 12 n = 14 n = 16

J1 2 7 21

Kh1 2 6 17

H1 1 3 4

K1 1 4 5

Among chiral P -undetectable alternating pretzel KLs with pluses, knots 2 1 1 1 1 1, 2 2, 2+, 2 2 2 2, 2 1, 2+ with n = 14, and 2 1 1 1 1 1 1 1, 2 2, 2+, 2 2 1 2, 2 1 2 1, 2+ with n = 16 can not be recognized as chiral by any P . Theorem 2.18. Every non-alternating pretzel KL with an odd crossing number is chiral. The next class of KLs considered are non-alternating pretzel KLs. The first chiral J- and H-undetectable 3-component link 4, 2, 2− with n = 8 crossings is followed by chiral P -undetectable KL with an odd number of crossings, which are Kh-detectable. Chiral knot 2 2, 3, 2− (942 ) is Kundetectable (the phenomenon called “ 942 syndrome” by C. Liang and K. Mislow, 1995), but Kh-detectable. For 8 ≤ n ≤ 16 we have the following results: n=8 n=9 n = 10 n = 11 n = 12 n = 13 n = 14 n = 15 n = 16

J1 2 3 3 14 6 40 14 83

Kh-

3 7 22

H1 1 1 1 6 3 11 6 27

K1 1 4 5

From the experimental results, we conjecture that K-polynomial always detects non-alternating chiral pretzel links with an even number of crossings, and Kh-polynomial always detects non-alternating chiral pretzel KLs

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with an odd number of crossings. We also conjecture that there is a one-toone correspondence between alternating chiral K-undetectable KLs with 2n crossings and non-alternating chiral K-undetectable KLs with 2n − 1 crossings (n ≥ 5). A non-alternating pretzel links of the length l with k minuses can be given by its non-minimal representation of the form r1 , r2 , ..., rl +−k, where ri (1 ≤ i ≤ l) are R-tangles. Definition 2.34. Two R-tangles of the form r = −r0 p and r = r0 (p−1) 1, where r0 is a R-tangle and p ≥ 2 are called opposite R-tangles. For example, 2 1 = −3, 2 1 3 = −2 1 2 1 = −2 − 1 − 2 − 1. A minimal representation of a pretzel KL with k minuses can be obtained by substituting k R-tangles by their opposites. For example, 2 2, 4 1, 2 2 1, 3, 2, 2 + −3 = −2 1 1, −5, −2 3, 3, 2, 2 = 2 2, −5, −2 3, −2 1, 2, 2 = ... = 2 2, 4 1, 2 2 1, −2 1, −2, −2. Every standard projection of a pretzel link of length 2l (l ≥ 2) can be represented by a regular (2l)-gon with vertices denoting its R-tangles, called pretzel link (2l)-diagram. Definition 2.35. A (2l)-diagram is antisymmetric if there exist a central antisymmetry (anti-inversion) or mirror antireflection which transforms each vertex ri into ri (1 ≤ i ≤ l), tangle 2 into tangle −2 and vice versa, and does not contain a pair of opposite tangles other then 2, −2. Conjecture 2.9. A non-alternating pretzel link is achiral iff it has an antisymmetric (2l)-diagram. Non-alternating links 3, 2, 2 1, 2, 2, 2 + −3 = 3, 2, −3, 2, −2, −2 with a mirror-antisymmetric diagram, and 3, 2, 2, 2 1, 2, 2+−3 = 3, 2, 2, −3, −2, −2 with a centro-antisymmetric diagram (Fig. 2.110) are both achiral. In the same way, non-alternating pretzel links L1 = 2 1, 3 1, 4 1, 3, 4, 5 + −3 = −3, −4, −5, 3, 4, 5 with a centro-antisymmetric (2l)-diagram, and L2 = 2 1, 3, 3 1, 5, 4 1, 4 + −3 = −3, 3, −4, 5, −5, 4 with a mirror-antisymmetric (2l)-diagram are achiral (Fig. 2.111a). The link L3 = 2 1, 4 1, 3, 4, 5, 3 1+−3 = −3, −5, 3, 4, 5, −4, a mutant of L1 is also chiral, since its antisymmetry axis contains opposite R-tangles 5 and −5 (Fig. 2.111b). If the proposed Conjecture holds, a pretzel link of the length 4 is achiral iff it consists from two pairs of opposite R-tangles. Non-detectable chiral KLs occur in almost all classes of algebraic KLs. For example, for n = 12 we have J- and Kh-undetectable alternating chiral

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Fig. 2.110 Link 3, 2, −3, 2, −2, −2 with a mirror-antisymmetric diagram, 3, 2, 2, −3, −2, −2 with a centro-antisymmetric diagram.

and

Fig. 2.111 (a) Achiral links −3, −4, −5, 3, 4, 5 with a centro-antisymmetric diagram, and −3, 3, −4, 5, −5, 4 with a mirror-antisymmetric diagram; (b) diagram of chiral link −3, −5, 3, 4, 5, −4.

3-component link (2, 4) (3, 3), for n = 16 J-, H-, and Kh-undetectable knots (2 1 2 1 1, 2) (3, 2), (2, 5) (3, 4), and J- and Kh-undetectable 2-component links (2, 3 1) (3, 4 1), (3 1 2, 2) (4, 2). All of them are K-detectable. In the class of K-, H-, and Kh-undetectable alternating knot (2, 2 1+) (3, 3 1+) with n = 14 crossings, for every even n ≥ 16 we have chiral P -undetectable KLs, which are undetectable by any P -polynomial. The situation is similar with non-alternating algebraic KLs. For example, for n = 12 there is non-alternating J-, H- and Kh- undetectable 3-component link (2, 2) (6, 2−), for n = 13 J- and Kh-undetectable 2component link (2, 2 1) (2 3, 3−), for n = 14 the same class contains 12 J-, 6 H-, and 5 Kh-undetectable chiral KLs. Those five KLs can be detected as chiral only by K-polynomial. All P -polynomials fail in the case of chiral knot (2 1, 2+) (3, 2−) with n = 11 crossings, etc.

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In the case of alternating polyhedral KLs we have infinite classes of chiral KLs undetectable for all P -polynomials: • • • • •

6∗ 2.2 1 1, 6∗ 2.2 1 1 1 1, 6∗ 2.2 1 1 1 1 1 1, 6∗ 2 1 1.2 1 1 1 1,... 6∗ 2 1 1.2.2.2, 6∗ 2 1 1 1 1.2.2.2, 6∗ 2 1 1 1 1.2.2.2 1 1,... 6∗ 2.(2, 2 2) 0, 6∗ 2.(2, 2 2 2 2) 0, 6∗ 2.(2, 2 2 2 2 2 2) 0,... 8∗ 2 0 : .2 1 1 0, 8∗ 2 0 : .2 1 1 1 1 0, 8∗ 2 0 : .2 1 1 1 1 1 1 0,... 9∗ 2 1 0.2, 9∗ 2 0.2 1 1 1, 9∗ 2 1 1 1 0.2 1 1, 9∗ 2 1 1 0.2 1 1 1 1 1,...

In general, we conclude that among all P -polynomials, Kauffman polynomial gives the best results in detection of chiral KLs. The list of alternating chiral K-undetectable Montesinios KLs and arborescent KLs of the form (r1 , r2 +) (r3 , r4 +), where r1 ,r2 ,r3 , and r4 are R-tangles is given in the following tables. Table 12 n = 10 n = 12 n = 14 n = 16 n = 18

n = 20

n = 16 n = 18 n = 20 n = 22

n = 24

2 2, 2 1, 2+ 2 1 1 1, 2 2, 2+ 2 1 1 1 1 1, 2 2, 2+ 2 2 2 2, 2 1, 2+ 2 1 1 1 1 1 1 1, 2 2, 2+ 2 2 2 1 1, 2 2, 2 1+ 2 1 1 1 1 1 1 1 1 1, 2 2, 2+ 2 1 1 1 1 2, 2 1 1 1 1, 2 1+ 2 2 2 2 2 2, 2 1, 2+ 2 1 1 1 1 1 1 1 1 1 1 1, 2 2, 2+ 2 1 1 1 1 2, 2 1 1 1 1, 2 1 1 1+ 2 2 2 1 1, 2 1 1 1 1 1, 2 2+

2 1 1 1 2, 2 1, 2 1+

2 1 1 2, 2 1 1, 2 1+

2 1 1 2, 2 1 1 1, 2 1 1+ 2 2 2 2, 2 1 1 1, 2+ 2 1 1 1 1 1, 2 1 1 2, 2 1 1+ 2 2 2 1 1, 2 1 1 1, 2 2+

2 2 1 2, 2 1 2 1, 2+

2 1 1 1 1 1 1 1, 2 1 1 2, 2 1 1+ 2 1 1 2 1 1, 2 2 1 1 1, 2 2+ 2 2 2 2 2 2, 2 1 1 1, 2+

(2 1, 2+) (2 1 1 1 1 1, 2+) (2 1 1 1 1 1 1 1, 2+) (2 1, 2+) (2 1 1 1, 2 1 1+) (2 1 1, 2 1+) (2 1 1 1 1 1 1 1 1 1, 2+) (2 1, 2+) (2 1 1 1 1 1, 2 1 1+) (2 1 1, 2 1+) (2 1 1 1 1 1 1 1 1 1 1 1, 2+) (2 1, 2+) (2 1 1 1 1 1 1 1, 2 1 1+) (2 1 1, 2 1+) (2 1 1 1 1 1, 2 1 1+) (2 1 1 1, 2 1 1+) (2 1 1 1 1 1 1 1 1 1 1 1 1 1, 2+) (2 1, 2+) (2 1 1 1 1 1 1 1 1 1, 2 1 1+) (2 1 1, 2 1+) (2 1 1 1 1 1 1 1, 2 1 1+) (2 1 1 1, 2 1 1+)

2 1 1 1 1 1 2, 2 1 1 1, 2 1+ 2 2 2 2, 2 1 1 1 1 1, 2+ 2 1 1 1 1 1 1 1, 2 2 2 2, 2+ 2 1 1 2 2 1 1 2, 2 1 1, 2 1+

(2 1 1 1 1 1, 2+) (2 1 1 1, 2+) (2 1 1 1 1 1 1 1, 2+) (2 1 1 1, 2+) (2 1 1 1 1 1 1 1 1 1, 2+) (2 1 1 1, 2+) (2 1 1 1 1 1 1 1, 2+) (2 1 1 1 1 1, 2+) (2 1 1 1 1, 2 1 1 1+) (2 1 1 1 1, 2 1+) (2 1 1 1 1 1 1 1 1 1 1 1, 2+) (2 1 1 1, 2+) (2 1 1 1 1 1 1 1 1 1, 2+) (2 1 1 1 1 1, 2+) (2 1 1 1 1 1, 2 1 1 1 1+) (2 1 1 1 1, 2 1+)

The following table contains alternating chiral K-undetectable KLs of the form 6∗ r1 .r2 : n= n= n= n=

12 14 16 18

6∗ 2 1 1 1 1.2 6∗ 2 1 1 1 1 1 1.2 6∗ 2 1 1 1 1 1 1 1 1.2 6∗ 2 1 1 1 1 1 1 1 1 1 1.2

6∗ 2 1 1 1 1.2 1 1 6∗ 2 1 1 1 1 1 1.2 1 1 6∗ 2 1 1 1 1 1 1 1 1.2 1 1

6∗ 2 1 1 1 1 1 1.2 1 1 1 1

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Chiral non-alternating pretzel K-undetectable KLs, which can be detected by Khovanov polynomial are given in the next table. There is one-toone correspondence between alternating chiral K-undetectable KLs with 2n crossings and non-alternating chiral K-undetectable KLs with 2n − 1 crossings (n ≥ 5), based on opposite R-tangles and their reversals (see Table 12 and the next table). n=9 n = 11 n = 13 n = 15 n = 17

n = 19

2 2, 3, 2− 2 1 2, 2 2, 2− 2 1 1 1 2, 2 2, 2− 2 2 2 2, 3, 2− 2 1 1 1 1 1 2, 2 2, 2− 2 2 2 1 1, 2 2, 3− 2 1 1 1 1 1 1 1 2, 2 2, 2− 2 1 1 1 1 2, 2 1 1 1 1, 3− 2 2 2 2 2 2, 3, 2− 2 1 1 1 1 1 1 1 1 1 2, 2 2, 2− 2 1 1 1 1 2, 2 1 1 1 1, 2 1 2− 2 2 2 1 1, 2 1 1 1 2, 2 2−

2 1 1 1 1 1, 3, 3−

2 1 1 2, 2 1 1, 3−

2 1 1 2, 2 1 2, 2 1 1− 2 2 2 2, 2 1 2, 2− 2 1 1 1 2, 2 1 1 2, 2 1 1− 2 2 2 1 1, 2 1 2, 2 2−

2 1 2 2, 3 1 2, 2−

2 1 1 1 1 1 2, 2 1 1 2, 2 1 1− 2 2 1 1 2, 2 1 2 1 1, 2 2− 2 2 2 2 2 2, 2 1 2, 2−

2 1 1 1 1 1 2, 2 1 2, 3− 2 2 2 2, 2 1 1 1 2, 2− 2 1 1 1 1 1 2, 2 2 2 2, 2− 2 1 1 2 2 1 1 2, 2 1 1, 3−

All considered chiral KLs can be detected as chiral (up to a chiral mutation) by n-colored Jones polynomial with n ≥ 3 colors. An example of chiral mutation for alternating knots is a chiral knot 10∗∗∗ 2 :: .2 0 : .2 0.2 (Fig. 2.112a) obtained by chiral mutation from achiral knot 10∗∗∗ .2 0 :: 2 0 : .2 0.2 (Fig. 2.112b). This knot is chiral undetectable for all polynomial invariants, including colored Jones polynomial.

Fig. 2.112

(a) Chiral knot 10∗∗∗ 2 :: .2 0 : .2 0.2; (b) achiral knot 10∗∗∗ .2 0 :: 2 0 : .2 0.2.

Chirality of hyperbolic KLs can be completely detected using the program SnapPea by G. Weeks (http://geometrygames.org/SnapPea/).

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A knot is called strongly achiral if it admits an embedding into S 3 pointwise fixed by the (orientation-reversing) involution (x, y, z) → (−x, −y, −z). Two additional statements about achiral knots, based on Conway polynomial, are the following: • suppose that K is an achiral knot. Then there is a polynomial F ∈ Z4 [z 2 ] such that F 2 = C(z)C(iz)C(z 2 ) ∈ Z4 [z 2 ] (Conant, 2006); • Conway polynomial of a strongly achiral knot must decompose as φ(z)φ(−z) (Hartley and Kawauchi, 1979). The first is conjecture by J. Conant (2006), proved for n ≤ 14 crossings, and the other is theorem proved by Hartley and Kawauchi (1979). If we use them as chirality criteria, the first can not detect chiral rational knot 5 2 (72 ) with C(z) = 1 + 3z 2 for which F 2 = (1 + z 4 )2 . Taken together, they can not detect chiral rational knot 8 2 (101 ), for which F 2 = 1, and C(z) = (1 − 2z)(1 + 2z), or any chiral Montesinos knot from the family (2k) 1, (2k + 1), 2+. After this discouraging result, comes a real surprise: both criteria are able to detect as chiral all aforementioned P -undetectable alternating knots! Unfortunately, this is not sufficient to accept them as reliable chirality criteria, since they fail for significant number of knots belonging to different worlds, even for rational or alternating pretzel knots. 2.11

A dream— new KL tables

We hope that future KL tables will follow the vertical structure (families of KLs), and not the minimal crossing number of KLs (“horizontal” structure). The concept of new KL tables, given in Appendix A (http://www.mi.sanu.ac.yu/vismath/Appendix.pdf), is now still restricted to generators of KL families with n ≤ 9 crossings, but we hope that very soon will be extended to generating KLs with a larger number of crossings. An (almost) complete derivation of alternating knots with n ≤ 11 crossings and non-alternating knots with n ≤ 10 crossings was given in the papers by P.G. Tait, T.P. Kirkman, and C.N. Little, at the end of 19th century. In knot theory books, the first classical knot tables appeared in K. Reidemeister’s book Knotentheorie (1932). All other similar tables are mostly copies of Reidemeister’s tables with some minor changes in knot projections. Standard knot tables contain knots with n ≤ 10 crossings,

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and the only tables containing links with n ≤ 9 crossings are given by D. Rolfsen (1976). Tables can be found in the books by D. Rolfsen (1976), G. Burde and H. Zieschang (1985), L.A. Kauffman (1987a), C. Adams (1994), A. Kawauchi (1996), K. Murasugi (1996), and V. Manturov (2004). All knot tables contain polynomial knot invariants: Alexander polynomials, Jones polynomials (Adams, 1994), Kauffman polynomials (Kauffman, 1987a), and data about some other knot invariants and properties– hyperbolic volumes (Adams, 1994), signatures (Burde and Zieschang, 1985), unknotting numbers (Kawauchi, 1996), chirality and invertibility (Burde and Zieschang, 1985; Kawauchi, 1996), symmetry groups of knots (Kawauchi, 1996), etc. The Table of Knot Invariants by C. Livingston and J.C. Cha (http://www.indiana.edu/ knotinfo/) and Knot Theory by D. Bar Natan (http://katlas.math.toronto.edu/wiki/ The Mathematica Package KnotTheory) give a survey of knot invariants, including some of the most recent ones (e.g., Khovanov polynomial, etc.) Knots are usually denoted by their ordering numbers in the classical notation as 31 , 41 , 51 , 52 , 61 , 62 , 63 , 71 - 77 , 81 - 821 , 91 - 949 , 101 - 10166 , without any common “vertical” ordering principle connecting knots with n and n + 1 crossings. The classical notation gives no information about a KL (except its place in knot tables), but it has been used up to the present time in most knot theory books. All knots up to n = 8 crossings are alternating, and non-alternating knots appear for n ≥ 8: 819 - 821 , 942 - 949 , 10124 - 10166 , etc. The development of computers stimulated new approach to KL tabulation, based on Dowker codes. All knots with n ≤ 16 crossings (Dowker and Thistlethwaite, 1983) were derived by constructing all permutations of n even numbers, checking their realizability as knot projections, and finding minimal Dowker codes. Since Dowker code depends on a projection and the choice of a beginning point, the mapping between knots and their Dowker codes is one-to-many. In order to have a unique correspondence it is necessary to find a minimal Dowker code for each knot. Hence, in Knotscape tabulation every knot is given by its minimal Dowker code and the signs of crossings (that are omitted for alternating knots). Dowker codes (or their equivalents: Gauss codes, P -data, etc.) are indispensable in every computer program working with KLs, but not very useful in attempt of KL classification.

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Using computer enumeration and Dowker algorithm, M.B. Thistlethwaite (by using the program Knotscape, http://www.math.utk.edu/ ∼ morwen/), and H. Doll and J. Hoste (1991), obtained tables of knots with n ≤ 16 crossings and tables of links with n ≤ 9 crossings. A similar program for deriving knot projections with n ≤ 10 crossings was developed by S. Jablan and V. Veliˇckovi´c in 1995. S. Rankin, O. Flint and J. Schermann (2004, 2004a) derived alternating knots up to n ≤ 23 crossings. Still unpublished results of M. Thistlethwaite contain alternating link tables up to n = 19 crossings, and non-alternating up to n = 12 crossings. Comparing the computer enumeration results for alternating KLs obtained in 1990-ties, that contained about 2 million of KLs, with the present ones, with over 6 billion KLs tabulated, we can conclude that greatest gain in tabulation resulted partly from progress in computer performances, but mainly from the more efficient algorithms. Using the basic tabulation scheme of J.A. Calvo (1997), KL realizability test is unnecessary, and significant savings are achieved by inductively generating n-crossing alternating diagrams from the k-crossing diagrams, where k < n. J. Conway (1970) used a similar idea: substituting vertices of basic polyhedra by chains of bigons in two possible positions. Calvo applied this recursive method not only to basic polyhedra, but to all KLs with k crossings. Using Calvo’s algorithm, no testing of Dowker codes for realizability, primality or nugatory crossings is ever required. The main problem, redundancy among the diagrams produced, is solved by flyping algorithms (also described by J.A. Calvo), based on the general flype structure of reduced prime alternating diagrams (Hoste, 2006). S. Rankin, O. Flint and J. Schermann (2004, 2004a) considered four diagrammatic operations, called D, ROTS, T, and OTS (Fig. 2.113) and their effect on prime alternating knot diagrams. D and ROTS are first used to build n + 1-crossing diagrams from n-diagrams by vertex substitutions, and then T and OTS are repeatedly applied to the obtained collection until new knots appear. In order to make process efficient and avoid the redundancy, the authors made a careful analysis of the flype cycle structure of a knot and introduced a refined knot encoding scheme that contains this information. Continuing the “geometrical” line (Kirkman-Conway-Caudron) and the classification of KLs proposed by S. Jablan (1999a,b), we introduced new KL tables, ordered according to KL families. We strongly believe that Conway notation and new KL tables based on the concept of families reflect structural properties of KLs. The main advantage is the possibility to

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Fig. 2.113

Diagrammatic operations D, ROTS, T, and OTS.

extend properties of individual KLs to infinite classes of KLs and obtain general results (e.g., general formulas for polynomial and other subfamilydependent KL invariants). New tables for prime knots with n ≤ 8 crossings were completed in 2002, and now we extend this result to all KLs with n ≤ 9 crossings (Appendix A, http://www.mi.sanu.ac.yu/vismath/Appendix.pdf). In the first version of new knot tables, based on knot families (Fig. 2.114) every family is defined by its general Conway symbol, i.e., Conway symbol with parameters denoting chains of bigons. Conway symbols of knots with n ≤ 10 crossings are given in the “Notation” subsection for each family, followed by the classical notation. The list of their particular Alexander polynomials is followed by a general formula for the Alexander polynomials of the family. The symmetry group, symmetry type, signature, and BJ-unknotting number of every family are determined in general form. All the corresponding data are first computed for individual knots with n ≤ 19 crossings using the program LinKnot. The obtained results are extrapolated to the whole families in order to derive general formulas for the Alexander polynomials, symmetry groups, symmetry types, signatures, and BJ-unknotting numbers. General Alexander polynomials derived in this way coincide with the formulas for general Alexander polynomials of the family p (p = 2k + 1) and subfamilies p 2, p 1 2 (p = 2k + 1), proved by A. Cavicchiolli, B. Ruini and F. Spaggiari (2001). All the general formulas in new knot tables belong to the realm of experimental mathematics: these results are estimated, extrapolated and conjectured, and they need to be proved or disproved. As an example, we are giving Dowker codes and the Jones polynomials for the knot family p, p = 2k + 1. The reader can find analogous complete

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Fig. 2.114

The family p q (2 ≤ p ≤ 5, 2 ≤ q ≤ 5, p ≥ q).

tables for knots with n ≤ 7 crossings at the addresses: http://www.mi.sanu.ac.yu/vismath/ http://members.tripod.com/vismath7/knotab/

Knot family: 2k + 1 Notation: 3 5 7 9

31 51 71 91

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Dowker codes: 3 5 7 9 11 13 15 17 19

4 6 8 10 12 14 16 18 20

6 8 10 12 14 16 18 20 22

2 10 12 14 16 18 20 22 24

2 14 16 18 20 22 24 26

4 2 18 20 22 24 26 28

4 2 22 24 26 28 30

6 4 2 26 28 30 32

6 4 2 30 32 34

8 6 4 2 34 36

8 6 4 2 38

10 8 6 4 2

10 8 6 4

12 10 12 14 8 10 12 14 16 6 8 10 12 14 16 18

Alexander polynomials: 3 5 7 9 11 13 15 17 19

[1 [1 [1 [1 [1 [1 [1 [1 [1

−1 −1 −1 −1 −1 −1 −1 −1 −1

1 1 1 1 1 1 1 1

−1 −1 −1 −1 −1 −1 −1

1 1 1 1 1 1

−1 −1 −1 −1 −1

1 1 −1 1 −1 1 1 −1 1 −1 ∆(p) =

2k X

(−1)i ti

i=0

Jones polynomials: 3 5 7 9 11 13 15 17 19

1 2 3 4 5 6 7 8 9

4 7 10 13 16 19 22 25 28

1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1

−1 −1 −1 −1 −1 −1 −1 −1 −1

1 1 1 1 1 1 1 1

−1 −1 −1 −1 −1 −1 −1 −1

1 1 1 1 1 1 1

Symmetry group: D1 Symmetry type: chiral, invertible.

−1 −1 −1 −1 −1 −1 −1

1 1 1 1 1 1

−1 −1 −1 −1 −1 −1

1 1 1 1 1

−1 −1 −1 −1 −1

1 1 1 1

−1 −1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 −1 1 −1

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Signature: 2k Unknotting number: k The concept of new knot tables, based on knot families, can be naturally extended to links. Working with the program LinKnot we succeeded to extend new knot tables to all KLs with n ≤ 9 crossings. Tables given in the Appendix A (http://www.mi.sanu.ac.yu/vismath/Appendix.pdf) contain generating KLs for every family of KLs, the Conway symbol of the family with conditions for parameters, the number of components, the Alexander polynomial given by general formula, general formulas for the number of different projections (only for alternating KLs), general formulas for BJunknotting (BJ-unlinking) numbers and signatures, data about period(s) of KLs, achirality, and BJ-unlinking gap. The concept of new KL tables is based on the notion of generating KLs and families originating from them. Hence, one of the possible future goals is a search for new KL invariants that will be able to recognize families. New “family” invariants have to be preserved by Reidemeister moves and nmoves, since all members of some family can be obtained from a generating KL by a sequence of n- moves. Unfortunately, the transition from one family member to another is only possible at the level of minimal canonical Conway symbols. In this way we have obtained KL tables given in the Appendix A, that consist only of generating KLs, families derived from them, and parametric data about families (i.e., the KL properties and invariants in a general form). Computational results imply that Alexander polynomials of two alternating KLs from the same family must be different– therefore all we need for distinguishing alternating KLs would be their family and Alexander polynomial. The solution is even simpler if we restrict our attention to alternating KLs given in Conway notation– we can use a minimal Dowker code (obtained directly from the Conway symbols by the LinKnot function MinDowAltKL). This function gives minimal Dowker code for all alternating KLs, except for those derived from basic polyhedra permitting flypes. If a projection of a basic polyhedron permits flypes, we are not able to compute the minimal Dowker codes for all alternating KLs derived from this basic polyhedron. Definition 2.36. A family of KL diagrams is obtained in the following way: consider three KL diagrams Dn , Dn+1 , and Dn+2 of the same link L, where every diagram is obtained from the preceding one by adding a bigon to the same single bigon or chain of bigons. Outside the region containing

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the chain of bigons, diagrams are identical. Diagrams D+ , D− , and D0 (Fig. 2.7) are related by Conway skein relation. Introducing orientation of a KL, every chain of bigons becomes parallel or antiparallel (Fig. 2.115).

Fig. 2.115 case.

Chain of bigons (a) after crossing change (b); (c) antiparallel case; (d) parallel

In a chain of bigons, transformation D+ → D− is equivalent to the transformation Dn+2 → Dn . Depending on two possible cases, parallel or antiparallel, D0 = Dn+1 or D0 = L0 = const, where L0 is a constant KL called KL of difference. Theorem 2.19. Conway polynomial is defined by recursive relations between KLs that belong to the same family. Proof.

Conway polynomial is defined by skein relation ∇(D+ ) − ∇(D− ) = x∇(D0 )

In the antiparallel case, it gives the recursive relation ∇(Dn+2 ) − ∇(Dn ) = x∇(L0 )

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and in the parallel case the recursive relation ∇(Dn+2 ) − ∇(Dn ) = x∇(Dn+1 ). Hence, in both cases Conway polynomials of successive KLs belonging to the same family satisfy recursive relations. The same line of reasoning can be extended to all polynomial invariants based on skein relations, so we propose the following conjecture: Conjecture 2.10. All polynomial invariants based on skein relations are defined by recursive relations between KLs that belong to the same family.

Fig. 2.116

Subfamily 2k with antiparallel orientation.

Fig. 2.117

Subfamily 2k + 1 with parallel orientation.

Let us consider the simplest family of KLs, denoted by Conway symbol n (n ∈ N ). For n = 2k we obtain the subfamily of links 2k (2, 4, 6,...) that can be used as the simplest example of the antiparallel case (Fig. 2.116).

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Fig. 2.118

Subfamily (2k) 2 with antiparallel orientation.

For this subfamily, the following relation holds: Dn+2 = 2k + 2, Dn = 2k, and the link of difference is the unlink 1. For every n (n = 2k), ∇(D+ ) − ∇(D− ) = ∇(Dn+2 ) − ∇(Dn ) = x∇(L0 ), i.e., ∇(2k + 2) − ∇(2k) = x∇(1) = x. Hence, general formula for Conway polynomial of the subfamily 2k is ∇(2k) = kx (Kauffman, 1987a, page 23). Conway polynomials of the knots with parallel orientation, belonging to the subfamily 2k + 1 (3, 5, 7,...) (Fig. 2.117) satisfy recursive relation: ∇(0) = 1, ∇(2k + 1) = x∇(2k) + ∇(2k − 1). General formula for the Conway polynomial of the subfamily 2k + 1 is k X k + i 2i ∇(2k + 1) = 1 + x . 2i i=1 This method can be applied to an arbitrary subfamily of KLs, for example, the subfamily (2k) 2 (Fig. 2.118). This subfamily has antiparallel orientation with Dn+2 = (2k + 2) 2, Dn = (2k) 2, and the link of difference is Hopf link 2. For every k ∇(D+ ) − ∇(D− ) = ∇(Dn+2 ) − ∇(Dn ) = x∇(L0 ), i.e., ∇((2k + 2) 2) − ∇((2k) 2) = x∇(2). Hence, general formula for the Conway polynomial of the subfamily (2k) 2 is ∇((2k) 2) = 1 − (2k − 1)x2 .

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In the antiparallel case computations of Conway polynomial are fairly simple, unlike in parallel case where complicated recursions can occur, especially for multi-parameter subfamilies. The same line of reasoning can probably be extended from Conway polynomial (Theorem 2.19) to all polynomial invariants based on skein relations: Alexander polynomial, Jones polynomial, A2 invariant, Links-Gould invariant, and Khovanov polynomials. For example, for the subfamily (2k) 2 Alexander polynomials of the successive members (2k + 2) 2 and (2k) 2 will differ by the constant polynomial 1 − 2x + x2 , and for Jones polynomial by x2k (1 − x − x3 + x4 ). Since the Alexander polynomial of the knot 2 2 is 1−3x+x2, and its Jones polynomial is 1 − x + x2 − x3 + x4 , the general formula for the Alexander polynomial of this family is 1 − 3x + x2 + (k − 1)(1 − 2x + x2 ), and the general formula for Pk the Jones polynomial is 1 − x + x2 − x3 + x4 + i=2 (1 − x − x3 + x4 )x2i−2 . This is the way how estimated general formulas for Alexander polynomials of KL families given in Appendix A (http://www.mi.sanu.ac.yu/ vismath/Appendix.pdf) are computed. Based on the obtained results, we strongly believe that all properties of KLs belonging to each particular family are well-ordered, and that it is possible to extend the particular results to a general form. So far this principle can be applied to all polynomials based on skein relation (Conway, Alexander, Jones,...), symmetry properties, signatures, BJ-unknotting numbers, braid family representatives, Dowker codes, etc. One of the main and the most intriguing open questions is BernhardJablan Conjecture (see page 83). For all KLs that we tested, BJ-unlinking numbers uBJ (L) obtained according to it coincide with the unlinking numbers determined using other methods. The most interesting conjectures indicated by the results obtained are that crossing number, minimum writhe, unlinking number, u∞ -unlinking number, signature, and genus of a link L are linear subfamily-dependent KL invariants (Conjecture 1.2), and that for every minimal link diagram of a non-alternating KL, all link diagrams belonging to its subfamily are minimal (Conjecture 1.9).

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Chapter 3

History of Knot Theory and Applications of Knots and Links

3.1

History of knot theory

The discovery of knots probably predates that of fire or wheel. Ropes, cords, and knots needed to secure them played an important role in the early technological development. The main reason for the lack of discovery of such artifacts is that they have been made from organic materials (vegetable fibres, sinews, thongs, hair, etc.), thus subject to decay. However, even certain wild gorillas are able to make complete knots, primarily Granny and Reef knots, so that the beginning of knot tying most likely preceded the evolution of mankind. The indirect testimony for an early use of cordage and knots are perforated objects, beads or pendants, dating some 300 000 years ago, and spherical stones found in Africa and China (about 500 000 years old), probably used as bola weights in hunting. More recently bows and arrows that required well-made cordage and secure knots, as well as Paleolithic figurine in soft limestone from Kostenki (Russia, 24 000 B.C.) show belts made from multiple twined flexible elements. Some actual Neolithic knots are preserved in North Zealand and Denmark. Sophisticated plaits made with strips of date palm leaf originate from Ancient Egypt (Turner and Van De Griend, 1995). Arrangements of knots served as a basis for mathematical recording systems in the Peruvian quipus or Zu˜ ni knots from the New Mexico, where the knots functioned as symbolic and mnemonic devices. Various examples of knot-art can be found in all ancient civilizations, in Japanese and Chinese art, Celtic art, ethnic Tamil and Tchokwe art, in Arabian, Greek or Smyrnian laces... Celts made extensive use of knot-work pictures created for decorative and religious purposes (G. Bain, 1973; I. Bain, 1990). Their art required a high level of mathematics, to geometrically create knotted curves even with zoomorphic ornaments.

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In his essay on orthopedic knots, a Greek physician named Heraklas (first century A.D.) described and explained, giving step-by-step instructions, eighteen ways to tie orthopedic slings. This essay, that survived because it was included (without drawings) in Medical Collections by Oribasius of Pergamum, was recovered, reillustrated, and translated to Latin during the Renaissance. It is the oldest testimony of a scientific application of knots (Przytycki, 2004). The idea to consider knots from the point of view of combinatorial topology (this means, Analysis Situs or Geometria Situs, the term introduced by G.W. Leibnitz in 16791) was first proposed by A.T. Vandermonde (1771). Describing braids, nets, or knots fashioned by craftsmans, he emphasized that there the questions of measurement are not important, but those of position, the manner in which the threads are interlaced. C.F. Gauss was the first to consider knots as mathematical entities. One of his oldest documents is a sheet of paper dated 1794, containing thirteen sketches of knots with names in English, probably an excerpt he copied from some English book. Gauss formulated the “crossing problem”, by assigning letters to crossings of a self-intersecting curve, trying to determine “words” describing a closed curve (page 16), and defined a linking number (1833) by giving its analytical definition– the Gauss integral2 . Another sketch from his notebooks is a drawing of a braid with strand permutation coding (Przytycki, 2004). Gauss’ work was continued by J.B. Listing, credited with the first usage of the word “topologie” in 1836. He represented knots, closed space curves, by their projections (diagrams) and made an attempt to derive and classify all projections up to 7 crossings (1847). Listing defined minimal or reduced diagrams of knots, diagrams with a minimal number of crossings (Definition 1.25), and proposed the first invariant named Complexions-Symbol for knots with minimal diagrams. Although his Complexions-Symbol had too many serious defects to be acceptable as a knot invariant, it posed a challenge to other researchers to try to find better invariants. Listing showed that the figure-eight knot, called Listing knot in honor of his accomplishment in knot theory, is equal to its mirror image (i.e., that figure-eight knot is achiral), recognized that the left trefoil is different from the right trefoil knot (i.e., that trefoil knot is chiral), and introduced a writhe, sum of all crossing signs in knot diagram (Definition 1.31). 1 The first application of geometria situs dates from 1736: it is L. Euler’s solution of the famous K¨ oningsberg bridges problem, that represents the beginning of graph theory. 2 H.K. Brunn observed in 1892 that the linking number of two-component link, considered by Gauss, can be read from a diagram of the link (Definition 1.32).

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The main inspiration for enumerations of knots come from Sir William Thomson (Baron Kelvin of Largs). In attempt to make a classification of chemical elements, in the mid-1860s he announced his model of the atom, called “vortex theory”. After Helmholtz’s paper (1858) on vortex motion, based on an experiment with vortex smoke rings, Kelvin developed the theory of particles as tiny topological twists, i.e., knots in the ether (1867). He believed that the variety of chemical elements can be explained by the kinds of different knots. He also thought that the ability of atoms to transform into each other, transmutation, was related to the cutting and recombining of knots. This theory was taken seriously until the wide acceptance of Mendeleev’s periodic system of elements by the scientific community. Vortex theory inspired P.G. Tait to start with the enumeration and classification of knotted structures and solve the census problem. For this, he developed Scheme-method, a representation of reduced knot diagrams by codes (already known to Gauss) and the Partition method (Turner and Van De Griend, 1995), an improvement of Listing’s attempt. In collaboration with the Reverend T.P. Kirkman and C.N. Little, they succeeded in making a list of all alternating knots up to 11 crossings. The derivation of knots with 10 crossings took them six years to complete. Tait also considered some of the fundamental problems in knot theory: chirality (Definition 1.26) and unknotting number (or Gordian number, called by Tait “beknottedness”) (Definition 1.56), and introduced the graph of a knot (page 24). He made a few conjectures on alternating knots, e.g., that the minimal number of crossings of an alternating KL is always realized in an alternating diagram (Kauffman-Murasugi Theorem, Theorem 1.6), and that two minimal diagrams of the same oriented alternating KL have the same writhe (Theorem 1.7). His famous Flyping Conjecture was recently proved by Menasco and Thistlethwaite (1991, 1993), about 100 years after it was formulated (Theorem 1.11). Kirkman’s geometrical system for the systematic derivation of knot projections (4-valent planar graphs) was closely related to the enumeration of basic polyhedra and, at the same time, represented a geometrical method for classifying knot projections. Kirkman derived the census of 1581 plane curves with 11 crossings from which Little distinguished 357 different alternating knots. Little also considered the derivation of nonalternating knots, and in addition to the flype (page 44), introduced a 2-pass: a KL transformation where a string is simply pulled over a tangle (page 47). After six years of work, Little produced a catalogue consisting of 43 non-alternating knots with n = 10 crossings and 551 drawings of their various minimal projections (with few omissions). The only serious error in

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his tables was the duplication discovered by K. Perko in 1974 (Fig. 1.26). Little observed that the writhe of a reduced knot diagram is invariant with respect to flypes and 2-passes, and proposed that it is a knot invariant, but it is not: the first known counterexample is Perko pair. Little erroneously believed that just two kinds of moves, flypes and 2-passes, are sufficient to pass between all minimal diagrams of the same knot3 . After Tait, Kirkman and Little, pertaining to knot tabulation, almost nothing important happened for a century, until the works of J.H. Conway and A. Caudron in 1970-80s, and the computer derivation of KLs. The principal problem in knot enumeration is deciding when two knots are ambient isotopic (Definition 1.20). Two KLs are isotopy equivalent if one of them can be transformed to the other by pushing and pulling, but not cutting, its string(s). The problem of isotopy, known as the Knot problem, became the main problem in knot theory. Closely connected to the knot problem is the problem of achirality, ambient isotopy of a KL to its mirror image (Definition 1.26). Thirty years after Tait’s first results in enumeration of achiral knots with n ≤ 10 crossings, M.G. Haseman in her dissertation partially extended knot tables, and described achiral knots with n ≤ 12 crossings. Tait conjectured that every achiral KL must have an even number of crossings. Therefore neither Tait nor Haseman considered the possibility of the existence of achiral knots with an odd crossing number. The first oriented achiral link 8∗ .−2 0.2 0.−2 0 with n = 11 crossings was discovered in 1998 (Liang, Mislow and Flapan, 1998). The achiral non-alternating knot 10∗∗ 2 0.2 : −2 0 : 2 0.−1.−1.−1.−2 0 with n = 15 crossings was found by M. Thistlethwaite, who also recognized several duplicates in Haseman’s tables. However, the Tait’s Conjecture about achiral KLs holds for alternating KLs: there is no alternating achiral KL with an odd number of crossings (Corollary 1.1). After the empirical phase, the emphasis in the theory of knots turned away from enumeration toward attempts to prove the completeness of knot lists and to show that they do not contain repetitions. The first steps in the development of the required mathematical apparatus were made by H. Poincar´e, who introduced several topological objects and tools, e.g., the concept of the complex and its fundamental group. The first proof of the existence of non-trivial knots is given by H. Tietze in 1908, using the fundamental group of knot complement (or knot group). W. Wirtinger in 3 In

the derivation of non-alternating knots and in the knot minimization program knotfind.c (the part of Knotscape) M. Thistlethwaite used 13 different diagrammatic moves (Hoste, Thistlethwaite and Weeks, 1998).

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1905 outlined a method of finding a presentation of knot group, called now the Wirtinger presentation (page 201). M. Dehn proved in 1914 that the oriented trefoil is not isotopic to its mirror image, i.e., that trefoil knot is chiral. The study of the fundamental group of a knot complement and the knot complement itself was one of the main topics of research in knot theory for the next 50 years, culminating in 1988 in the proof of Tietze conjecture (1908) stating that a knot is determined by its complement (Theorem 1.5, Gordon and Luecke, 1989). In the 1930s, after the discovery of the first polynomial knot invariant by J.W. Alexander, knot theory became a branch of topology, leaving its roots, geometry. Using Betti numbers and torsion coefficients, J.W. Alexander and G.B. Briggs (1926-27) distinguished all knots with n ≤ 9 crossings, except three pairs which were resolved five years later by K. Reidemeister. L-polynomial (normalized Alexander polynomial) was used as a tool for distinguishing knots. In Reidmeister’s book Knotentheorie (1932), each knot is represented by one minimal projection. All books about knot theory follow Reidmeister’s book: knot projections were merely redrawn and never changed into a different minimal projection of the same knot. Reidemeister introduced Reidemeister moves (Theorem 1.10), three fundamental local transformations of a KL diagram sufficient to represent any ambient isotopy at the level of projections. Since links have a higher degree of complexity, they were rarely considered. The complete list of non-oriented links for n ≤ 9 crossings, based on the paper by J. Conway (1970), can be found only in the book Knots and Links by D. Rolfsen (1976). Soon after Reidemeister’s book appeared, H. Seifert introduced an algorithm to construct a special surface whose boundary is a KL (Theorem 1.37). This surface, named after Seifert, can be used to study KLs and their invariants. After Alexander, progress in the study of knots can be referred to as “era of knot invariants”. Fundamental groups could not be used to distinguish the Reef Knot (3#3) from the Granny Knot (3# − 3). H. Seifert proved in 1934 that they are different, since their complements are non-isomorphic. Also, different diagrams of equivalent knots can yield different presentations of the knot group. Since there is no general algorithm enabling us to decide whether two presentations represent isomorphic groups, mathematicians continued to search for a simpler invariant. The long history of polynomial knot invariants began with the discov-

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ery of the Alexander polynomial in 1928. This polynomial was able to distinguish 76 out of 84 knots with n ≤ 9 crossings. There are several interesting properties of Alexander polynomial: every alternating knot has an alternating Alexander polynomial (proved by K. Murasugi in 1958); the Alexander polynomial of a composite knot is the product of the Alexander polynomials of its prime factors. Alexander also discovered a relationship between the polynomials of three oriented knots whose projections are identical except in a neighborhood of one fixed crossing (later introduced by J. Conway as the skein relation, which plays fundamental role in developing recursive definitions of knot invariants). In spite of the early discovery of this recursive relation, it was not used prior to Conway’s approach. As a result, Alexander polynomials were calculated by means of determinants till 1970. Alexander polynomial can not distinguish a knot from its mirror image, and its power to distinguish different knots considerably decreases as the number of crossings increases. In 1923 Alexander proved the theorem that every KL can be obtained as closure of a braid, and inspired E. Artin to introduce the braid group (1925). A.A. Markov gave equivalence moves for closed braids. Markov theorem (proved by J. Birman) was used to express Knot problem in purely algebraic terms, through the classification of Markov classes, i.e., as the Algebraic link problem. Establishing connection between knot group presentations (Wirtinger presentations) and Alexander polynomials was facilitated by the discovery of free differential calculus by R.H. Fox. After introducing bridge number (Definition 1.64), H. Schubert proved in 1949 that any KL can be decomposed uniquely as a connected sum (or direct product) of prime KLs (Theorem 1.33). J.H. Conway in 1967 introduced a fundamentally new idea. After introducing the basic concept of a tangle (proposed earlier by Tait), a portion of a knot diagram with four free-end strands, he gave a concise generic geometrical notation for describing KLs in terms of their construction from tangles. He discovered a remarkable connection between rational KLs and continued fractions, and defined skein relations which made possible a recursive computation of polynomial KL invariants. Using skein relations (page 211), polynomial KL invariants can be computed recursively, by a kind of “unknotting process” which reduces the number of crossings in each step. J. Conway checked and extended the knot tables of Tait, Kirkman and Little. The first attempt to classify KLs in certain larger classes (called worlds)

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came from A. Caudron. In his fundamental work, a 361-page preprint Classification des nœuds et des enlancements published in 1982, he corrected and extended Conway’s results, and made the first “periodic tables” of KLs. His results marked the end of the hand calculation era. In 1982 H. Dowker and M. Thistlethwaite computerized the derivation of all knot diagrams using Dowker algorithm. Until now, knot tables were extended by M. Thistlethwaite, J. Hoste, and G. Weeks up to n = 19 crossings. Every knot in their tables, included in the program Knotscape for n ≤ 16 crossings, is denoted by its minimal Dowker code. S. Rankin, O. Flint and J. Schermann (2004, 2004a) derived alternating knots up to n = 23 crossings. Their tables contain, for example, 25182878921 alternating knots with n = 23 crossings (Rankin, 2006). H. Doll and J. Hoste (1991) tabulated oriented links up to n = 9 crossings (Cerf, 1998). In his unpublished results, M. Thistlethwaite made computer tabulation of alternating links with n ≤ 19 crossings and nonalternating links with n ≤ 12 crossings. The most complete information about the present state of computer KL tabulation is given in the paper The enumeration and classification of knots and links by J. Hoste (2006). Today, with the development of computers, the notation and enumeration of KLs is very similar with the situation occurring in different structures with hardly recognizable ordering principles: prime numbers, polyominoes etc., resisting attempts of classification. Following the line of Kirkman, Conway and Caudron, we have attempted to present a consistent geometrical, combinatorial and graph-theoretical approach to the derivation and classification of KLs. One of our main ideas was to avoid standard classification of KLs according to number of components and number of crossings. We hope that this goal is, at least partially, achieved by implementing the Conway notation into our computer program LinKnot. Links have always played a subordinated secondary role to knots. The name of the program itself (proposed by R. Sazdanovi´c) underlines the important role given to the links in this program. It is primarily dedicated to an experimental work with a large series of KLs (i.e., families) and derivation of new conjectures. Therefore, the program LinKnot is a tool for experimental mathematics. In the 1980s V. Jones discovered new polynomial and established connections between von Neumann algebras, statistical mechanics, braid theory and knot invariants (Jones, 2005). Jones began with two statistical models, Ising model and Potts model, studying their corresponding parti-

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tion functions. Hence, Jones did not only introduced a refined invariant for distinguishing and analyzing KLs, but related knot theory to other fields of mathematics and theoretical physics, in particular statistical mechanics and quantum field theory. This revolutionary discovery was followed by new more powerful polynomials: HOMFLYPT and Kauffman polynomial. However, none of these polynomials is a complete invariant, able to distinguish all KLs, or to completely recognize chirality. Khovanov homology (Khovanov, 1997, 2001), which categorifies one-variable Jones polynomial is a strictly stronger invariant then the Jones polynomial itself (Bar Natan, 2002; Manturov, 2004). The introduction of Vassiliev’s invariants gave rise to the hope that a complete KL invariant can be found. Working in the space of knots, Vassiliev invariants are essentially different from all other previously mentioned KL invariants: instead of associating to each KL an mathematical quantity or polynomial, they assign to a KL a numerical value depending on a set of initial conditions. Many of the invariants introduced before, such as Alexander, Jones, and Kauffman polynomials, are Vassiliev invariants. On the other hand, none of the classical KL invariants: the minimal crossing number, unknotting (unlinking) number, signature, bridge number, braid index and genus of a KL, are Vassiliev invariants. The initial interest in knot theory was stimulated by Kelvin’s theory of atomic structure (1867). By the turn of the century, after scientific confirmation of Mendeleev’s periodic tables, it was clear that Kelvin’s theory was incorrect. Chemists were no longer interested in classifying knots. However, topologists continued to study knots. The focus of chemists turned towards attempts to synthesize molecular KLs. The first pair of linked rings in a form of the Hopf link, a catenane, was synthesized by H. Frisch and E. Wasserman in 1961. The first molecular knot, a trefoil made out of 124 atoms was produced by C. DietrichBushecker and J.-P. Sauvage in 1989. They refer to stereochemical topology, synthesis, characterization, and analysis of topologically interesting molecular structures (Flapan, 2000). Construction of numerous KLs become possible after the synthesis of first molecular M¨obius ladder with three rungs by D. Walba, R. Richards and R.C. Haltiwanger in 1982, and addition of twists to the M¨ obius ladders managed by Q.Y. Zheng in 1990 (Fig. 3.1). In fact, after breaking the rungs, M¨obius multi-strand twisted ladders became a molecular closed braid representation of a KL. In the 1950s F.H.C. Crick and J.D. Watson unravelled the double helix

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Obtaining Hopf link from twisted M¨ obius ladders.

structure of DNA. A molecule of DNA can also take the form of a ring and become knotted. In the process of recombination, a DNA knot is temporarily broken, physically changed, and then reconnected. In the 1970s it was discovered that an enzyme, topoisomerase, is responsible for this process. The first electron microscope pictures of knotted DNA were produced in 1985 (Wasserman, Dungan and Cozzarelli, 1985). The linking number and its splitting into average writhe W r and twist T w is used as a basic tool to analyze the geometry of supercoiled DNA. C. Ernst and D.W. Sumners (1990, 1999) reconstructed the actions of enzyme (TN3 Resolvase) by solving tangle equations. Distances of rational knots and links were calculated by I.K. Darcy and D.W. Sumners (2000). Mathematical models or descriptions of large particle systems, such as the Ising and Potts model, or Yang-Baxter equation, give rise to knot invariants generated by partition function. Different applications of knot theory in physics, chemistry, and biology are considered in books by L. Kauffman (1991), C. Adams (1994), K. Murasugi (1996), E. Flapan (2000), and in the collection of papers edited by D.W. Sumners (1993). In this book we will try to emphasize the beauty, universality and diversity of knot theory through its various, non-standard applications to ornamental art, fullerenes, self-referential systems, and KL automata.

3.2

Mirror curves

Before proceeding to the detailed discussion of applications of knot theory to ornamental art, we shall first describe the construction of mirror curves. Start with any connected edge-to-edge tiling of a part of a plane by polygons.

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Connect the midpoints of adjacent edges to obtain a 4-regular graph: every vertex is incident to four edges, called steps. Every closed path in this graph, where each step appears only once, is called a component. A mirror curve is the set of all components. Since the graph is 4-valent, at each vertex we have three choices of edges to continue the path: to choose the left, middle, or right edge. If the middle edge is chosen the vertex is called a crossing. Every mirror curve can be converted into a knotwork design by introducing the relation “over-under”. The name “mirror curves” can be justified by visualizing them on a rectangular square grid RG[a, b] of dimensions a, b (a, b ∈ N ), whose sides are mirrors, and additional internal two-sided mirrors are placed between the square cells, coinciding with an edge, or perpendicular to it at its midpoint. In this grid, a ray of light, emitted from one edge-midpoint at an angle of 45◦ , will close a component after a series of reflections. Beginning from a different edge-midpoint, and continuing until the whole step graph is used, we trace a mirror curve. This construction can be extended to any connected part of a regular triangular, square or hexagonal tessellation, this means to any polyiamond, polyomino or polyhexe, respectively. 3.2.1

Tamil treshold designs

“During the harvest month of Margali (mid-December to mid-January), the Tamil women in South India used to draw designs in front of the thresholds of their houses every morning. Margali is the month in which all kinds of epidemics were supposed to occur. Their designs serve the purpose of appeasing the god Siva who presides over Margali. In order to prepare their drawings, the women sweep a small patch of about a yard square and sprinkle it with water or smear it with cow-dung. On the clean, damp surface they set out a rectangular reference frame of equidistant dots. Then the curve(s) forming the design is (are) made by holding rice-flour between the fingers and, by a slight movement of them, letting it fall out in a closed, smooth line, as the hand is moved in the desired directions. The curves are drawn in such a way that they surround the dots without touching them.” (Gerdes, 1989). The (culturally) ideal design is composed of a single continuous line. Names given to designs formed of a single “never-ending” line are normally pavitram, meaning “ring” and Brahma-mudi or “Brahma’s knot” (Fig. 3.2). The purpose of the pavitram is to scare giants, evil spirits, or devils away. Is it not strange that a design composed of two or several superimposed

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Tamil designs.

closed paths, is nevertheless called pavitram? Maybe designs formed of a few never-ending lines are just degraded versions of single closed path figures? Is it always possible to construct a similar design, but made out of only one line? Yes, minor changes transform some imperfect, multi-linear designs into the ideal ones. 3.2.2

Tchokwe sand drawings

“The Tchokwe people of northeast Angola are well known for their beautiful decorative art. When they meet, they illustrate their conversations by drawings on the ground. Most of these drawings belong to a long tradition. They refer to proverbs, fables, games, riddles, etc. and play an important role in the transmission of knowledge from one generation to the other.” (Gerdes, 1990) “...Just like the Tamils of South India, the Tchokwe people invented a similar mnemonic device to facilitate the memorization of their standardized drawings. After cleaning and smoothing the ground, they first set out with their fingertips an orthogonal net of equidistant points. The number of rows and columns depends on the motif to be represented. Applying their method, the Tchokwe drawing experts reduce the memorization of a whole design to that of mostly two numbers and a geometric algorithm.

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Most of their drawings display bilateral and/or rotational (90◦ or 180◦ ) symmetries. The symmetry of their pictograms facilitates the execution of a drawing. This is important, as the drawings have to be executed smoothly and continuously. Any hesitation or stopping on the part of the drawer is interpreted by the audience as an imperfection and lack of knowledge, and assented with an ironic smile.” (Gerdes, 1990) Tchokwe sand drawings called sona (singular: lusona) played an important role in transmitting knowledge and wisdom from one generation to the next. Young boys enjoyed making sand drawings with their fingers and in stories about them. They have learned how to make simple drawings and their meaning during the period of intensive schooling, the mukanda initiation rites. The more difficult sona were only known by the story tellers, who were real akwa kuta sona (those who know how to draw), highly estimated and forming a part of an elite in Tchokwe society (Gerdes, 1993).

Fig. 3.3

Tchokwe sand drawings.

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“Leonardo spent much time in making a regular design of a series of knots so that the cord may be traced from one end to the other, the whole filling a round space...” (Bain, 1973).

Fig. 3.4

Leonardo’s Concatenation.

Two of the greatest painters-mathematicians: Leonardo and D¨ urer were interested in constructing knot designs, closely related to mirror curves (Bain, 1973). They knew and very effectively used the fact that for a rectangular square grid RG[a, b] of dimensions a, b, where a and b are relatively prime, mirror curve is always a single closed curve uniformly covering the rectangle. Moreover, there is one more beautiful geometrical property: mirror curves can be obtained using only a few different prototiles. In particular, only three prototiles are sufficient for construction of all mirror curves with internal mirrors incident to the cell-edges of a regular triangular tiling, five for square (Fig. 3.5), and 11 for hexagonal regular tiling (Jablan, 1995). Using the combinations of polygons from 11 uniform Archimedean tilings (Gr¨ unbaum and Shephard, 1986), or prototiles producing an impression of space structures and colored prototiles, we may obtain artistic interlacing patterns, examples of modular design: the use of a few initial

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Fig. 3.5

Knot design obtained from a uniform tiling.

elements (modules - prototiles) for creating an infinite collection of designs. The mirror curves obtained from Archimedean tilings resemble the optical phenomenon: change in direction of a light ray which transfers from one to the other physical environment. 3.2.3

Construction of mirror curves

“The imitation of the three-dimensional arts of plaiting, weaving and basketry was the origin of interlaced and knotwork interlaced designs. There are few races that have not used it as a decoration of stone, wood and metal. Interlacing rosettes, friezes and ornaments are to be found in the art of most people surrounding the Mediterranean, the Black and Caspian Seas, Egyptians, Greeks, Romans, Byzantines, Moors, Persians, Turks, Arabs, Syrians, Hebrews and African tribes. Their highlights are Celtic interlacing knotworks, Islamic layered patterns and Moorish floor and wall decorations.” (Bain, 1973) The common geometrical construction principle of these designs, discovered by P. Gerdes, is the use of (two-sided) mirrors incident to the edges of a square, triangular or hexagonal regular plane tiling, or perpendicular to the edges in their midpoints (Gerdes, 1990, 1996, 1997, 1999). In the ideal case, after the series of consecutive reflections, the ray of light reaches its beginning point, defining a single closed curve. In other cases, the result consists of several closed curves. For example, the following mirror-schemes (Fig. 3.6) correspond to the Celtic designs from G. Bain’s book Celtic Art (1973). Can we find a mathematical principle behind constructing a perfect

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Celtic knot designs.

curve– single line placed uniformly in a regular tiling. How can we arrange generating mirror sets and classify curves obtained? In principle, any polyomino (polyiamond or polyhexe) (Golomb, 1994) with mirrors on its border, and two-sided mirrors between cells or perpendicular to the internal cell-edges in their midpoints, can be used for creating perfect curves. We propose the following construction from a polyomino (polyiamond or polyhexe): Construction 3.1. First, construct all different curves in a polyomino containing lines that connect different cell-edge midpoints until the polyomino is uniformly covered by k curves. Then, in order to obtain a single curve, place internal mirrors and use “curve surgery”, according to the following rules: (1) any mirror placed in a crossing point of two distinct curves connects them in one curve; (2) depending on the position of a mirror, a mirror placed into a self-

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crossing point of an (oriented) curve either does not change the number of curves, or breaks the curve in two closed curves (Fig. 3.7). In every polyomino we may place k −1, k, k +1,. . ., 2A− P2 internal twosided mirrors, where A is the area and P is the perimeter of the polyomino. Placing the minimal number of mirrors k − 1, we need to obtain a single curve, and to preserve this property when we add other mirrors.

Fig. 3.7

A mirror placed in a crossing point of (a) two different curves; (b) one curve.

In the case of a rectangular square grid RG[a, b] of dimensions a, b, the initial number of curves, obtained without internal mirrors is k = GCD(a, b) (GCD– greatest common divisor), so in order to obtain a single curve, the possible number of internal two-sided mirrors is k − 1, k,. . ., 2ab − a − b. According to the rules for placing internal mirrors, we propose the following algorithm for creating mono-linear designs: in every step one of k − 1 internal mirrors is placed in a crossing point belonging to different curves. After this, when the curves are combined and transformed into a single line, we can add other mirrors according to the rules described in Construction 3.1, taking care about the number of curves (Fig. 3.8).

Fig. 3.8 The successive introduction of internal mirrors in the RG[2, 2] that preserves a single curve.

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P. Cromwell (1993) used symmetry of mirror curves for the classification of the Celtic frieze designs, and P. Gerdes (1989, 1995) for the reconstruction of Tamil designs. Looking to the ornamental art, at the first glance it seems that symmetry is the mathematical basis for the construction and possible classification of perfect curves (Gerdes, 1989, 1990; Cromwell, 1993). The existence of asymmetric curves suggests another approaches. The first approach is the geometrical, symmetry-oriented one: Definition 3.1. Two mirror curves are equal iff there is a similarity transforming one into the other. In other words, one mirror curve can be obtained from the other by a combined action of proportionality and isometry. Instead of considering equality of curves, we may consider the equality of mirror arrangements defined in the same way. In Subsection 3.2.4 we will try to find the number of different perfect curves (i.e., their corresponding mirror arrangements) which can be derived from RG[a, b] for a given number of mirrors m (m = k − 1, k, . . . , 2ab − a − b). Definition 3.2. A transformation S of the Euclidean n-dimensional space E n is called isometry if for every two points X, Y and their images X ′ = S(X), Y ′ = S(Y ) holds XY ∼ = X ′Y ′. A figure f is any non-empty subset of points of space. Definition 3.3. A figure is called invariant with regard to a transformation S if S(f ) = f , and S is called a symmetry of f . Theorem 3.1. Symmetries of a figure form a group called symmetry group of f and denoted Gf (see, e.g., Gr¨ unbaum and Shephard, 1986; Martin, 1980; Jablan, 2002). Isometric symmetry groups of the space E n can be classified according to a sequence of maximal proper (sub)spaces invariant with respect to the action of transformations of the groups in question. Symmetry groups of friezes G21 , bands G321 , plane ornaments G2 , and layers G32 can be used for the classification of knot-work patterns. Isometric symmetry groups will be denoted according to the crystallographic notation (or Hermann and Maugin notation). Theorem 3.2. There exist exactly 7 symmetry symmetry groups of friezes, 31 symmetry groups of bands, 17 symmetry groups of plane ornaments,

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Fig. 3.9

Celtic friezes with the same symmetry group of bands p1a1.

Fig. 3.10

Celtic friezes with the same symmetry group of bands p121.

and 80 symmetry groups of layers (see, e.g., Shubnikov and Koptsik, 1974; Coxeter and Moser, 1980; Gr¨ unbaum and Shephard, 1986; Martin, 1980; Jablan, 2002). Definition 3.4. The fundamental region of a symmetry group of an object or pattern is the smallest part of the pattern, which, based on the symmetry, determines the whole object or pattern. In all symmetry-oriented classifications of interlaced patterns, i.e., infinite knotwork patterns (e.g., in Cromwell (1993) or in Gr¨ unbaum and Shephard, 1986), symmetry is used as the only criterion for the classification. Linear knotwork patterns are classified according to 7 symmetry groups of friezes, or 31 symmetry groups of bands (Washburn and Crowe, 1988; Gr¨ unbaum and Shephard, 1980, 1983, 1986) without taking in account their topological or knot-theoretical properties. In the same way, plane symmetry patterns are classified according to 17 symmetry groups of ornaments or 80 symmetry groups of layers. In all these cases we have an asymmetric fundamental region multiplied by symmetries belonging to the symmetry group, without taking in consideration that the fundamental region can be any asymmetric tangle with its particular knot-theoretical

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properties. For example, according to the symmetry oriented classification, two bands (Fig. 3.9) will be considered as equivalent because their symmetry group is p1a1, in spite of the fact that the first is based on the tangle 2 1 giving as the numerator closure Hopf link 221 (2), and the other is the direct product of knots 2 1 1 2. For the classification of infinite symmetric interlaced patterns we propose the following two criteria: (1) isometric symmetry group of the pattern; (2) tangle belonging to a fundamental region. These criteria are not always sufficient, and we also need to consider other knot-theoretical properties such as whether a pattern represents prime or composite KL arrangement. In the case of an n-tangle (n = 2, 3, ...) representing a fundamental region, it is possible to construct all its closures and use them for a further classification. For example, by joining left and right ends of the tangles (Fig. 3.10) that are the basic elements of two friezes with the same symmetry p121 we obtain links 3#2 and 2 1 2, 2, 2+, respectively. This classification, proposed by S. Jablan and Lj. Radovi´c in 2001, is similar to the approach proposed earlier by I. Emery (1995). 3.2.4

Enumeration of mirror curves

In this section we will investigate the enumeration of different monoloinear curves (i.e. the corresponding mirror arrangements) which can be derived from a rectangular square grid RG[a, b] of dimensions a, b, covered by k curves, for a given number of mirrors m (m = k − 1, k, . . . , 2ab − a − b). Unfortunately, the general solution of this problem is far: placing every new internal mirror changes the whole structure. It behaves like a kind of Game of Life or cellular automata, where a local change results in the global change. So far, we have only a few combinatorial results based on the P ET (P´olya Enumeration Theorem) (Harary and Palmer, 1973), obtained for particular cases by S. Jablan, and generalized by G. Baron. The following list gives the number of perfect curves obtained from a rectangular grid RG[a, b], k = GCD(a, b) with the minimal number k − 1 of two-sided internal mirrors incident to the cell-edges, where t = (ab − a LCM (a, b)) : (k(k − 1)) = 4xy (LCM – least common multiple), x = 2k , b y = 2k .

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Fig. 3.11 curve.

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Different arrangements of edge-incident mirrors in RG[6, 3] giving a single

(I) If only k − 1 edge-incident mirrors are placed in RG[a, b], such that a 6= b, the number of perfect curves is k−3

k−1

(1) (4k)k−2 tk−1 + 2(4k) 2 t 2 for k odd; k−2 k−2 (2) (4k)k−2 tk−1 + (4k) 2 zt 2 , with z = x for a ≡ 0 (mod 2k), b ≡ k (mod 2k), and z = x + y, for a ≡ b ≡ k (mod 2k), for k even. (II) the number of perfect curves for k − 1 edge-incident or edgeperpendicular mirrors, and a 6= b is k−3

k−1

(1) 2(8k)k−2 tk−1 + 4(8k) 2 t 2 for k odd; k−2 k−2 (2) 2(8k)k−2 tk−1 + 2(8k) 2 zt 2 , with z = x for a ≡ 0 (mod 2k), b ≡ k (mod 2k), and z = x + y for a ≡ b ≡ k (mod 2k), for k even. For a = b, we have to put t = 1, z = 1, divide the numbers by 2, and get the following results (i) the number of perfect curves for k − 1 only edge-incident mirrors is k−3

(1) 22k−5 k k−2 + 2k − 3k 2 for k odd; k−2 (2) 8k 2k−5 k k−2 + 2k − 3k 2 for k even;

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(ii) the number of perfect curves for k − 1 edge-incident or edgeperpendicular mirrors is k−3

(1) (8k)k−2 + 2(8k) 2 for k odd; k−2 (2) (8k)k−2 + (8k) 2 for k even. Even for relatively small RGs (e.g., with a = 6, b = 3), and minimal number of mirrors (k−1 = 2), the number of the different curves obtained is large. For example, there are 52 different arrangements of two edge-incident mirrors in a rectangle 6 × 3 producing perfect curves. Among them, only 8 are symmetric– 4 mirror-symmetric and 4 point-symmetric (Fig. 3.11). 3.2.5

Lunda designs

Lunda design is obtained from a mirror curve by numbering small squares in the order in which a curve passes through and then reducing all numbers modulo 2. The result is a 0-1 sequence, i.e., “black”-“white” mosaic (Gerdes, 1997, 1999). Lunda designs have the local equilibrium property: the sum of the integers on every two border unit squares with the joint vertex is the same (Fig. 3.12a), and the sum of the integers in the four unit squares between two arbitrary neighboring grid points is always twice the previous sum (Fig. 3.12b). This gives the global equilibrium property: the sums in all rows are equal, and the same holds for the columns. Local and the resulting global equilibrium property hold even if the reduction is made modulo 4. In particular, enumerating a regular curve (with the mirrors incident to the grid edges) and reducing all the numbers modulo 4, we obtain four-colored Lunda designs, where every vertex is orderly surrounded by numbers 0,1,2,3 and the disposition of the sequences around the points is alternately clockwise and anti-clockwise. The correspondence between monolinear mirror-curves (i.e., the corresponding arrangements of mirrors) and Lunda designs is many-to-one, so the same Lunda design can originate from different mirror arrangements (Fig. 3.12c). Classification of mirror arrangements according to Lunda designs they produce is an open question. 3.2.6

Polyominoes

A plane region without “holes”, formed by n edge-to-edge adjacent squares is called a polyomino (Golomb, 1994). If instead of squares we use n equilat-

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Fig. 3.12 (a) Possible border situations; (b) possible situations between vertical and horizontal neighboring grid points; (c) Lunda designs.

eral triangles or n regular hexagons, we obtain, respectively, polyiamonds or polyhexes. We will restrict our discussion to polyominoes (although it also can be applied to polyiamonds and polyhexes). For polyominoes not having a reflective symmetry, we may distinguish or not their “left” and “right” form. Hence, we have two possible equality criteria for polyominoes:

(1) considering only the shape (without distinguishing “left” or “right” form); (2) considering both shape and orientation.

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So far, there is no formula for calculating the number of different polyominoes. The number of various polyominoes is computed for n ≤ 15 (http://mathworld.wolfram.com/Polyomino.html).

Fig. 3.13

Polyominoes in 0-1 notation.

In a polyomino with the borders consisting of mirrors, in every border cell we can place two-sided mirrors perpendicular to the internal edges in their midpoints. After a series of reflections, the ray of light will “describe” a shape called closed Dragon curve (or self-avoiding curve). If we denote a reflection in a border mirror by 0, and a reflection in an internal mirror by 1, we have 0-1 words (or symbols) for polyominoes (Fig. 3.13), where these words are cyclically equivalent. For n = 1 we will have only one polyomino 0000, for n = 2 the polyomino 00010001, for n = 3 two polyominoes: 000101000101 and 000100100011, for n = 4 five of them: 0001010100010101, 0001010001100011, 001001001001, 0001001100010011, and 0001001010001011, etc. From their binary symbols we can make conclusions about the symmetry: every reversible word denotes a polyomino with a sense-reversing symmetry (it does not have “left” and “right” form); irreversible symbols correspond to the polyominos appearing in the “left” and “right” form (e.g., 0001001100010011, or 0001001010001011).

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We can translate these symbols (or binary numbers) into hexadecimal numbers and assign one number to each polyomino, in order to establish one-to-one correspondence between the numbers and polyominoes. For example, this number can be the minimum of all cyclic-equivalent symbols (e.g. to the polyomino 00010001 correspond cyclically equivalent symbols 00100010, 01000100, 10001000 and the minimum of them is 00010001 = 11 in the hexadecimal system). Hence, we have a notation for polyominoes where exactly one number corresponds to every polyomino, and vice versa. Open question is: find the general algebraic form of the number determining a polyomino? Namely, some numbers will determine “open polyominoes”, “hollow polyominoes” or “overlapping polyominoes”, that are not included in our definition, and other will determine “real” polyominoes. Every (n + 1)-omino can be derived from some n-omino by adding a single square to it. The addition operation is a positional one, i.e., the result depends on the position where the new square is added. We have the following addition rules: (1) a0 + 0000 = a10001 (1-edge contact); (2) a0110 + 0000 = a1001 (2-edge contact); (3) a0110110 + 0000 = a1010 (3-edge contact), where a never ends with 1. These rules can be efficiently used for the computer enumeration of polyominoes. In each step we need to derive (n + 1)-minoes from n-minoes by adding a square, then check the equality of obtained polyominoes and make the list of all (n + 1)-minoes. The main problem are “undesired” edge contacts (e.g., contacts in parallel edges, producing “hollow” and “overlapping” polyominoes). 3.2.6.1

Lunda polyominoes and Lunda animals

Polyominoes (either black or white) appearing in Lunda designs will be called Lunda polyominoes (Gerdes, 1996). The possible shape of Lunda polyominoes is restricted by the local equilibrium condition for Lunda designs. Therefore, some polyominoes are inadmissible (e.g., 001001001001). On the other hand, Lunda polyominoes also include “hollow” polyominoes. In his book Lunda geometry: Designs, Polyominoes, Patterns, Symmetries P. Gerdes (1996) introduced the concept of Lunda-animals and obtained the first approximation of the total number of different Lunda n-ominoes. Lunda-animal is a Lunda m-omino with a unit square at one of its ends,

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representing a head. A Lunda-animal walks in such a way that after each step the head occupies a new unit square, and every other cell occupies the position previously taken by the preceding cell. In other words, two subsequent positions of a Lunda-animal have a Lunda (m−1)-omino in common. How many different positions p5 (n) of a Lunda 5-omino are possible after n steps? P. Gerdes proved that: pm (n) = f (n+3) for m = 1, 2, 3, . . . , 8, where f (n) is the famous Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . given by the recurrence formula: f (0) = 0, f (1) = 1, f (n+ 1) = f (n)+ f (n− 1). It is interesting that for every Lunda m-omino for m < 9 the result is the same, so pm (n) = f (n+3) for 1 ≤ m ≤ 8. From m = 9 onwards, pm (n) < f (n+3). Finding the general formula for pm (n) is an open problem.

3.2.7

KLs and mirror curves

Now let us take a look of the classification of mirror curves through knot theory glasses! Every mirror curve can be simply transformed into an interlacing knotwork design, that is, into a projection of some alternating knot. Such curves appear in the history of ornamental art, more frequently as knotworks, then as plane curves. Even the name Brahma-mudi (Brahma’s knot) denoting Tamil curves refers us to knots. Therefore, the classification of mirror curves goes via proper reduced minimal knot projections. Two projections or knot diagrams are equal if they are isotopic as graphs, where the isotopy ensures that relations “over”-“under” are preserved. In order to classify our curves, treated as knot projections, we can use the invariants of KL projections (page 312). The rectangular square grid RG[2, 2] is the minimal RG from which we can derive some non-trivial alternating KLs (different from the unknot)– the trefoil knot 31 (or 3 in the Conway notation) and the 2-component link 221 (or 2) (Fig. 3.14). From RG[3, 2] we obtain the knots 74 , 62 , 31 #31 , 51 , 52 , 41 and 31 (or 3 1 3, 3 1 2, 3#3, 5, 3 2, 2 2, and 3 in the Conway notation), where different mirror-arrangements may give the same projection. Is it possible to derive every knot projection from some RG with a large enough number of crossings? What is the upper bound for this number? Which knot projections can be obtained from a particular RG? Which mirror-arrangements in some RG give the same knot projection? Find the minimal RG for a given knot! Can you obtain several non-isomorphic projections of some knot from the same RG? These and many other problems connected with mirror curves represent an open field for research.

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Fig. 3.14

3.2.8

Hopf link 211 (2) and trefoil knot 31 (3) in RG[2, 2].

Mirror curves on different surfaces

The construction of mirror curves is independent from the metric properties or the geometry of the surface, so the same principle of construction can be applied to any tiling (e.g., on a sphere) (Gerdes, 1996, 1999) or in the hyperbolic plane (Dunham, 2000; Sazdanovi´c and Sremˇcevi´c, 2002a,b). Algorithm 3.1 Let us consider any edge-to-edge tiling of a part of an arbitrary surface. First connect midpoints of adjacent edges to obtain a 4-regular mid-edge graph with k components. Using the rules for adding two-sided mirrors, it can be converted in a single mirror curve in a finite number of steps (Fig. 3.15). It is easy to prove that this simple algorithm is finite. If k = 1, we have a single mirror curve, so no additional two-sided mirrors are needed. If the number of components is k (k ≥ 2), we place first two-sided mirror in the crossing of two different curves, connect them and obtain k −1 components. Continuing in the same way, a single mirror curve will be obtained after introducing k − 1 mirrors. Our game becomes more interesting if we allow adding mirrors in self-crossing points of the same component. This move can either preserve the number of curves or increase it by 1, so we can end up with a single or multi-component curve. Open question is: find a general formula for the number k of curves for any tiling, before mirrors are placed. From mirror curves on different surfaces we could obtain the corresponding Lunda designs. All non-isomorphic Lunda designs on a regular octahedron are given in Fig. 3.16. Try to enumerate non-isomorphic Lunda designs obtained from regular polyhedra!

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Fig. 3.15 Construction of a single mirror curve from the tiling (a) by connecting edge mid-points (b), tracing components (c) and introducing a mirror (d).

3.2.9

Mirror curves in art

Let us consider mirror curves from different cultures, distant in space and time, and try to discover some common principles used for construction of mirror curves. We will compare mirror curves from Tamil art, Tchokwe sand drawings and Celtic art, try to discover the common properties of the constructions used, and establish some hierarchy with regard to their complexity. As the final result, we will describe a kind of algorithmic approach used by these cultures for the construction of knotwork designs and compare it with similar approaches used in knot theory. At the beginning of knotwork art, every culture probably used plates– rectangular square grids RG[a, b] of dimensions a, b (a, b ∈ N ) without internal mirrors4. Plates have been recognized as the basis of all Celtic knotworks by the antiquarian J. Romilly Allen whose twenty years’ work is summarized in the book Celtic Art in Pagan and Cristian Times (1904). The initial number of mirror curves for plates without internal mirrors is k = GCD(a, b) (GCD– greatest common divisor), so a single curve is ob4 A cylinder seal from Ur, Mesopotamia, representing a snake with interlacing coil dates from 2600-2500 B.C. (Przytycki, 2004, Fig. 1.3)

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Fig. 3.16 Lunda designs on the octahedron. Small triangles in uppermost right corners show the coloring of an outside region.

Fig. 3.17

KLs obtained from RG[a, 2] for a = 3, 4, 5.

tained iff a, b are mutually prime numbers. From the knot theory point of view, every single-curve plate, turned into an alternating knot by introducing the relation “over-under”, represents a Lissajous knot (Bogle, Hearst, Jones and Stoilov, 1994). The infinite series of plates, obtained for an arbitrary a (a ≥ 3) and b = 2, consists of the rational KLs of the form 3 1 3, 3 1 2 1 3, 3 1 2 1 2 1 3, 3 1 2 . . . 2 1 3 (Fig. 3.17). Notice that for every odd b we obtain a knot, and for every even b a 2-component link.

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Fig. 3.18

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KLs obtained from RG[a, 3] for a = 3, 4.

The number of different projections of these KLs is: 1, 4, 13, 68, 346,. . ., respectively, but in knotworks, only one of them– the most symmetric, is used for each a. Most symmetric projections can be found and drawn using the LinKnot function MaxSymmProjAltKL. For a = 3 we have the projection 3 1 3, for a = 4 the projection (((1, (3, 1), 1), 1), 1, 1, 1), for a = 5 the projection (((1, ((1, (3, 1), 1), 1), 1), 1), 1, 1, 1), for a = 6 the projection (((1, ((1, ((1, (3, 1), 1), 1), 1), 1), 1), 1), 1, 1, 1) etc. The sequence 1, 4, 13, 68, 346,. . . is not included in the Encyclopedia of Integer Sequences; in fact, it is possible to obtain many new infinite sequences defined by numbers of different projections of specific classes of KLs. For an arbitrary a (a ≥ 3) and b = 3 we obtain plates with polyhedral KLs: for a = 3 we have 3component link 8∗ 2 : 2 : 2 : 2, for a = 4 the knot 1312∗ : 2 0 ::: 2 0.2.2 0, etc. (Fig. 3.18). Definition 3.5. Any monolinear mirror curve placed in some polyomino without internal mirrors is called a plate design. Let us now describe four general rules for combining plate designs and/or mirror curves. The first three rules are given by P. Gerdes (1999), and the fourth is proposed by S. Jablan. We will restrict our consideration to mirror curves placed in polyominoes with square cells. Construction rules: (1) The first rule defines a combination of two mirror curves that share one edge of an open cell on their borders (Fig. 3.19a). Such a composition corresponds to the direct product of KLs, and it was probably one of the most exploited constructions in knotwork art. For given mirror curves M1 and M2 , this kind of direct product we will call ×-direct product and denote

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Fig. 3.19

Rules for composing plate designs and mirror curves.

it by M1 × M2 . If we combine two mirror curves in this way, first with c1 , and the other with c2 components, the result is a new mirror curve with c1 + c2 − 1 components. Hence, the ×-direct product of two 1-component mirror curves is a new 1-component mirror curve. This idea was used, for example, in the Tchokwe design from Fig. 3.20 and in many Celtic friezes. As a particular application of the first rule, we can add a single square to the border of any monolinear mirror curve. This transformation corresponds to adding an external loop to a KL diagram. It does not change the number of components and can be repeated, since it has a decorative function in knotwork art. For example, the Tamil (unknot) design from Fig. 3.2a is created by a series of external loop additions, beginning from the RG[1, 1]; the knot design from Fig. 3.2b by adding loops to the RG[4, 3]; and the knot design from Fig. 3.2c by adding loops to the RG[5, 3]. The same construction is used for Tchokwe designs (Fig. 3.3a). (2) The second rule is the one defining the direct product K1 #K2 in knot theory (Fig. 3.19b). In the language of mirror curves M1 and M2 , it means that we cut one external edge of each mirror-curve M1 and M2 , and reconnect them again to obtain a new mirror-curve, that will be denoted by M1 k M2 .

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Fig. 3.20

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× and k direct product in Sona drawings.

Fig. 3.21

Rule 3 in Sona drawings.

(3) The third rule is restricted to plate designs: two monolinear plate designs whose overlapping contains exactly two cells will give a new monolinear plate design. The schematic interpretation of the third rule is given in Fig. 3.19c.

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Fig. 3.22

Fig. 3.23

A multiple application of the Rule 4 in Tamil drawing.

Algorithm for creating monolinear symmetric mirror curves.

In order to introduce the fourth rule we need to define new operation, addition. The addition of a plate design P1 to plate design P2 is an edgeto-edge identification of their border cells belonging to rectilinear borders (Fig. 3.19d). In the same way, we can add a plate design P1 to some mirror curve M placed in some polyomino. (4) The fourth rule is: an RG[a, b] for which b|a, added to any monolinear mirror curve M (or monolinear plate design P2 ) along the edge b, will

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Fig. 3.24

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Basic elements of Celtic knotworks.

Fig. 3.25

Celtic tangles.

give a monolinear design (Fig. 3.19d). In particular, any square RG added to a monolinear design gives a new monolinear design. Rule 4 can be applied to mirror curves: we can add a mirror curve M2 to a monolinear mirror curve M1 in such a way that every curve contact point along edge b of the polyomino in which M2 is placed belongs to a different component of M2 . The new mirror curve M1 + M2 will be monolinear. These four rules are sufficient for creating monolinear plate designs and extend the monolinearity from RGs to plate designs (Figs. 3.20-3.22). For the further derivation of monolinear mirror curves from monolinear plate designs we can use the rules described in the Subsection 3.1.3 for adding internal mirrors, illustrated in Fig. 3.7. Since symmetry is desirable visual property, in knotwork art symmetric mirror curves prevail over asymmetric ones. This means that most of the mirror arrangements are not aesthetically appealing: as we mentioned before, only 8 out of 52 two-mirror arrangements from RG[6, 3] are symmetrical. For a construction of symmetric mirror curves we propose the following

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Fig. 3.26

Celtic knots, friezes and plane knotwork ornaments.

algorithm. Let a symmetric monolinear plate design P be given. We place an internal mirror in some crossing A of P and trace an oriented mirror curve M . Now we have two possibilities: (1) if P is not completely covered by M , choose a not self-crossing point on M , symmetric to A and put a mirror symmetric to the mirror in A (Fig. 3.23a1). If a symmetric point with this property does not exist, rotate the mirror in A for 90◦ around its midpoint and then place the mirror symmetric to it (Fig. 3.23a2 ); (2) if P is completely covered by M , place a new mirror symmetric to the mirror in A (Fig. 3.23b1 ). If monolinearity is destroyed, rotate the mirror in A for 90◦ around its midpoint and then place the mirror symmetric to it (Fig. 3.23b2 ). This algorithm is applied until the maximum number of internal mirrors that preserve monolinearity is used. This approach will be used to explain the construction of different mirror curves occurring in Tamil, Tchokwe and Celtic knotworks. We already explained and illustrated knotwork designs that represent a single monolinear RG, which were derived from a single monolinear RG by adding a series of external loops, as well as designs obtained as a ×-direct product of monolinear RGs (Rule 1).

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Fig. 3.27

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Celtic circle and square knot designs.

The first rule, ×-direct product, is very frequently used in Celtic knotwork art for the construction of friezes, as well as the k-direct product (Rule 2). Both of them are the standard tools for obtaining translational repetitive structures: friezes or even plane symmetry groups. Applying the second rule actually connects two monolinear RGs in their corners, i.e., it is the k-direct product of the corresponding knots. Another possibility is using (more or less) “open” RGs and their k-direct product. Although we obtain the same composite KLs, in the visual sense obtained patters will be different. The ×-direct product was used in Celtic knot art as well, mainly for the construction of frieze knotworks (or bordures). In order to analyze Celtic knotworks based on k-direct product first we need to insert some internal mirrors perpendicular to the edges in basic (monolinear) RGs, in order to obtain parts or “tangles” of Celtic knotworks with an appropriate placement of incoming and outgoing strands. The possible choices for their positions are two top (or bottom) corners of an elementary RG, two diagonal (ascending or descending) corners, or all four corners forming a tangle. In the first case we place internal mirrors perpendicular to “vertical” and “horizontal” edges of border cells, forming an L-shape form (Fig. 3.21). Furthermore, cutting the long edge(s) of the design and reconnecting them, we obtain different frieze designs (direct

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Fig. 3.28 (a) k-direct product in the Tchokwe design; (b,c) the application of Rule 3 in Tchokwe designs.

Fig. 3.29 Derivation of Celtic monolinear cross knot design from plate design obtained using the Rule 4.

products of basic KLs) with incoming and outgoing strands appropriately placed (Fig. 3.25). The other possibility is creating “tangles” from RGs (Fig. 3.25) and composing them into a chain (or a closed circle) (Fig. 3.263.27). In the case of Tchokwe sand drawings a similar strategy was used in order to obtain “open” RGs that can be composed by k-direct (Fig. 3.28a) or ×-direct product (Fig. 3.3b) in larger knotworks.

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Fig. 3.30

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Celtic monolinear knot cross design.

Fig. 3.31

Celtic mirror curves.

The third rule was one of the favorite rules in the construction of Tchokwe sand drawings. A whole series of “social” monolinear plate designs representing a leopard with cubs (Fig. 3.21a), a design called kambava wamulivwe that represents an animal called kambava that died inside a rock (Fig. 3.28b), or lusona drawing called tambwe that represents a lion (Fig. 3.28c) is composed in this way. The fourth rule offers the highest degree of freedom and often gives symmetric plates in knotwork art (Fig. 3.29). Various designs can be obtained by adding along edge b any RG(a, b) with the property b|a, or square RG, in a symmetric or asymmetric way to a monolinear plate design. In this way, we can create perfect curves of a desired shape. Creating a variety of monolinear plate designs opens the door to artis-

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Fig. 3.32

Celtic monolinear design with broken symmetry.

Fig. 3.33 Construction of Celtic monolinear knot design (b) by breaking the symmetry of the two-component symmetric design (a).

tic creativity and play: there is a huge number of ways for introducing internal edge-incident and edge-perpendicular mirrors in order to preserve monolinearity (Figs. 3.30-3.33). Together with the remarkable example of a monolinear cross knot design

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(Fig. 3.30), another interesting example is a complex monolinear design (Figs. 3.32-3.33). Because the symmetric version of the same design (Fig. 3.33a) is a two-component knot design, the Celtic master constructed an almost symmetric monolinear design by breaking symmetry (Fig. 3.33b).

Fig. 3.34 (a) Tchokwe sand drawings; (b) the geometric construction of the corresponding basic polyhedra.

Fig. 3.35

Torus knot [12,5].

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Fig. 3.36

Fig. 3.37

Michelangelo’s plaza.

Celtic circular knot-designs and basic polyhedra.

In the preceding chapter we investigated various properties of basic polyhedra and their families (page 295). Several families of basic polyhedra appear in knotwork. The family of basic polyhedra starting with 8∗ , up to 162∗, yields monolinear designs in Tchokwe sand drawings (Fig. 3.34a). These series can be obtained from the shadows of torus knots of the form [4, b], GCD(4, b) = 1. Construction of these basic polyhedra is inspired by a pattern from nature– the cobweb of a large spider, a series of inscribed squares (Fig. 3.34b).

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Fig. 3.38 (a) A tangle T , its numerator N (T ), and denominator closure D(T ); (b) the pair of links (2 1 2, 2 1 2) obtained as the numerator and denominator closure of the Celtic tangle; (c) the closure of the tangle 3; (d) the bands (p121,3) and (p121,5).

In the same way we can obtain different basic polyhedra derived from shadows of torus knots [a,b], GCD(a, b) = 1. They represent the same geometric structure: a series of inscribed n-gons (n ≥ 3). Similar infinite series of basic polyhedra inspired by patterns from nature, such as the growth patterns of certain plants, can be found in artworks. For example, a shadow of the torus knot [12,5] (Fig. 3.35) appears in Michelangelo’s plaza (Fig. 3.36). Celtic masters used friezes without bigons to construct basic polyhedra (Fig. 3.37b) by identifying opposite sides of friezes. This method has quite a general character and can be used for creating other circular knot designs (Fig. 3.27, 3.37a). In order to classify complex periodic knotworks (e.g., Celtic friezes or plane ornaments from Fig. 3.26, or laces), we will recognize basic patterns– tangles, equivalent to fundamental regions and combine two approaches: the theory of symmetry and knot theory. First we determine the symmetry group, and then add the information about tangles. For this description,

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friezes or plane symmetry groups are treated as 3-D objects, by taking into consideration the relation “over-under”. Their symmetry groups can be found among 31 symmetry groups of bands, or 80 symmetry groups of layers. However, bands or layers that have the same symmetry group can be, visually and topologically, very different. They can be composed of different generating elements, repeated according the same symmetry rules. Therefore, together with the symmetry classification, we can use the classification of their “building blocks”, tangles (Fig. 3.38a). Using numerator and denominator closures (N (T ), D(T )), from the Celtic tangles from the upper row of the Fig. 3.25 we obtain the ordered pairs of alternating KLs: (2 1 2, 2 1 2) (Fig. 3.38b), (3, 2 2), (1, 3), (8∗ , 2 1 2 1 2), (2 1 1 1 2, 3#3), (2 1 1 1 2, 2 2 1 2), and (3#3, 2 1 2). Similarly, tangles with two open ends can be closed (Fig. 3.38c). In order to distinguish knotwork symmetry patterns, we will use the notation consisting of a symbol of the symmetry group and closure(s) of the tangle. For example, two bands with the same symmetry group p121 have the symbols (p121,3) and (p121,5) (Fig. 3.38d). In order to obtain more precise classification, instead of KLs obtained as closures, we can use their projections. 3.2.10

KLs and self-avoiding curves

This part of the work is inspired by a series of sculptures titled Viae Globi, created by Carlo Sequin (2001) (Fig. 3.39), and by a conversation with Haresh Lalvani, who proposed to identify vertices of a polygon, in particular two vertices of a triangle in order to obtain “a triangle with two vertices” (Fig. 3.40). This simple idea is a part of his extensive unpublished work. Using this idea in knot theory, we have established correspondence between KL shadows with n crossings and 2n-gons with n pairs of collapsed points. The unicursal curves related to knot theory, atoms and d-diagrams are considered in papers by V. Manturov (2000a, 2000b) and his book Knot Theory (Manturov, 2004, Chapter 15). Given KL shadow can be transformed into a single closed mirror curve by placing a two-sided mirror in an appropriate position in every vertex (Fig. 3.7). This mirror curve is a self-avoiding path, dividing the plane ℜ2 or surface of a sphere S 2 into two regions, interior and exterior, which are equivalent on a sphere. Figure 3.41 shows two self-avoiding curves derived from Borromean rings (represented as a Schlegel diagram of an octahedron), and a self-avoiding curve derived from the fullerene C60 by

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Fig. 3.39

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Sculpture Lombard by Carlo Sequin.

Fig. 3.40

Triangle with two vertices.

the mid-edge truncation. The number and type of mirrors necessary to convert any minimal KL shadow of a given KL into a self-avoiding curve is an invariant of a KL. For both self-avoiding curves derived from Borromean rings, the number of mirrors is {3, 3}. Let us number points where a self-avoiding curve touches mirrors: for n mirrors there will be 2n points. We can pair points corresponding to the same mirror and think of a self-avoiding curve is an 2n-gon with n pairs of identified vertices. This fact will be used to establish connection between self-avoiding curves and chord diagrams. In order to avoid loops, we never identify adjacent points. If we denote points belonging to internal mirrors by underlined numbers, and overline numbers belonging to external mirrors, the self-avoiding curve (Fig. 3.42) can be denoted by the code

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Fig. 3.41 (a) Two self-avoiding curves derived from Borromean rings; (b) self-avoiding curve derived from the fullerene C60 by the mid-edge truncation.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Analogously to Gauss codes, this code will depend from a beginning point and orientation. In order to have one-toone correspondence between self-avoiding curves and codes we will define short codes and for every curve choose the minimal code as the representative. If we think of the exterior and interior of a curve as equivalent, then a code and its dual (the code with inverted underlinings and overlinings) are considered to be the same. In the same way as with Gauss and Dowker codes, the proposed codes for self-avoiding curves can be written in a more concise form. First, we can write our code as a sorted list of ordered pairs {{1, 4}, {2, 11}, {3, 6}, {5, 8}, {7, 10}, {9, 12}}. If we agree to replace every pair of overlined numbers by the same numbers without overlinings, to replace every pair of underlined numbers by these numbers in opposite (descending) order, and sort the obtained list, the result is the list {{1, 4}, {5, 8}, {6, 3}, {9, 12}, {10, 7}, {11, 2}}. The list of second elements in each pair gives the short code {4, 8, 3, 12, 7, 2}. The complete code can be recovered from the short code by reversing the procedure described above. A different agreement: replacing every pair of overlined numbers by these numbers in opposite (descending) order, every pair of underlined numbers by the same pair of numbers and sort the obtained list, gives the dual list {{2, 11}, {3, 6}, {4, 1}, {7, 10}, {8, 5}, {12, 9}}, and the dual short code {11, 6, 1, 10, 5, 9}. Every self-avoiding curve can be graphically interpreted by a chord diagram: a regular 2n-gon, where points belonging to internal mirrors are connected by full, and points belonging to external mirrors by broken diagonal lines (or by black and white lines). For example, the first self-avoiding curve from Fig. 3.41a will be described by the chord diagram from Fig.

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Fig. 3.42 (a) Coding of the first self-avoiding curve from Fig. 3.41a and its chord diagram; (b) chord diagram of the second self-avoiding curve from Fig. 3.41a and its dual.

3.42a, or by its dual obtained by inverse bicoloring, where full (black) lines are replaced by broken (white) lines and vice versa. In the both cases (Fig. 3.42) chord diagrams are equal to their duals, i.e., they are self-dual. From every chord diagram we can easily obtain a code of the corresponding selfavoiding curve and vice versa, and to recover an original KL from which the curve is derived. Two stages of the recovering are illustrated in Fig. 3.43.

Fig. 3.43

Recovering a self-avoiding curve from its chord diagram.

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In order to derive and enumerate all self-avoiding curves with n mirrors, one should first derive all different non-colored chord diagrams, and then impose the appropriate coloring. The following rule holds for noncolored chord diagrams: every vertex belongs to exactly one diagonal (chord). In fact, we are searching for all different minimal sets of diagonals that span a regular 2n-gon, where sets that can be obtained one from another by symmetries of 2n-gon are considered to be the same. For n = 2, 3, . . . , 7 we obtain, respectively 1, 2, 7, 29, 176, 1788 such sets. The sequence obtained is A003437 from the On-line encyclopedia of integer sequences (http://www.research.att.com/∼njas/sequences/), which represents the number of unlabeled Hamiltonian circuits on n-octahedron (Singmaster, 1975). An n-octahedron is the complete n-partite graph K2,2,...,2 (n pairs of opposite vertices with edges connecting each vertex to every other vertex except its opposite). Singmaster notes that such a Hamiltonian cycle can be viewed as a way of seating n couples around a circular table so that no man is next to his wife. The number of cases is given by the following formula (Pratt, 1996): n X (−1)k

k=0

n k

2n k [ 2n−k ]2 (2n − k)! 2n n!

.

Chord diagrams derived for n = 2, 3, 4 are given in Fig. 3.44. Among all chord diagrams we can distinguish 2-vertex connected graphs (containing the edges of an 2n-gon as well), corresponding to non-prime KLs, and others, 3-connected, corresponding to prime KLs. For coloring of chord diagrams we have the rule: every two diagonals crossing each other must have different colors. A chord diagram will be colorable iff it is planar. The other, purely visual, criterion for colorability is the following: a chord diagram is colorable iff crossings of its diagonals do not form a polygon with an odd number of edges, and three or more diagonals do not have a common point (Fig. 3.45). Coloring of a (colorable) chord diagram represents a projection of a polyhedron enclosed in an 2ngon, with proper visibility of all edges. In the case of 2-vertex connected chord diagrams, coloring is not unique: from the same uncolored chord diagram we can obtain several different colored diagrams (Fig. 3.44). KL shadows, their corresponding self-avoiding curves and colored chord diagrams for n = 2, 3, 4 are given in Fig. 3.46, and for n = 5 in Fig. 3.47. In the case of 3-vertex connected chord diagrams, a coloring is completely forced by the coloring of one edge: by

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Fig. 3.44

Chord diagrams for n = 2, 3, 4.

Fig. 3.45

Non-colorable chord diagrams.

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choosing its color we can obtain only one colored chord diagram, or its dual. Hence, in the case of 3-vertex connected planar diagrams, an uncolored chord diagram provides a complete information about the corresponding self-avoiding curve. Every uncolored chord diagram can be given as a list of unordered pairs of numbers denoting chords. For example, the uncolored chord diagram from the Fig. 3.48a can be denoted as {{1, 3}, {2, 6}, {4, 9}, {5, 8}, {7, 10}}. The same figure illustrates its bicoloring (a), the reconstruction of its corresponding self-avoiding curve (b-e), and KL shadow obtained (f).

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Fig. 3.46 KL shadows, self-avoiding curves, and colored chord diagrams obtained for n = 2, 3, 4.

Restricting our attention to 3-vertex connected planar chord diagrams corresponding to prime KLs, for n = 2, 3, . . . , 8 we obtain, respectively, 1, 1, 3, 7, 33, 148, 923 chord diagrams corresponding to self-avoiding curves derived from prime KLs. For n = 2 we have one chord diagram {{1, 3}, {2, 4}}, and for n = 3 one diagram {{1, 3}, {2, 5}, {4, 6}}. For n = 4 there are three diagrams, given in the following table:

{{1, 3}, {2, 5}, {4, 7}, {6, 8}}, {{1, 4}, {2, 7}, {3, 6}, {5, 8}}

{{1, 3}, {2, 6}, {4, 8}, {5, 7}}

For n = 5, the seven chord diagrams are given in the following table:

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Fig. 3.47 n = 5.

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KL shadows, self-avoiding curves, and colored chord diagrams obtained for

Fig. 3.48 (a) Uncolored chord diagram and its bicoloring; (b-e) reconstruction of its corresponding self-avoiding curve; (f) the corresponding KL shadow.

{{1, 3}, {2, 5}, {4, 7}, {6, 9}, {8, 10}} {{1, 3}, {2, 5}, {4, 9}, {6, 8}, {7, 10}} {{1, 3}, {2, 6}, {4, 9}, {5, 8}, {7, 10}} {{1, 4}, {2, 8}, {3, 7}, {5, 10}, {6, 9}}

{{1, 3}, {2, 5}, {4, 8}, {6, 10}, {7, 9}} {{1, 3}, {2, 6}, {4, 9}, {5, 7}, {8, 10}} {{1, 3}, {2, 7}, {4, 10}, {5, 9}, {6, 8}}

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Fig. 3.49 n = 6.

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Chord diagrams and self-avoiding curves corresponding to prime KLs for

For n = 6, thirty three chord diagrams and their corresponding selfavoiding curves given by mirror placements are illustrated in Fig. 3.49. Different shadows of the same KL can give different self-avoiding curves, as in the case of the link 2 2 2.

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Fig. 3.50 Three pairs of equal self-avoiding curves (a) shown as shapes (b) and their chord diagrams (c).

Fig. 3.51

KL families and chord diagrams of self-avoiding curves.

Visual recognition of self-avoiding curves, either direct or from shapes (colored plane regions defined by self-avoiding curves) (Fig. 3.50b), is complicated even for a small number of mirrors, but it is almost immediate from chord diagrams (Fig. 3.50c). It is interesting to mention a possible connection between shapes originating from self-avoiding curves and some biological forms. From every KL shadow can be derived one or several self-avoiding curves. Some conclusions about original KLs can be made based on the chord diagrams of their corresponding self-avoiding curves. For example, to every diagonal connecting two vertices separated by one vertex, and to every pair of parallel adjacent diagonals corresponds a bigon in the original KL shadow; diagrams without them correspond to basic polyhedra. In this way, we can follow a process of bigon collapsing (see page 8) directly in chord diagrams.

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Among all chord diagrams, we can distinguish antisymmetric diagrams, preserved under opposite coloring of chords. In respect to self-avoiding curves, this means that the external region is equivalent to the internal one. Such diagrams are called self-dual. For example, for n = 6, eleven among 33 chord diagrams are self-dual. Idea of families of KLs again plays an important role– we will consider families of KLs, their corresponding chord diagrams and self-avoiding curves. Chord diagrams corresponding to the same family can be visually recognized (Fig. 3.51).

Fig. 3.52

Via tori that can be obtained by identifying opposite sides of the rectangle.

Self-avoiding curves can be embedded on different surfaces, so together with Viae Globi on a sphere S 3 , we can consider Viae Tori on a torus (introduced in analogy to Sequin’s Viae Globi), or on any other surface (Fig. 3.52). For a given number n the LinKnot function fDiffViae derives all different self-avoiding curves with n mirrors that can be obtained from prime KLs with n crossings. The other way around, the LinKnot function fVia ToKL finds basic prime KL of every self-avoiding curve given by its (uncolored) chord diagram.

3.3

KLs and fullerenes

Among the chemical elements, carbon C is the basis of all life. A whole branch of chemistry, organic chemistry, is devoted to the study of C-C bonds and different molecules originating from them. Carbon is the only known 4-valent element able to produce long homoatomic stable chains or different 4-valent nets. Another candidate is silicon, whose homoatomic chemistry is rapidly developing.

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In addition to long-known diamond and graphite, a new form of carbon, the fullerene C60 , was first synthesized by H.W. Kroto, R.F. Curl and R.E. Smalley in 1985. Along with its structure, that of a spherical closed pentagonal/hexagonal monoatomic shell, it has the remarkable property of rotational symmetry of order 5 (which according to Barlow’s crystallographic restriction theorem is forbidden in crystallographic space or plane symmetry groups) and the highest possible, icosahedral, point-group symmetry. Since the discovery of C60 , different fullerenes (e.g., C70 , C76 , C78 , C82 , C84 , etc.) have been synthesized, opening a new field for research of potentially different possible fullerene structures from point of view of geometry, graph theory, or topology. The most complete discussion of fullerenes is given by P.D. Fowler and D.E. Manolopoulos (1995). Definition 3.6. Fullerene is a 3-planar graph with pentagonal and hexahonal faces. Theorem 3.3. Every fullerene has exactly 12 pentagonal faces. For every even n ≥ 24 there exists at least one fullerene Cn (Gr¨ unbaum and Motzkin, 1963; Voytekhovsky and Stepenshchikov, 2005). Chemical fullerenes are obtained from fullerene graphs by substituting vertices with carbon atoms. 3.3.1

General fullerenes, graphs, symmetry and isomers

Since carbon is 4-valent, there are four possible vertex configurations shown in Fig. 3.53a, denoted as 31, 22, 211 and 1111. The configurations 31 and 22 are obtained by adding carbon atom(s) between any two others connected by a double bond (Fig. 3.53b). Therefore, we can restrict our consideration to the remaining two non-trivial cases: 211 and 1111. On the other hand, deleting 31 and 22 vertices we obtain a reduced 4-valent graph, where at most one double bond (bigon), which can be denoted by colored (bold) edge, occurs in each vertex (Fig. 3.53a). First, we can consider all 4-valent graphs on a sphere. In chemistry, vertices of type 1111 are only theoretically acceptable. In knot theory, 4-valent graphs on a sphere with all vertices of the type 1111 are basic polyhedra. If all the vertices of such 4-valent graph are of the type 211, such graph we will be called a general fullerene. Every general fullerene can be derived from a basic polyhedron by a vertex bifurcation, this means, by substituting vertices with bigons in one of two possibile positions (2 and

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2 0) (Fig. 3.53c). In knot theory, general fullerenes are polyhedral source links. An edge-colored 3-valent graph (with bold edges denoting bigons), unique up to isomorphism, can be assigned to every general fullerene. Hence, we have two complementary ways for the derivation of general fullerenes: the vertex bifurcation method applied to basic polyhedra, and the edge-coloring method applied to 3-valent graphs, where each vertex has exactly one colored edge. For every general fullerene we can describe its geometrical structure (i.e., the positions of C atoms) by a non-colored 3valent graph, while its chemical structure (i.e., positions of C atoms and their double bonds) is described by the corresponding edge-colored 3-valent graph. Likewise, for every general fullerene we can distinguish two symmetry groups: a symmetry group G corresponding to the geometrical structure and its subgroup G′ corresponding to the chemical structure. Therefore, we will distinguish geometrical and chemical isomers. For example, for C60 , G = G′ = [3, 5] = Ih = S5 of order 120 (Coxeter and Moser, 1980), but for C80 with the same G, G′ is always a proper subgroup of G, and its chemical symmetry is lower than the geometrical. Hence, the first fullerene with G = G′ = [3, 5] = Ih = S5 after C60 is C180 , then C240 , etc. Without restrictions on the number of edges of fullerene faces, 7 general fullerenes can be obtained from the first (nontrivial) basiv polyhedron 6∗ , i.e., regular octahedron {3, 4}. From the basic polyhedron 8∗ with v = 8 we derive 30, and from the basic polyhedron 9∗ we obtain 4 general fullerenes, etc. In fact, this list of general fullerenes derived from basic polyhedra is identical with the list of source links with the maximal number of bigons derived from basic polyhedra (see the Section 2.5). 3.3.2

5/6 fullerenes

Among general fullerenes we can distinguish the class consisting of 5/6 fullerenes having only pentagonal and hexagonal faces. If n5 is the number of pentagons, and n6 the number of hexagons, from the relation 3v = 2e and the Euler theorem it follows that n5 = 12. Hence, the first 5/6 fullerene will be C20 with n6 = 0, the regular dodecahedron {5, 3}. It has two non-isomorphic edge-colorings, resulting in two chemically different isomers of the same geometrical dodecahedral form (Fig. 3.54a). The first basic polyhedron generating 5/6 fullerenes is the one with v = 10 vertices. For v = 10, there are three basic polyhedra, but only 10∗ and 10∗∗ generate 5/6 fullerenes, each only one of them (Fig. 3.54b,c). On the other hand, they

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Fig. 3.53 (a) The four possible vertex configurations; (b) the addition of carbon atom(s) between two others connected by a double bond; (c) the introduction of bigons in 1111 vertices.

generate, respectively, 78 and 288 general fullerenes. Their number is equal to the number of source links with the maximal number of bigons derived from the basic polyhedra 10∗ and 10∗∗ in the Section 2.5. There are two mutually dual methods for the derivation of fullerenes: (1) edge-coloring of a 3-regular graph, with one colored edge in each vertex; (2) introducing bigons in every vertex of a 4-regular graph. This provides a double check of the obtained results. The duality of these methods is illustrated in the example of two C20 chemical isomers, both derived from the same geometrical dodecahedral form with G = [3, 5] = Ih = S5 of order 120. However, the first has G′ = D5d = [2+, 10] = D5 × C2 of order 20, and the other G′ = [2, 2]+ = D2 of order 4 (Fig. 3.54a,b). In this case, the symmetry of chemical isomers derived by the vertex bifurcation is preserved from their generating basic polyhedra (Fig. 3.54b). For the enumeration of general fullerenes (i.e., source links derived from basic polyhedra in the Section 2.5) we used the Polya Enumeration Theorem (PET), applied to basic polyhedra, knowing their automorphism groups (see

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Fig. 3.54 (a) Two non-isomorphic edge colorings of the regular dodecahedron; (b) 5/6 fullerene derived from 10∗ ; (c) 5/6 fullerene derived from 10∗∗ .

Section 2.5), but its application to 5/6 fullerenes is not possible. The same restriction holds for the other derivation method, because of the condition that in every vertex exactly one edge of a 3-regular graph must be colored. The 3-valent graphs with n < 13 vertices and their edge-colorings producing 4-valent graphs are considered by A.Yu. Vesnin (1991). Similarly, we can prove that 5/6 fullerenes with 22 atoms can not exist, and there are seven 5/6 fullerenes C24 with the same geometrical form and G = D6d = [2+, 12] = D12 (Fig. 3.55). Often, chemical symmetry group G′ is not sufficient for distinguishing chemical isomers. They can be distinguished using polynomial invariants of KL projections (see Section 2.8). 3.3.3

Knot theory and fullerenes

The function fKLfromGraph converts any 4-valent graph into the corresponding alternating KL projection and calculates its Dowker code in the Knotscape format. From this Dowker code, the function fPDataFrom Dow computes P -data. For example, from the graph G = {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {1, 5}, {6, 7}, {7, 8}, {7, 8}, {8, 9}, {9, 10},

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Fig. 3.55

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Seven 5/6 fullerenes C24 with the same geometrical form.

{10, 11}, {10, 11}, {11, 12}, {12, 13}, {13, 14}, {14, 15}, {15, 16}, {16, 17}, {16, 17}, {19, 20}, {17, 18}, {18, 19}, {19, 20}, {6, 20}, {20, 21}, {21, 22}, {22, 23}, {23, 24}, {24, 25}, {25, 26}, {26, 27}, {27, 28}, {28, 29}, {29, 30}, {30, 31}, {31, 32}, {32, 33}, {33, 34}, {34, 35}, {35, 36}, {36, 37}, {37, 38}, {38, 39}, {39, 40}, {21, 40}, {7, 22}, {8, 25}, {10, 26}, {11, 29}, {13, 30}, {14, 33}, {16, 34}, {17, 37}, {19, 38}, {40, 41}, {41, 42}, {42, 43}, {43, 44}, {44, 45}, {45, 46}, {46, 47}, {47, 48}, {48, 49}, {49, 50}, {50, 51}, {51, 52}, {52, 53}, {53, 54}, {54, 55}, {41, 55}, {24, 44}, {27, 45}, {28, 47}, {31, 48}, {32, 50}, {35, 51}, {36, 53}, {39, 54}, {55, 56}, {56, 57}, {57, 58}, {58, 59}, {59, 60}, {56, 60}, {13, 14}, {1, 9}, {1, 9}, {2, 12}, {2, 12}, {3, 15}, {3, 15}, {4, 18}, {4, 18}, {5, 6}, {5, 6}, {21, 40}, {22, 23}, {24, 25}, {26, 27}, {28, 29}, {30, 31}, {32, 33}, {34, 35}, {36, 37}, {38, 39}, {41, 42}, {44, 45}, {47, 48}, {50, 51}, {53, 54}, {55, 56}, {43, 57}, {43, 57}, {46, 58}, {46, 58}, {49, 59}, {49, 59}, {52, 60}, {52, 60}, {23, 42}} of the fullerene C60 , we obtain the Dowker code of the corresponding link in the Knotscape format

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{{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}, {28, 36, 10, 18, 42, 6, 48, 24, 12, 54, 30, 16, 60, 2, 22, 4, 72, 66, 68, 78, 8, 74, 84, 14, 82, 90, 20, 88, 64, 26, 34, 96, 58, 32, 38, 102, 40, 44, 106, 112, 50, 46, 118, 56, 52, 62, 100, 116, 70, 108, 92, 114, 76, 98, 120, 80, 104, 94, 86, 110}} with the carbon rings as the components. Polynomial invariants of KL projections can be applied to alternating KL projections corresponding to fulleren isomers and efficiently used for their recognition. The LinKnot functions JablanPoly and LiangPoly calculate these invariants (see page 312). For example, let us show that two isomers of C20 (Fig. 3.54b,c) are different. After converting their (chemical5 ) Schlegel diagrams into alternating KL diagrams, denoting their generators, and calculating the corresponding projection polynomials, we obtain dD ′ (t) = t20 − 10t18 + 45t16 − 120t14 + 200t12 − 197t10 + 105t8 − 40t6 + 25t4 − 10t2 , dD ′′ (t) = t20 − 10t18 + 45t16 − 120t14 + 208t12 − 250t10 + 217t8 − 130t6 + 49t4 − 10t2 ,

proving their difference. Using the same multivariable invariant for link projections, we can distinguish seven non-isomorphic diagrams obtained from the fullerene C24 (Fig. 3.55). The same results can be obtained using the Liang polynomial. All 4-valent (chemical) Schlegel diagrams of fullerenes can be converted into alternating KL diagrams. For example, two chemical isomers of C20 will give knots, and from 7 isomers of C24 we obtain four knots, one 3component, one 4-component and one 5-component link. Among the links obtained, two of them (3-component and 5-component link) contain a minimal possible component: hexagonal carbon ring. Notice that C60 consists only of regularly arranged hexagonal carbon rings, which is maybe the additional reason for its stability (Fig. 3.56). Therefore, it will be interesting to consider the infinite class of 5/6 fullerenes with this property, called perfect fullerenes. Some perfect fullerenes were modelled with hexastrips6 by P. Gerdes (1998). Similar structures, buckling patterns of shells and spherical honeycomb structures have been considered by different authors (e.g., T. Tarnai (1989)). 5 Schlegel

diagrams with bigons denoting double bonds. tangle” from Gauss’ notebook represents a weaving with hexastrips (Przytycki, 2004, Fig. 3.5). 6 “Framed

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Fig. 3.56

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Derivation of C60 from C20 and carbon rings in C60 .

Fig. 3.57

Non-perfect and perfect (C180 ) fullerenes.

Let us describe how to obtain perfect fullerenes from any 5/6 fullerene. Given 5/6 fullerene in geometrical form (i.e., by a 3-valent graph), apply mid-edge-truncation and vertex bifurcation in all vertices of the obtained triangular faces, transforming them into hexagons with alternating bigonal edges. For example, from C20 , connecting the midpoints of all adjacent

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Fig. 3.58

Fullerenes C70 and C80 .

edges we obtain the 3/5 fullerene covered by connected triangular net and pentagonal faces preserved from C20 . Then we place bigons in all vertices of the truncated polyhedron, to turn all triangles into hexagonal faces. In this way, we derive C60 (in its chemical form) from C20 (Fig. 3.56). Mid-edge-truncation can be applied to any 5/6 (geometrical) fullerene, giving a new perfect (chemical) fullerene formed by carbon rings. Similarly, from a 5/6 fullerene with v vertices we can always derive new perfect 5/6 fullerenes with 3v vertices (Fig. 3.58). Moreover, symmetry of the generating fullerene is preserved. According to the theorem by Gr¨ unbaum and Motzkin (1963), for every non-negative n6 6= 1, there exists a 3-valent convex 5/6 polyhedron having n5 = 12 pentagonal and n6 hexagonal faces. Hence, from the infinite class of 3-valent 5/6 polyhedra with v = 20 + n6 vertices, we obtain the infinite class of perfect fullerenes with v = 60 + 3n6 vertices. Perfect fullerenes satisfy two important stability conditions: (1) the isolated pentagon rule (IPR); (2) the hollow pentagon rule (HPR). The IPR rule means that there are no adjacent pentagons, and HPR means that all pentagons are “holes”, i.e., every pentagon has only external double bonds. The first 5/6 fullerene satisfying IPR is C60 , and it also

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Fig. 3.59

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Fullerenes obtained for different values of n6 .

satisfies HPR. The IPR is well known as the stability criterion: all fullerenes of lower order (less than 60) are unstable, because they do not satisfy IPR. On the other hand, C70 satisfies IPR, but not HPR (Fig. 3.58). The same holds for C80 (Fig. 3.58), which has the same icosahedral geometrical symmetry as C60 , but since HPR can not be satisfied, its symmetry will be reduced due to edge-coloring. Therefore we conclude that only perfect fullerenes with G = G′ = [3, 5] = Ih = S5 , satisfying both IPR and HPR, are C60 , C180 , C240 , etc. We need also to notice that for n6 = 0, 2, 3 there are always exactly one 3-valent 5/6 polyhedron (i.e., the geometrical form of C20 , C24 , C26 ), but for some larger values (e.g. n6 = 4, 5, 7, 9) there are several geometrical isomers of generating fullerenes, and consequently, the same number of derived perfect fullerenes (Fig. 3.59). Hence, considering fullerene isomers, we can distinguish geometrical isomers, that

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is, different geometrical forms of some fullerene treated as 3-valent 5/6 polyhedra, and chemical isomers– different arrangements of double bonds, obtained from the same 3-valent graph by its edge-coloring. 3.3.4

Nanotubes, conical and biconical fullerenes and their symmetry

In this section, Bohm symbols will be used for denoting different categories of symmetry groups, (Bohm and Dornberger-Schiff, 1966). The first subscript n in a symbol Gnst... represents the maximal dimension of space in which the transformations of the symmetry group act, while the following subscripts st . . . represent the maximal dimensions of subspaces remaining invariant under the action of transformations of the symmetry group, which are properly included in each other. General fullerenes belong to the category of point symmetry groups G30 . The category G30 consists of seven polyhedral symmetry groups without invariant planes or lines: [3, 3] or Td , [3, 3]+ or T , [3,4] or Oh , [3, 4]+ or O, [3+, 4] or Th , [3,5] or Ih , [3, 5]+ or I, and from seven infinite classes of point symmetry groups with the invariant plane (and the line perpendicular to it in the invariant point): [q] or Cqv , [q]+ or Cq , [2+, 2q+] or S2q , [2, q+] or Cqh , [2, q]+ or Dq , [2+, 2q] or Dqd , [2, q] or Dqh , belonging to the subcategory G320 (Coxeter and Moser, 1980). The point symmetry groups G30 were mentioned when we were talking about symmetry of KL diagrams (see page 63). For the groups of the subcategory G320 , in the case of rotations of order q > 2, the invariant line (i.e., the rotation axis) may contain 0, 1 or 2 vertices of a general fullerene. Therefore, from the topological point of view, among all general fullerenes with a geometrical symmetry group G belonging to G320 we can distinguish cylindrical fullerenes (nanotubes), conical and biconical ones. Symmetry group of polyhedral 5/6 fullerenes G can be only [3, 3] (Td ), [3, 3]+ (T ), [3, 5] (Ih ), or [3, 5]+ (I), since their topological structure (n5 = 12) is incompatible with the octahedral symmetry group [3, 4] (Oh ) or its polyhedral subgroups. In the case of nanotubes (or cylindrical fullerenes) we have infinite classes of 5/6 fullerenes with the geometrical symmetry group [2, q] (Dqh ) and [2+, 2q] (Dqd ), and the same chemical symmetry. The first infinite class of cylindrical nanotubes C30 , C50 , C70 ,. . . with G = G′ = D5h is obtained from a cylindrical 3/4/5 four-valent graph with two pentagonal bases, 10 triangular and 5(2k + 1) quadrilateral faces (k = 0, 1, 2, . . .) and with the same symmetry group (Fig. 3.60). The infinite

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Fig. 3.60

Fig. 3.61

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Cylindrical nanotubes.

C70 isomers with the same geometrical structure.

class of nanotubes C30 , C50 , C70 ,. . . is obtained by the vertex bifurcation, preserving symmetry, where C70 is the first nanotube satisfying IPR. The geometrical structure of C70 admits different edge colorings (i.e., chemical isomers). Starting from arbitrary two chemical isomers and reducing the length of bigon chains we obtain different source links. The example of two

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Fig. 3.62

Fig. 3.63

Isomers of C70 .

Fullerenes C36 , C60 , C84 .

different C70 isomers with the same geometrical structure (Fig. 3.61) and the same G and G′ , shows that symmetry is not sufficient for distinguishing fullerene isomers, so we need additional tools (see the Subsection 3.2.3). In the same way, from 4-valent graphs with two hexagonal bases, 12 triangular and 6(2k + 1) quadrilateral faces (k = 0, 1, 2, . . .) we obtain the infinite class

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of fullerenes C36 , C60 , C84 ,. . . with the symmetry group G = G′ = D6h (Fig. 3.63).

Fig. 3.64

Fig. 3.65

Fullerenes C20 , C40 ; C24 and C48 .

Fullerene C40 obtained from two C20 .

The next series of symmetry groups [2+, 2q] (Dqd ) with q = 5, 6 can be obtained in the same way, from 4-valent graphs with q-gonal bases, 2q triangular and 2kq quadrilateral faces (k = 1, 2, . . . for q = 5; k = 0, 1, 2, . . . for q = 6) (Fig. 3.64). As the limiting case, for q = 5 and k = 0, we obtain C20 with the icosahedral symmetry group G and G′ = D5d . C20 can be used as a building block of the whole class of nanotubes C40 , C60 , C80 ,. . . with G = D5d , (Fig. 3.65). Nanotubes C48 , C72 , C96 ,. . . can be obtained in the same way, by “gluing” the pentagonal bases, from the fullerene C24 (q = 6, k = 0) (Fig. 3.65). The geometrical structure of the nanotube class with G = Dqd (q = 5, 6) permits the edge coloring that preserves symmetry, so there always exist isomers with G = G′ . If 3-rotation axis contains the opposite vertices of a fullerene, we have

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Fig. 3.66

Fullerenes C26 and C56 .

biconical fullerenes (e.g., C26 , C56 ) with G = D3h , G = D3d , respectively (Fig. 3.66). After the edge coloring, symmetry must be disturbed, and for the biconical fullerenes G′ is always a proper subgroup of G. For example, for C26 (Fig. 3.66), G = D3h , G′ = C2v .

Fig. 3.67

Fullerene C42 .

Fullerene representatives of other symmetry groups from the category G320 can be constructed in the same manner: biconical C32 with G = D3 , biconical C38 or conical C34 with G = C3v , conical C46 with G = C3 (Boo, 1992), or the infinite class of cylindrical fullerenes C42 , C48 , C54 ,. . . with G = D3 (Fig. 3.67). In general, edge coloring of 3-valent graphs changes symmetry of all conical or biconical fullerenes mentioned, so their geometrical symmetry is always higher than the chemical.

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All nanotubes, conical and biconical fullerenes described in this subsection can be analyzed by converting them to corresponding alternating KLs, where the geometrical symmetry properties can be related with the corresponding symmetries of KLs. 3.3.5

Fullerenes on other surfaces

Different regular homoatomic carbon plane nets are discussed by T. Balaban (1989). They can be derived in the same way as the general fullerenes: by introducing bigons in the vertices of 4-valent graphs or by an edge-coloring of a 3-valent graph, resulting in a 4-valent graph. For example, we can start from the square regular tessellation {4, 4} (Fig. 3.68a), Archimedean tiling (3, 6, 3, 6) (Fig. 3.68b) or 2-uniform tiling (3, 42 , 6; 3, 6, 3, 6) that are all 4-valent (Gr¨ unbaum and Shephard, 1986), and place bigons in their vertices, or from the regular tiling {6, 3} that is 3-valent and color its edges. Similarily, perfect plane nets can be derived from arbitrary 3-valent tilings.

Fig. 3.68

Nets derived from 4-regular tilings.

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Fig. 3.69

Fig. 3.70

Fig. 3.71

Tessellation on torus.

Perfect fullerene with 8-gonal holes on a double torus.

The infinite perfect 6/7 fullerene in H 2 with heptagonal holes.

The necessary condition for general fullerenes on other surfaces follows from the Euler theorem v − e + f = 2 − 2g, where g is the genus of the surface. Since genus of torus is g = 1, under 5/6 restriction we obtain that for 3-valent graphs number of pentagonal faces is n5 = 0. Hence, we can only get the regular tessellation {6, 3}, consisting of b2 + bc + c2 hexagons

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(b ∈ N , c ∈ N ) (Coxeter and Moser, 1980). This tessellation is obtained by identifying opposite sides of the rectangle (Fig. 3.69). Perfect hexagonal fullerenes on torus can be derived from a finite {6, 3} tessellation. The proposed approach can be extended to double, triple, etc. torus with g = 2, 3, . . . Similar transformations of carbon nets from one surface to the other (e.g., from a plane to a cylinder, and then to torus) can perhaps explain how certain fullerenes are formed and describe the way they grow (Kroto, 1989). Allowing heptagons or octagons for faces, from the relationship 2e = 3v and the Euler formula, it follows that n5 − n7 − 2n8 = 12(1 − g). For a sphere without octagons, n5 − n7 = 12, and for a torus without octagons n5 = n7 (Mackay and Terrones, 1993). General fullerenes with a higher degree of symmetry can be obtained from various vertex-transitive structures (e.g., uniform polyhedra, stellated regular and semi-regular polyhedra or infinite polyhedra) (Mackay and Terrones, 1993). Perfect fullerenes with q-gonal holes on a double torus (g = 2) (Fig. 3.70, q = 8 (Bilinski, 1985)) can be derived from different uniform 4-valent polyhedra of the type (3, q, 3, q) (q = 7, 8, 9, 10, 12, 18), using regular vertexbifurcation of triangular faces, which transforms them into hexagons. In the same way, the uniform tessellations of the type (4, q, 4, q), (q = 5, 6, 8, 12) or (5,10,5,10) on a double torus may result in different finite general fullerenes. Interesting classes of infinite general fullerenes with non-euclidean plane symmetry groups can be derived from the tessellations of the hyperbolic plane H 2 . For example, from the uniform tessellation (3,7,3,7) we derive the infinite perfect 6/7 fullerene in H 2 with heptagonal holes (Fig. 3.71) (Mackay and Terrones, 1993).

3.4

KLs and logic

Building set theory on intuitive concepts of membership and collection leads to paradoxes due to misussage of self-membership (or self-reference). One of them is the famous Russell paradox. The Russell set is defined to be the set of all sets that are not members of themselves. X is a member of X exactly when X is not a member of X (Kauffman, 1995). In the language of mathematical logic, this means that pp = 1, where p denotes ¬p, and 1 stands for “True”. The sentence “This statement is false” is an example of a similar para-

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dox based on self-reference, which implies the logical equation p = p (Epimenides paradox). Whitehead and Russell (1927) posed the problem in terms of selfmembership (or self-reference), and avoided it by prohibiting mixing different levels of discourse. In G¨odel-Bernays set theory these paradoxes are solved by making a distinction between set and class, i.e., by introducing a hierarchy of levels. A class is a set if it is a member of another class. In this system the Russell class is R = {X|X ∈ / Xand X is a set}, so R is a class, but not a set. This concept accepts self-membership, or self-reference. Namely, a self-reference can be used as a tool for creating interesting ascending chains of membership. Definition 3.7. Let us denote the empty set by a line segment − . A finite expression E in line segments is well-formed if: (1) E is empty set, or (2) E = F G, where F and G are well-formed. A finite ordered multi-set S is an expression in the form S = T , where T is any well-formed expression. It follows that T = A1 A2 . . . An , where n ∈ N , and each Ai (i = 1, 2, . . . , n) is a finite ordered multi-set. The terms Ai are the members of S. In this way, we obtain different multi-sets, e.g., S = − − − − , where the members of S are −, −, − − , respectively. The term S can be encoded by a sequence 0 0 1 0 1 0 0 1 0 1 1 1, where 0 stands for left, and 1 for right ends of line segments. Somewhat more complex encoding is shown in Fig. 3.72.

Fig. 3.72

Encoding a bracket arrangement by 0-1 code.

Two finite ordered multi-sets are equal iff they have the same members in the same order, i.e., iff their 0-1 encodings are identical. An isomorphic

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visual interpretation of S can be obtained by taking rectangles instead of line segments (Fig. 3.73). Ordered finite multi-sets are isomorphic to the class of rooted planar tries, by graphical dualities illustrated in Fig. 3.74. A depth of a level to which a member belongs, directly visible from S, can be obtained by counting nodes from the root of tree.

Fig. 3.73

A visual interpretation of S by rectangles (Kauffman, 1994).

Overlining can be extended to infinite sets S represented by line segment arrangements which satisfy two rules: (1) each arrangement S has a single top line segment, and (2) the collection of line segments overlined by that top line segment is a disjoint union of the members of S. Any finite or infinite collection of line segments in which there is no ambiguity in any pair of line segments that one is overlined by the other or not is a form. From every form S we can obtain S by the operation called overlining, and for every two forms F1 and F2 we can define their product (or juxtaposition) as F1 F2 . For example, an infinite collection of forms can be created as − , − , − − , − − − , . . . Some interesting infinite forms can be created from recursive systems of equations. For example, a successive use of rules A = − B, B = A gives − −

−

−

− −

−

A= -... A= B= A= The simplest recursive form F = F , beginning from F = − , results in the sequence of natural numbers − , − , − , − , . . . Recursive form F = F F can be called Fibonacci form because the number of line segments at depth n is nth Fibonacci number. If F (n) denotes the number of line segments (or number of nodes in the corresponding rooted tree) at depth n of the form F , for any two forms F1 and F2 holds:

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Fig. 3.74

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(a) Line segment interpretation of Fibonacci form; (b) rooted tree as its dual.

(1) (F1 F2 )(n) = F1 (n) + F2 (n) and (2) F (n) = F (n − 1). For the Fibonacci form F = F F , F (0) = F (1) = 1 and F (n) = F (n − 2)+F (n−1). Defining the growth rate as µ(F ) = limn→∞ FF(n+1) (n) , we obtain √

that the growth rate of the Fibonacci form is the golden ratio ϕ = 5−1 2 Many fractals, in particular L-systems (Lindenmayer systems), can be described by recursive forms. For example, the structure of Koch fractal can be expressed by a recursive form K = KKKK. Definition 3.8. A lattice is an algebraic structure (L, ∧, ∨) consisting from a set L and two algebraic operations ∧ and ∨ (“meet” and “join”, or “and” and “or”), such that for any a, b, c ∈ L hold: (1) a ∧ a = a, a ∨ a = a (idempotent laws),

(2) a ∧ b = b ∧ a, a ∨ b = b ∨ a (commutativity laws),

(3) (a ∧ b) ∧ c = a ∧ (b ∧ c), (a ∨ b) ∨ c = a ∨ (b ∨ c) (associativity laws),

(4) a ∨ (a ∧ b) = a, a ∧ (a ∨ b) = a (absorption laws).

Definition 3.9. A Boolean algebra is a lattice (B, ∧, ∨) that satisfies four additional properties:

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(1) there exists an element 0 ∈ B that a ∨ 0 = a for all a ∈ B (lower bound), (2) there exists an element 1 ∈ B that a ∧ 1 = a for all a ∈ B (upper bound), (3) for all a, b, c ∈ B, (a ∧ b) ∨ c = (a ∨ c) ∧ (b ∨ c) (distributivity law)

(4) for every a ∈ B there exists an element ¬a in B such that a ∧ ¬a = 0 and a ∨ ¬a = 1 (complement law).

Instead of using classical Boolean algebra with only two discrete values 0 and 1, we can consider more general algebra whose values belong to the continuous set [0, 1], and construct a new model of polyvalent Boolean algebra called the square-free polynomial model. We define p ∧ q and ¬p in the following way: p ∧ q = pq, ¬p = 1 − p. Definition 3.10. A square-free polynomial is any polynomial in which a degree of each variable is 1. Introducing the idempotent law p2 = p for all variables, we model a polyvalent Boolean algebra on the continuous interval [0, 1], with squarefree polynomials, standard polynomial multiplication, and idempotency. This model is very similar to a Boolean ring, but ¬p is defined as ¬p = 1−p, and not as ¬p = 1 + p. In this way the structure of standard truth tables is preserved and extended to the set [0, 1]. A concept of square-free polynomials is implicitly given in the original works of G. Boole (2003). De Morgan laws imply p ∨ q = p + q − pq, and we can check if our model is consistent with regard to the whole set of axioms. The following equalities are obtained by straightforward computations: a ∧ a = a2 = a, a ∨ a = a + a − a2 = a + a − a = a (idempotency), a ∧ b = ab = ba = b ∧ a, a ∨ b = a + b − ab = b + a − ba = b ∨ a (commutativity), (a ∧ b) ∧ c = (ab)c = a(bc) = a ∧ (b ∧ c), (a ∨ b) ∨ c = (a + b − ab) + c − (a + b − ab)c = a + b − ab + c − ac − bc + abc = a + b + c − bc − ab − ac + abc = a + (b + c − bc) − a(b + c − bc) = a ∨ (b ∨ c) (associativity), a ∨ (a ∧ b) = a + ab − a2b = a + ab − ab = a, a ∧ (a ∨ b) = a(a + b − ab) = 2 a + ab − a2 b = a + ab − ab = a (absorption), a ∨ 0 = a + 0 − a0 = a, a ∧ 1 = a1 = a (lower and upper bound), (a ∧ b) ∧ c = (ab)c = a(bc) = (a ∨ c) ∧ (b ∨ c) (distributivity), a ∧ ¬a = a(1 − a) = a − a2 = a − a = 0, a ∨ ¬a = a + (1 − a) − a(1 − a) = a + 1 − a − a + a2 = a + 1 − a − a + a = 1 (complement).

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Hence, our model is compatible with Boolean logic. In order to simplify notation, we denote ¬p by p, p ∧ q by pq, and continue to work in the line segment notation. This is a model of SpencerBrown calculus of indications, based on one symbol q, where this symbol is replaced with a single line segment (Spencer-Brown, 1969). Hence, two forms of equality will be expressed as: (1)

−−

(2)

−

=

=

−

(form of condensation), and

(form of cancellation).

An unmarked state appears in the form of cancellation. These two rules can be replaced with a a = a and a = a. Here are examples of visual proofs in calculus of expressions that Spencer-Brown calls the primary arithmetics, expressed in the line segment notation: e=

−−

−

−

−

(1) =

−

−

−

−

(2) =

−

−

− (2) =

−

− (2) = − (2) =

−

Denoting ¬p by p, and p ∧ q by pq, using the line segment notation and square-free polynomials, and interpreting unmarked state as 0, various tautologies are almost obvious. For example, p p = (1 − p)p = p − p2 = p − p = 0, pr qr = 1 − (1 − pr)(1 − qr) = pr + qr − pqr2 = pr + qr − pqr = (p + q − pq)r = (1 − (1 − p)(1 − q))r = p qr. The proposed method is very powerful, used even in proofs of more complex tautologies, e.g., a crosstransposition q r p r x r y r = r pq rxy . The left side gives: qrprxryr = 1 − (1 − (1 − q)(1 − r))(1 − (1 − p)(1 − r))(1 − (1 − x)r)(1 − (1 − y)r) = 1 − pq − pr − qr + 4pqr − r2 + 3pr2 + 3qr2 − 6pqr2 + 2r3 − 3pr3 − 3qr3 + 4pqr3 − r4 + pr4 + qr4 − pqr4 − pqrx − pr2 x − qr2 x + 3pqr2 x − r3 x + 2pr3 x + 2qr3 x − 3pqr3 x + r4 x − pr4 x − qr4 x + pqr4 x − pqry − pr2 y − qr2 y + 3pqr2 y − r3 y + 2pr3 y + 2qr3 y − 3pqr3 y + r4 y − pr4 y − qr4 y + pqr4 y − pqr2 xy − pr3 xy − qr3 xy + 2pqr3 xy − r3 xy + pr4 xy + qr4 xy − pqr4 xy = 1 − pq + pqr − rxy.

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The right side results in: r pq rxy = (1 − (1 − r)pq)(1 − rxy) = 1 − pq + pqr − rxy + pqrxy − pqr2 xy = 1 − pq + pqr − rxy,

so this proves the tautology. Duality (De Morgan laws) holds for ∧ and ∨, after replacing 0 by 1 and vice versa, and each variable p by its inverse p. For example, a dual tautology for the crosstransposition q∨r∨p∨r∨x∨r∨y∨r = r∨p∨q∨r∨x∨y

can be proved in the similar way, working with p = 1−p and p∨q = p+q−pq. All proofs and calculations hold in the continuous set [0, 1], i.e., in the polyvalent Boolean logic. In order to simplify further computations, we can calculate square-free polynomials corresponding to other logic operations and work with them in the same way as before. For example, p ⇒ q = 1 − p + pq, p ⇔ q = 1 − p − q + 2pq, p ⊻ q = p + q − 2pq, etc. The Sheffer stroke NAND, p|q = pq, which can be algebraically expressed as p|q = 1 − pq will be the most interesting for future use in logical gates. Operations (¬, ∧), or (¬, ∨) are a base of Boolean polyvalent logic: all logical operations can be expressed in terms of (¬, ∧), or (¬, ∨). Sheffer operation NAND (| or ↑ in his original notation) and its dual, Lukasiewicz’s operation NOR (↓), are single-operation logical bases. After computing square-free polynomial P corresponding to a complicated logical expression L, we are interested to find from P the logical expression L′ = L′ (P ), which is the simplification of L. It is sufficient to take all possible values of variables belonging to P from the discrete set {0, 1} and write the corresponding conjuctive or disjunctive normal form (CNF or DNF). For example, from P = 1 − p + pq, by taking (p, q) values from the set {(0, 0), (0, 1), (1, 0), (1, 1)} we obtain DNF ¬p ∨ q. 3.5

Waveforms

After interpreting the marked state − as 1, and unmarked state as 0, we can introduce a waveform arithmetic, based either on the discrete set {0, 1}, or continuous set [0, 1]. Treating f = f as a recursive form defined on the set {0, 1}, and taking successive substitutions of f in its equivalent form

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f = 1 − f , we obtain a sequence with the period two, 0,1,0,1,0,1,0,1 . . .. On the other hand, the equation f = 1 − f has the fixed point f = 21 of the recursion f = f in the continuous set [0, 1]. It is a solution of Liar paradox in polyvalent logic. In order to generate sequences of period greater than 2, it is necessary to apply recursions on more then one variable. The recursion T (x, y) = (y, x y) produces a sequence with the period three (0,0),(1,1),(0,1),(0,0),(1,1),(0,1). . . in the discrete set {0, 1}2 (Fig. 3.75). Solving the system of equations x = 1 − y and y = 1 − (1 − x)y obtained from x = y and y = x y, gives the fixed point (x, y) = (1 − ϕ, ϕ) on the [0, 1]2 , where ϕ denotes the golden ratio.

Fig. 3.75

Three periods of the sequence (0,0),(1,1),(0,1). . ..

Duality holds for periodic sequences generated by recursions, and for their fixed points as well. Solving the dual recursion T (x, y) = (y, x ∨ y) for a fixed point, we obtain the result (x, y) = (ϕ, 1 − ϕ), which can also be obtained from the previous result by duality. Similarly, the recursion T (x, y, z) = (xz, xyz, xyz x y z) generates a periodic sequence with the period five, (0,1,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1), (0,1,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1), (0,1,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1) . . . on a discrete set {0, 1}3. Its fixed point (x, y, z) = √ √ √ 3+ 2 3− 2 ( 7 , 2 , 2 − 2) that belongs to the [0, 1]3 is obtained by solving the system of equations x = 1 − xz, y = 1 − xy + xyz, z = 1 − y + xy + yz − 2xyz. The dual recursion T (x, y, z) = (x ∨ z, x ∨ y ∨ z, x ∨ y ∨ z x ∨ y ∨ z) generates again a dual periodic sequence with the period five on {0, 1}3. Solving the system of equations x = 1 − x − z + xz, y = z − xz − yz + xyz, z = 1 − x − y √+ xy 2xz + yz − 2xyz we obtain the dual fixed point √− z + 4− 2 2−1 √ (x, y, z) = ( 7 , 2 , 2−1), that belongs to [0, 1]3 . In some cases, a set of fixed points of a recursive operator can be the whole interval [0, 1]. For example, the set of fixed points of the operator T (x, y) = (y, x) is (x, 1 − x),

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x ∈ [0, 1]. This operator produces a stable state for every y = 1 − x, and a periodic sequence with the period 2 otherwise. L.H. Kauffman and F.J. Varela (1980) described an algorithm for producing recursive operators which generate periodic sequences of a desired period. Let a periodic sequence of period p with n variables be given, where 2n−2 < p ≤ 2n . The condition 2n−2 < p implies no variable is constant. Usually, we are trying to express all periodic sequences with a minimal number of variables n. A period of such a sequence can be represented in the form of an n × p array. For example, the sequence of period p = 5 with n = 3 variables considered before, can be represented as xyz 010 110 101 011 111 For every 0 in h-column we are associate its preceding row to h, where h is a variable denoting the head of a column (h = x, y, z). Notice that the last column is the preceding for the first, since we are working with cyclic order. In the x-column, 0 appears in the first and fourth row, so the fifth and third row will be associated to x, i.e., x = (1 1 1)(1 0 1). In the y-column, 0 appears in the third row, so we associate the second row to y and obtain y = (1 1 0). In the z-column 0 appears in the first and second row, so we associate the last and first row to z and obtain z = (1 1 1)(0 1 0). Replacing every 1 by h, and every 0 by h, and overlining each n-tuple, we obtain x = xyz xyz, y = xyz, z = xyz xyz, which gives the recursive operator T (x, y, z) = (xyz xyz, xyz, xyz xyz). Using the square-free polynomial method, we reduce obtained terms, and look for fixed points. Hence, T (x, y, z) = (xyz xyz, xyz, xyz xyz) = T (1√− xz, √ 1 − xy + xyz, 1 − y + xy + √ yz − 2xyz). Its fixed point (x, y, z) = ( 3+7 2 , 3−2 2 , 2 − 2) which belongs to [0, 1]3 we obtain in the same way as before, by solving the system of equations x = 1 − xz, y = 1 − xy + xyz, z = 1 − y + xy + yz − 2xyz. Now that we have an algorithm for deriving all sequences with a given period p, we need tools for distinguishing different sequences: nonisomorphic, not mutually reverse, and not mutually dual sequences. It is obvious that two sequences given by a cyclic permutation of rows in the array will be equal. Also, sequences that are the same up to a permutation of variables can be identified. Two dual sequences, where one

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can be obtained from the other by replacing 0 by 1 and vice versa are equivalent as well. Cyclically equivalent sequences will have the same fixed points; sequences with permuted variables will have fixed points permuted in the same way; dual sequences will have dual fixed points from the set [0, 1]n , where n is the number of variables. Representatives of equivalence classes of periodic sequences are called basic sequences. For p = 2 and n = 1 we have one basic sequence 0,1,0,1. . . defined by the operator T (x) = x, with no fixed points on a discrete set {0, 1} and one fixed point x = 12 on the interval [0, 1]. This sequence embodies the Liar paradox. For p = 2 and n = 2 we obtain two basic sequences: (0,0),(1,1). . ., defined by the operator T (x, y) = (xy, xy) = (1 − xy, 1 − xy), with the fixed point (x, y) = (ϕ, ϕ) on [0, 1]2 , and (0,1),(1,0). . ., defined by T (x, y) = (xy, xy) = (1 − x + xy, 1 − y + xy), with the fixed point (x, y) = (1, 1). For p = 3, the minimal number of variables n is n = 2. From the set (0,0), (0,1), (1,0), (1,1) we can produce 24 different sequences of period 3. Each of them satisfies the necessary condition that no variable is constant. Taking one representative from each equivalence class, we obtain three basic sequences of period 3: (0,0),(0,1),(1,0). . ., (0,0),(0,1),(1,1). . ., and (0,0),(1,1),(0,1). . . The first is defined by the operator T (x, y) = (x y xy, xy xy) = (y, xy xy) = T (y, 1 − x − y + 2xy), the second by T (x, y) = (xy x y, xy) = T (x + y − 2xy, 1 − xy), and the third by T (x, y) = xy xy = y xy. The fixed points of the first are (x, y) = (1, 1) and (x, y) = ( 12 , 12 ), of the second (x, y) = ( 12 , 23 ), and of the third (x, y) = (1 − ϕ, ϕ). For p = 4, the minimal number of variables n is n = 2, and we obtain three basic sequences: (0,0),(0,1),(1,0),(1,1). . . with T (x, y) = (xy xy, xy xy) = (xy xy, y) = (x + y − 2xy, 1 − y); (0,0),(0,1),(1,1),(1,0). . . with T (x, y) = (x y xy, xy xy) = (y, x) = (y, 1 − x); (0,0),(1,1),(0,1),(1,0). . . with T (x, y) = (xy xy, xy xy) = (x, xy xy) = (1 − x, 1 − x − y + 2xy). All of them can not be stabilized for any pair of values from the discrete set {0, 1}2, and in [0, 1]2 they have the same fixed point ( 12 , 12 ). The same line of reasoning can be applied to basic periodic sequences with higher periods and compute their number. For example, for n = 3 there are 45 sequences of the period p = 4, 160 for p = 5, 382 for p = 6, 840 for p = 7, 840 for p = 8, etc. T 2 (x) = xm n m n = xm n = T (x) can be obtained by iterating from an infinite self-referential form T (x) = xm n with two fixed parameters m

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and n. Moreover, we obtain T (x) = T 2 (X) = T 3 (x) = . . ., which implies x⋄ = . . . m n m n. This form contains a copy of itself, so x⋄ = x⋄ m n. As the result, every point 1−n+x0mn, where x0 is an initial state (x0 ∈ [0, 1]), will be the fixed point of T . The LinKnot function fBoolean calculates square-free polynomial of a given logic formula. The function fKauffAlg produces all different periodic sequences of the period p with n variables according to Kauffman algorithm. The function fDiffSeq produces all different periodic sequences of the period p with n variables, giving as the output their complete list, where every periodic sequence is given by one period, Kauffman code, and the corresponding square-free polynomials. The function fBalanced calculates stable (balanced) states of a given periodic sequence.

3.6

Knot automata

A class of circuit automata based on knot theory is considered by Kauffman (1994). The basic circuit element for these automata has an equation of the form z = xRy or z = xLy with box depictions as shown in Fig. 3.76. These equations correspond to “left” or “right” crossings of a KL, i.e., to the crossings with the sign 1 or −1, respectively. Translated into equations, Reidemeister moves (Fig. 3.77) become: (1) aRa = a, aLa = a; (2) (aRb)Lb = a, (aLb)Rb = a; (3) (aRb)Rc = (aRc)R(bRc), (aLb)Lc = (aLc)L(bLc). The resulting algebraic structure, a quandle (Definition 2.5), can be obtained from different solutions of the equations defined by Reidemeister moves. The simplest example of a quandle is the structure aRb = aLb = 2b − a, where a and b are elements of an additive abelian group G. In the case of a trefoil knot automaton (Fig. 3.78) the feedback loop forces the conclusion 3(b − a) = 0. Hence, 3 must divide the order of G in order for the trefoil automaton to have any balanced states. If G = ({0, 1, 2}, +3) is the abelian group with addition modulo 3, for a = 1, b = 2 we obtain a stable state of the trefoil automaton, i.e., the three-coloring of a trefoil. It distinguishes a trefoil from the unknot, or from the figure-eight knot. If the values of the knot automaton lie in a module over the ring Z[t, t−1 ], then for aRb = ta + (1 − t)b, aLb = sa + (1 − s)b with s = t−1 we obtain the

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Fig. 3.76

Fig. 3.77 1994).

Right and left crossing relations (Kauffman, 1994).

Reidemeister moves translated to the language of knot automata (Kauffman,

Alexander polynomial of a KL. The quandle considered earlier, defined by aRb = aLb = 2b − a, is a particular case (t = −1) of this general solution. In a digital circuit model the basic element is a NAND gate, or a simple inverter. For input p it produces p, and p1 p2 . . . pn gives as the output p1 p2 . . .. NOR gates work in the same way, producing p for p, and p1 ∨ p2 ∨ . . . for p1 ∨p2 ∨. . .. In a circuit diagram, a state is the edge-coloring of the directed graph in which the colors are chosen from the discrete set {0, 1}. All edges emanating from a given inverter have the same color in a given state. If z = p1 p2 . . . denotes the equation that defines the operation of a given inverter, a state is called balanced if the equation z = p1 p2 . . . is satisfied at every inverter in the diagram. For example, the circuit defined by equations x = y and y = x has exactly two balanced states in the dis-

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Trefoil knot automaton (Kauffman, 1994).

crete set {0, 1}. In general, if we take values from continuous interval [0, 1], all the values (x, y) = (x, 1 − x) (x ∈ [0, 1]) give balanced states of this circuit. A transition consists of reassigning the value of z for the outgoing edges z of one inverter that is unbalanced. Transition may or may not result in a balanced state (Kauffman, 1994). The equation x = x describes the circuit that embodies the Liar paradox. In bivalent Boolean logic, it has no solutions for a balanced state, so it demands a polyvalent logic in order to achieve a stable state. Such a solution is x = 12 . In a similar way, the automaton defined by equations x = y, y = x y (Fig. 3.79) has no stable states for the discrete values from the set {0, 1}, yet stabilizes for (x, y) = (1 − ϕ, ϕ) from the continuous set [0, 1]2 .

Fig. 3.79

Automaton x = y, y = x y.

A reductor with 6 gates and 13 leads proposed by G. Spencer-Brown is an example of automaton with an input z to the system (Fig. 3.80).

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For z = 0 it has two balanced states for (a, b, c, d, e, f ): A = (1, 1, 0, 1, 0, 1) and C = (1, 1, 1, 0, 1, 0), and two balanced states B = (1, 0, 1, 0, 1, 1) and D = (0, 1, 0, 1, 1, 1) for z = 1. The alternating sequence of input values z = 0, 1, 0, 1 . . . produces a sequence of states A, B, C, D, A, B, C, D . . . of the period 4. In the extended set [0, 1] for z = 0 the system has an infinite number of stable points determined by the parametric equation (a, b, c, d, e, f ) = (1, 1, c, 1 − c, c, 1 − c), and for z = 1 we have one additional fixed point (a, b, c, d, e, f ) = (ϕ, ϕ, ϕ, ϕ, ϕ, ϕ). In general case, we can take any z ∈ [0, 1] as an input z and obtain a system of equations with two parameters z and c. This system has a continuum of solutions. Kauffman (1994) conjectured that a determinate (asynchronous) reductor with less than six inverters does not exist.

Fig. 3.80

Minimal reductor with 6 gates.

Every circuit that has more then one stable state in the discrete set of values {0, 1}, has a continuum of stable states (i.e., fixed points) in [0, 1], defined by resulting parametric equations. On the other hand, a circuit with no stable states in the set {0, 1}, always has a stable state in the extended set [0, 1], i.e., in polyvalent logic. The principle of duality holds for automata with NAND and NOR gates: a system remains unchanged with regard to stable states if all NAND gates are replaced by NOR gates or vice versa. This holds also for automata that contain NAND and AND, or NOR and OR gates. The LinKnot function fAutoSigInp calculates stable states of an automaton given by a list of outgoing edges, signs of vertices, and inputs in vertices. For vertices with the sign 1 the operation NAND is used, while for vertices with sign −1 the operation AND. The result is a list of edge colorings corresponding to stable states and a list of stable states according to

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gates (see Kauffman, 1994), with values taken from the discrete set {0, 1}. If the list of signs is empty, it is treated as (1, ..., 1), and computations are made with NAND gates in all vertices of a graph. Inputs in vertices can be included by giving the corresponding list of vertices. Because only an input 0 produces a change in NAND or AND gates, all inputs are treated as 0. The function fAutoKL calculates stable states of an automaton obtained from a KL given in Conway notation, followed by a list of signs of vertices, and inputs in vertices. The logical operation NAND is used in vertices with the sign 1, and AND in vertices with the sign −1. The result is the oriented graph corresponding to a given KL, the list of edge colorings corresponding to its stable states, and the list of stable states ordered according to gates (see Kauffman, 1994). If the list of signs of a given KL is empty, the original list of the signs of a given KL is used for the computation. Inputs in crossings are the same as for the function fAutoSigInp.

Fig. 3.81

Figure-eight knot automaton.

The function fAutoKL can be used for analyzing the behavior of KLs with regard to stable states, this means, their edge colorings with two colors 0 and 1, compatible with the requirements of NAND and AND logical gates. For example, an automaton obtained from the figure-eight knot 2 2 (or 41 ) with the signs 1 in all crossings (i.e., with NAND gates in all crossings), defined by equations x = yz, y = xu, z = yu, u = xz, has no stable states in the discrete set {0, 1}. The corresponding operator T (x, y, z, u) = (yz, xu, yu, xz) generates the periodic sequence (0,0,0,0),(1,1,1,1). . . for (x, y, z, u) ∈ {(1, 1, 1, 1), (1, 1, 0, 0), (0, 0, 1, 1)} and for every other initial

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state (x, y, z, u) with more than two zeros. In all other cases, it generates the sequence of the period eight (0,1,1,0),(1,1,1,1),(0,1,0,1),(1,1,0,1), (1,0,0,1),(1,0,1,1),(1,0,1,0),(1,1,1,0). . . Using the original signs of the knot 2 2 (this means, NAND gates in the crossings with the sign 1, and AND gates in the crossings with the sign −1), we obtain the stable state (x, y, z, u) = (1, 1, 0, 0) (Fig. 3.81a). The mirror image −2 − 2 of the same knot gives the same result, showing that the figure-eight knot is achiral. For a figure-eight knot with all signs equal to 1, we have the stable state (x, y, z, u) = (ϕ, ϕ, ϕ, ϕ) on the set [0, 1]4 (Fig. 3.81b), where ϕ denotes golden ratio. A figure-eight knot with the signs −1 in all vertices (this means, with AND gates in all vertices) has two stable states, where all edges are labelled by 0, or all edges are labelled by 1. On the other hand, a trefoil 3 has no stable states with NAND gates in all crossings, but its mirror image −3 with AND gates in all crossings has two stable states from the discrete set {0, 1}, where all edges are colored by 0, or by 1. This shows that the trefoil knot is chiral.

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Tarnai, T. (1989) Buckling Patterns of Shells and Spherical Honeycomb Structures, in Symmetry: Unifying Human Understanding 2, (Ed. I. Hargittai), 639–652, Pergamon Press, Oxford, New York. Thistlethwaite, M.B. (1985) Knots tabulations and related topics, in Aspects of Topology, (Eds. I.M. James and E.H. Kronheimer), 1–76, Cambridge University Press, Cambridge. Thistlethwaite, M.B. (1987) A Spanning Tree Expansion of the Jones Polynomial, Topology, 27, 7, 297–309. Thistlethwaite, M.B. (1988) Kauffman’s Polynomial and Alternating Links, Topology, 27, 311–318. Thistlethwaite, M.B. (1998) On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications, 7, 7, 981–1004. Thistlethwaite, M. (1999) Knotscape 1.01, http://www.math.utk.edu/∼ morwen/knotscape.html Thomson, W.H. (1867) Hydrodynamics, Proceedings R. Soc. Edin., 6, 94–105 (also in Knots and Applications, Ed. L. Kauffman, World Scientific, Singapore, New Jersey, London, Hong Kong, 111–192). ¨ Tietze, H. (1908) Uber die topologichen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte f¨ ur Mathematik and Physik, 19, 1–118. Torisu, I. (1998) The determination of the pairs of two bridge knots or links with Gordian distance one, Proceedings of Amer. Math. Soc., 126, 5, 1565–1571. Trace, B. (1983) On the Reidemeister moves of a classical knot, Proceedings of Amer. Math. Soc., 89, 4, 722–724. Traczyk, P. (2004) A combinatorial formula for the signature of alternating diagrams, Fundamenta Mathematicae, 184, 311–316. Treybig, L.B. (1968) A characterization of the double point structure of the projection of a polygonal knot in regular position, Trans. Amer. Math. Soc., 130, 223–247. Trotter, H.F. (1963) Non-invertible knots exist, Topology, 2, 275–280. Trotter, H.F. (1969) Computations in knot theory, in Computational Problems in Abstract Algebra, Proc. Conf. Oxford 1967 (Ed. J. Leech), 359–364, Pergamon Press, New York. Turaev, V.G. (1994) Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, 18, Walter de Gruyter and Co., Berlin. Turner J.C. and Van De Griend P. (Eds.) (1995). History and Science of Knots, Series on Knots and Everything 11, World Scientific, Singapore. Vandermonde, A.T. (1771) Remarques sur les probl`emes de situation, Memoir´es de l’Academie Royale des Sciences (Paris), 566–574. van Mill, J. and Reed, G. (1991) Open Problems in Topology, Topology Appl., 42, 301–307. Vassiliev, V.A. (1990) Cohomology of knot spaces, in Theory of singularities and its applications, Amer. Math. Soc., Translations of Math., 98. Veblen, O. (1905) Theory on plane curves in non-metrical analysis situs, Transactions of the American Mathematical Society, 6, 83-98. Vesnin, A.Yu. (1991) Cubical graphs and generaton of alternating links, Math. methods in chem. inform., 140, 63–86 (in Russian).

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Vogel, P. (1990) Representation of links by braids: A new algorithm, Comment. Math. Helvetici, 65, 104–113. Voytekhovsky, Y.L. and Stepenshchikov, D.G. (2005) On the Motzkin-Gr¨ unbaum theorem, Acta Cryst., A61, 584–585. Washburn, D. and Crowe, D.W. (1988) Symmetries of Culture, University of Washington Press, Seattle. Wasserman, S., Dungan, J. and Cozzarelli, N. (1985) Discovery of a predicted DNA knot substantiates a model for site-specific recombination, Science, 229, 171–174. Watson, L.T. (2004) Knots, Tangles and Braid Actions, http://www.cirget.uqam.ca/∼ liam/mythesis.pdf Watson, L. (2005) Knots that cannot be distinguished by the Jones polynomial: New examples, http://www.cirget.uqam.ca/∼ liam/cv.pdf Watson, L. (2006) Knots with identical Khovanov homology, arXiv:math.GT/0606630v3. Watson, L. (2006a) Any tangle extends to non-mutant knots with the same Jones polynomial, J. Knot Theory Ramifications, 15, 9, 1153–1162. Weeks, G. (2004) Snappea 3.0, http://geometrygames.org/SnapPea/ Welsh, D.J.A. (1993) Complexity: Knots, Colourings and Counting, London Math. Soc., Lecture Note Ser. 186, Cambridge University Press, Cambridge. Wendt, H. (1937) Die Gordische Aufl¨ osung von Knoten, Math. Zeit., 42, 680–696. Whitehead, J.H.C (1937) On doubled knots, J. Lond. Math. Soc., 12, 63–71. Whitehead, A.N., and Russell, B.A.W. (1925/27) Principia Mathematica, Cambridge University Press, Cambridge, 3 Vols. Wilson, R.A. (2002) Graphs, Colourings and the Four-Color Theorem, Oxford University Press, New York. Wilcox, S. (2006) Cellularity of Twisted Semigroup Algebras of Regular Semigroups, http://hdl.handle.net/2123/720 ¨ Wirtinger, W. (1905) Uber die Verzweigungen bei Funktionen von Zwei Ver¨ anderlichen, Jahresbericht d. Deutschen Mathematiker Vereinigung, 14, 517. Zaslavsky C. (1973) Africa Counts: Number and Pattern in African Culture, Weber & Schmidt, Boston. Zinn-Justin, P. and Zuber J.-B. (2004) Matrix Integrals and the Generalizations and Counting of Virtual Tangles and Links, J. Knot Theory Ramifications, 13, 3, 325–355. Zinn-Justin, P. (2006) The alternating virtual link database, http://ipnweb.in2p3.fr/∼ lptms/membres/pzinn/virtlinks/

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Index

2-pass, 47, 377 BF R, 168 DT -code, 69 KL, 2 P -data, 36, 66 P -word, 37 R-tangle, 145 ∞-unknotting number, 121 n-diagram, 296 n-move, 110 n-octahedron, 420 n-tangle, 282, 283 elementary, 284 t−1 -tangle, 149

FromContinued Fraction, 182 GetBraidRep, 163 GetKnotLink, 304 GetMirrorImageKnot, 67 GetPdatabyTracking, 57, 66, 192 JablanPoly, 318, 432 JonesPolynomialbyBraid, 218 Kh, 219, 341, 353 KnotFromBraid, 163 KnotbyDT, 69 KnotbyDowkerThistlethwaite Notation, 69 LiangPoly, 318, 432 LinkingNo, 35 LinksGould, 341 LinksGouldInv, 341 ListOfOneFactors, 299 MSigRat, 77 MaxSymmProjAltKL, 185, 403 MinDowAltKL, 182, 305 MinDowProjAltKL, 182 MultTan, 299 MutationOfTangle, 47 NMoveRat, 112 NoSelfCrossNo, 109 NumberOfKL, 304 PLDataFromSnapPeaData, 203 PeriodAltKL, 184 PeriodProjAltKL, 183 PrimeGraph, 32 PrimeKL, 32 R, 72

fGraphKL, 25 A2, 219, 341 AlexanderPolynomialby BurauRep, 218 AllStatesRational, 91 AmphiAltKL, 183, 322, 338 AmphiProjAltKL, 183, 322, 338 BR, 341 BreakCoAll, 192 ChromaticPolynomial, 218 ColouredJones, 219, 341 ContinuedFraction, 182 CuttNo, 192, 193 Dow, 68 DowfromPD, 67 DrawPlanarEmbGraph, 7 DrawPlanarEmbKL, 7 475

indexfinal

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476

RK, 72 RL, 72 RatGenSourKL, 73 RatKnotGenU0, 107 RatKnotGenU1, 107 RatLinkU0, 108 RatLinkU1, 107 RatReduce, 77, 182 RatSourceKLNo, 72 RationalAmphiK, 80 RationalAmphiL, 80 RationalKL, 71 ReductionKnotLink, 42, 91, 163, 306 SameAltConKL, 183, 195, 212 SameAltProjKL, 182 SeifertMatrix, 165 ShowBraid, 163 ShowKnotbyOpengl, 67 ShowKnotfromPdata, 67 SkeinPolynomial, 218 SnapPeaDataFromPLData, 203 SplittNo, 186 Symm, 185 ThreeParallelPolynomial Invariant, 218 UnKnotLink, 86 UnR, 86, 91, 108 UnRFixProj, 92 WritheKnotFromPdata, 34, 67 fAdd Dig, 39 fAlexPoly, 218 fAllClosures, 284 fAutoKL, 457 fAutoSigInp, 456, 457 fBalanced, 453 fBasicPoly, 192, 193 fBasicPolyTan, 291 fBasicTan, 291 fBoolean, 453 fBraidW, 341 fBreakComp, 192 fClassicToCon, 52 fColTest, 201 fComponentNo, 15, 74 fCompositePoly, 275

LinKnot

fConwayToPD, 219, 341 fCreateGraphics, 67 fCreatePData, 57, 66, 69, 164 fCuttRealKL, 194 fDToDDirect, 122 fDiffProjectionsAltKL, 182, 183, 316 fDiffSeq, 453 fDiffViae, 426 fDowCodes, 24 fDowkerToPD, 219, 341 fEdmonds, 9 fFindCon, 193 fForSourceLinks, 264 fGap, 92 fGapRat, 92 fGaussExtSigns, 35 fGenKL, 267 fGenSet, 299 fGenSign, 35 fGenSignDirProd, 122 fGenerators, 200 fGraphInc, 38 fGraphKL, 218 fKLfromGraph, 38, 430 fKLinGraph, 39 fKauffAlg, 453 fKnotscapeDow, 36, 69 fKnotscapeDowToPD, 219, 341 fMakeType, 133 fMidEdgeGraph, 38 fOrientedLink, 76 fPDataFromDow, 69, 430 fPdataToPD, 219, 341 fPlanarEmb, 38 fPlanarEmbGraph, 7 fProdTangles, 274 fProjections, 182 fSeifert, 165 fSignat, 125, 165 fSignsKL, 35, 38 fSourceDow, 264 fStPlusNo1, 150 fStUnNo1, 150 fStellar, 231 fStellarBasic, 231

indexfinal

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477

Index

fStellarNalt, 235 fStellarPlus, 231 fSumTangles, 274 fTangleType, 133 fTorusKL, 193 fViaToKL, 426 Knotscape, 42 Knot Atlas, 304 Knot2000, 34, 36, 42, 57, 66, 69, 163, 203, 218 KnotLinkBase.m, 304 KnotPlot, 67, 304 KnotTheory, 218, 341, 353 Knotilus, 304 Knotscape, 32, 36, 42, 127, 304, 365, 381 LinKnot, 6, 9, 15, 24, 25, 35, 57, 66, 67, 71, 77, 80, 86, 91, 107, 109, 119, 122, 165, 182, 186, 192, 200, 212, 218, 231, 235, 245, 264, 304, 316, 338, 381, 403, 453, 456 Links-Gould Explorer, 219, 341 Mathematica, 42, 182 string, 57, 67, 305 OpenGL, 67 PET, 247 PolyBase.m, 58 SnapPea, 127, 203, 322, 327, 353 Table of Knot Invariants, 304 plantri, 58 2-bridge knot, 55 3-coloring, 196 achiral, 27, 28, 77, 126, 130, 183, 237, 312, 317, 333, 376, 378, 463 actuality table, 224 adjacency list, 2 adjacent, 1 Alexander polynomial, 211 trivial, 209 algebra Conway, 213 algebraic, 55

indexfinal

BF R, 168 alternating, 25 ambient isotopy, 12, 42, 202, 378 for singular KLs, 219 amphicheiral, 27 antiprism, 169, 193, 252 antiprismatic belt, 292 antisymmetric presentation, 127 antisymmetry, 63, 77, 127, 237 concealed, 321 discernible, 321 arborescent knot, 64 non-invertible, 140 Arf equivalent, 203 invariant, 203 associativity, 160 automaton, 453 balanced state, 455 basic polyhedron, 55, 57, 241, 245, 262, 268, 428 achiral, 326 antiprismatic, 345 code, 289 minimal, 289 composite, 270 derivation, 288 elementary, 270 family, 292 generating sequence, 292 list, 246 notation, 288 source link, 263, 264 Bernhard-Jablan Conjecture, 83, 195 bigon, 5, 39, 55, 64, 189, 241, 263, 264, 311, 428 collapse, 8, 55, 59, 64 Boolean algebra, 446 logic, 448 Borromean links, 188 rings, 16, 20, 28, 187 boundary, 122 Brahma-mudi, 399

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478

braid, 157, 163 alternating, 166 closed, 159 closure, 158 elementary, 160 group, 160 index, 163 isotopic, 158 isotopy, 160 minimal, 163 minimal presentation, 163 product, 159 relation, 160 universe, 166 word, 160 braid family representative, 166, 303 braid word extension, 166 generating, 166 Brauer semigroup, 299 bridge, 121, 122 number, 121, 122 two-bridge, 228 Brunnian links, 190 Burau representation, 218 Catalan number, 284 catastrophe, 219 catenane, 382 checker-board coloring, 24 chiral, 27, 28, 183, 376–378 non-detectable, 333 chirality detection, 318 chord diagram, 222, 296, 418 chromatic polynomial, 218 circle, 12 class, 444 classical definition, 82 classical notation, 16, 30, 52 classification, 59 closedness, 160 closure, 283 coloring, 197 k-coloring, 197 3-coloring, 196

LinKnot

number, 198 perfect, 197 commutativity distant braids, 160 complement, 12 Complexions-Symbol, 376 component, 3, 10, 59, 74, 384 number, 62, 66, 74, 296 Component algorithm, 15, 17 composition, 119, 229 tangle, 295 conditions, 98 conjugacy class, 206 conjugate, 206 conjugation, 161, 198 connected sum, 119 consistency, 205 continued fraction, 76, 195 contraction, 8 elementary, 8 conversion, 69 Conway notation, 51, 52, 307, 381 symbol, 35, 50 Conway algebra, 213 Conway notation, 51, 52, 307, 381 braid modified, 166 Conway polynomial, 207 trivial, 209 Crazy Spider Algorithm, 288 crossing, 384 number, 25, 121 problem, 376 virtual, 23 curve monolinear, 393, 408 self-avoiding, 397, 416 cutting number, 187 cycle, 3 De Morgan law, 447 denominator closure, 54 derivation, 234, 264, 306, 364 diagram, 25

indexfinal

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World Scientific Book - 9in x 6in

479

Index

alternating, 25 crossing number, 37 generator, 299 hard, 48 isomorphic, 315 minimal, 26, 147 proper, 25 reduced, 25 Schlegel, 262 vertex-bicolored, 311 virtual, 23, 50 dichromatic polynomial, 217, 219 diffeomorphism, 11 digital circuit model, 454 digraph, 1 DNA, 383 dot, 58 Dow, 36 Dowker algorithm, 365, 381 Dowker code, 19, 32, 37, 364, 368 from P -data, 67 minimal, 37, 182, 212, 381 realizable, 22 with signs, 32 Dowker notation, 305 Dragon curve, 397 DT-code, 32 dual, 5 duplicates, 239 edge, 1 coloring, 9 connected, 32 connectivity, 4 Edmonds algorithm, 9 elementary isotopy, 42 embedding, 4, 12 adjacency matrix, 5 equilibrium global, 395 local, 395 Euler’s characteristic, 7 circuit, 3 formula, 7 experimental mathematics, 366, 381

indexfinal

experimenting, 225 extending operation, 173 face, 5 factor link, 119 families undetectable, 342 family, 59, 61, 62, 83, 87, 109, 231, 305, 366, 381 P -undetectable, 342 Alexander-undetectable, 342 Conway-undetectable, 342 of basic polyhedra, 291 of braid words, 166 Fibonacci form, 446 number, 445 sequence, 72 flip, 211 flype, 43, 44, 182, 271, 377 diagram, 311 Flyping Conjecture, 377 fullerene, 427 5/6, 430 biconical, 440 general, 427 perfect, 434 function continuous, 12 gap, 93 conditions, 98 Gauss code, 16, 27, 32 with signs, 32 diagram, 222, 296 general formula, 366 generating link, 250 generation, 205 generator, 196, 200 genus, 123 graph, 1, 2, 217 k-regular, 2 k-valent, 2 4-valent, 15

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480

automorphism, 4 bipartite, 2 coloring, 218 complete, 2 complete bipartite, 2 isomorphic, 4 mid-edge, 38 non-planar, 4 of link, 25 oriented, 1 planar, 4, 8, 24 proper, 1 reduced, 1 simple, 1 vertex-bicolored, 77 weighted, 9 grid rectangular square, 387, 390 group, 160, 201 automorphism, 5 cyclic, 201 dihedral, 201 fundamental, 378 isomorphic, 202 of link, 201 presentation, 201, 380 isomorphic, 201 minimal, 201 relation, 201 symmetric, 205 Hamiltonian circuit, 420 hard diagram, 48 hexastrip, 432 hollow pentagon rule, 434 homeomorphism, 12 HOMFLYPT polynomial, 212 homoathomic, 426 idempotency, 166, 447 immersion planar, 16 incident, 1 infinity change, 48 invariant, 182, 306, 364 finite-order, 219, 221

World Scientific Book - 9in x 6in

LinKnot

polynomial, 207, 379 quantum, 225 subfamily-dependent, 62, 205 linear, 63 Vassiliev, 220 inverse, 160 inverter, 454 invertible, 80, 126 isolated pentagon rule, 434 isomer chemical, 436 isomorphic, 29 isotopy elementary, 13 Jacobstahl sequence, 80 Jones polynomial, 212 Jordan Curve Theorem, 7 Kauffman Conjecture, 321 Kauffman algorithm, 451 Kauffman polynomial, 214 Kauffman-Murasugi Theorem, 27, 87, 121 Khovanov detectable, 349 Kirkman’s method, 243 knot, 10 2-bridge, 55 achiral, 34, 179 non-alternating, 323 almost alternating, 307 almost mutant, 347 alternating, 305 arborescent, 148 art, 375 classification, 59 derivation, 377 design, 387 diagram, 376 distance, 383 enumeration, 19, 377 equivalent, 200 graph, 377 invariant, 379 Kinoshita-Terasaka, 210

indexfinal

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World Scientific Book - 9in x 6in

481

Index

large algebraic, 148 Lissajous, 403 Listing, 376 mathematical, 10 molekular, 382 Montesinos, 228 Neolithic, 375 non-alternating, 25, 233, 240, 305, 307, 308 othopedic, 376 Paleolithic, 375 pretzel, 148 prime, 119, 380 problem, 378 projection, 376 rational number, 71 real, 10, 193 singular, 219 table, 19, 29, 304, 305, 363, 378 virtual, 22 wild, 11, 14 knotwork, 410 Kontsevich’s theorem, 224 labelling, 198, 205 theorem, 198 lattice, 446 length pretzel link, 149 line segment arrangement, 445 link, 10, 379 achiral, 28, 179, 237, 323 algebraic, 55, 60, 64, 169 almost alternating, 307 alternating, 25, 305 ambient isotopic, 11 Borromean, 188 Brunnian, 188 chiral, 28 classification, 59, 63 composite, 32, 119 diagram, 25 distance, 87 generating, 61, 366 genus, 123

indexfinal

graph signed, 47 group, 201 Montesinos, 228 non-algebraic, 55 non-alternating, 25, 233, 240, 305, 307, 308 oriented, 76 polygonal, 13, 42 polyhedral, 55, 60 prime, 32, 119 projection, 25 rational, 54, 64, 70, 128 generating, 73 number, 70 real, 10, 193 shadow basic, 60 singular, 35, 219 source, 55, 73, 228, 246 stellar, 64 table, 363, 381 tame, 13 virtual, 22 wild, 13 linking number, 34, 62, 204, 376, 383 load, 67 logic, 443 loop, 1, 19, 25 Lunda design, 395, 400 Lunda-animal, 399 M¨ obius ladder, 382 map, 5 Markov’s move, 161 theorem, 380 mid-edge graph, 38 mid-edge-truncation, 434 minimality of reduced braids, 167 minimization, 48, 195 minimum braid, 165 minimum crossing number, 25 minimum writhe, 46, 62 minus, 57

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World Scientific Book - 9in x 6in

482

mirror curve, 216, 384, 388 art, 401 number, 393 image, 27, 67 module, 10 monoid, 299 monolinear, 409 Montesinos knot, 149 link, 149 multigraph, 1 mutant, 210 mutation, 47 Nakanishi-Bleiler example, 83, 91 NAND, 449, 456 nanotube, 436 net, 441 neutral element, 160 non-algebraic tangle, 270 non-alternating, 25 non-invertible, 80, 126 achiral, 128, 140 arborescent knot, 140 chiral, 131 families, 129 polyhedral knot, 143 pretzel knot, 136 NOR, 449 normal form conjuctive, 449 disjunctive, 449 normalization, 211 notation Alexander-Briggs, 16 classical, 16, 30, 52 number of components, 296 numerator closure, 54 octahedron, 187, 246 operation, 270 orientation, 125 overlining, 445

LinKnot

overpass, 121 pair ordered, 2 unordered, 2 pair-class, 349 paradox Russel, 443 partition, 231, 243 P -partition, 231 Partition-method, 377 pass-move, 203 path, 3 pavitram, 384 period, 79, 183, 451 periodic sequence, 450 Perko pair, 19, 34, 378 permutation, 22, 205 matrix, 314 physic, 383 statistical model, 215 plait, 375 planar isotopy, 42 plane, 8 plate, 403 combine rules, 404 combining, 403 plus, 57, 235, 237 point group, 63 Polya Enumeration Theorem, 247, 263, 333, 429 polygonal link, 42 polyhedral BF R, 168 polyhedral knot, 59 non-invertible, 143 polyhedron basic, 55, 57, 241, 245, 262, 268, 428 regular, 2 polynomial Alexander, 207, 211, 226, 368, 380 general formula, 227 trivial, 209, 226

indexfinal

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World Scientific Book - 9in x 6in

483

Index

chromatic, 218 Conway, 208, 211, 226 trivial, 209, 226 dichromatic, 217 HOMFLYPT, 207, 211, 382 Jones, 208, 213, 232, 368, 381 colored, 218 trivial, 208 two-variable, 212 Kauffman, 208, 382 bracket, 214, 215 Khovanov, 218 Laurent, 211 Liang, 432 square bracket, 216 square-free, 447 polyomino, 384, 389, 396 Lunda, 398 Pontryagin-Kuratowski’s Theorem, 7 presentation centro-antisymmetric, 319 pretzel link, 134, 150 pretzel knot non-invertible, 136 prime, 32, 119 prime decomposition theorem, 120 prism, 190, 228 projection, 25, 182 isomorphic, 29, 182, 315 minimal, 40, 48, 82, 311 non-isomorphic, 182 oriented, 32 polynomial, 315 special, 35 prototile, 387 quandle, 198, 453 Alexander, 200 quantum invariant, 219, 225 quipu, 10, 375 ramification, 53, 182 realizable, 22 recognition, 195

indexfinal

problem, 195 recursion, 370, 450 recursive form, 445 reduction, 145, 306 rational, 146 reductor, 455 Reidemeister move, 42, 161, 215, 379, 453 for special crossings, 219 for virtual KLs, 50 relation equivalence, 3 four-term, 220 one-term, 220 representative, 253 reversible, 126 rigid representation, 126 sand drawing, 386 Scheme-method, 377 Schlegel diagram, 432 Seifert surface, 122, 203 Seifert circle, 123, 163 nested, 123 Seifert matrix, 165 self-avoiding curve, 416 self-reference, 444 self-referential form, 453 set, 12, 444 empty, 444 multi-set, 444 open, 12 theory, 443 shadow generic, 14 proper, 19 realizable, 22 reduced, 19 regular, 14 Sheffer stroke, 449 signature, 125, 165, 366 Murasugi, 77 skein relation, 208, 211, 213, 380 smoothing, 211 solid, 64

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World Scientific Book - 9in x 6in

484

LinKnot

solid knot, 268 sona, 386 source braid, 167 source link, 55, 62, 248, 251, 253, 263 space, 57 special crossing, 219 projection, 219 sphere, 8 essential, 148 spin, 215 splitting gap, 186 number, 185, 186 stabilization, 161 standard definition, 82 state, 216 stellar tangle, 134 step, 384 subdivision, 8 subfamily, 62 subfamily-dependent invariant, 62 subgraph, 4 sublink, 191 subsolid, 64 subworld, 59 sum connected, 119 surface, 8, 22, 122, 400, 443 boundary, 9 genus, 9 non-orientable, 8 orientable, 8 surgery, 81, 211 symmetry, 63, 231, 391 group, 439 table, 16, 29, 305 knot, 304 periodic, 381 Tait coloring, 25 Tait’s Flyping Theorem, 43, 44, 84 Tamil, 384 tangle, 43, 132, 380 algebraic, 55

closure, 283 composition, 295 equivalent, 44 generalized, 282 hyperbolic, 269, 270, 272 integer, 54 non-algebraic, 269, 270, 272 product, 53, 57 rational, 54, 76 reduction, 145 stellar, 134 substitution, 55 sum, 53 type, 132, 296 tangle-type computation, 301 tautology, 449 Tchokwe, 385 tessellation, 443 theta-curve, 282 tiling, 384, 400 uniform, 387 topoisomerase, 383 topological space, 12 topology, 12 torus, 9, 22 knot, 193 link, 193 trail, 3 tree, 4 trefoil, 28, 220 twist, 383 undetectable, 341 P -undetectable, 341 HOMFLYPT-undetectable, 344 Jones-undetectable, 344 Khovanov-undetectable, 344 unknot, 107 P -unknot, 341 unknotting gap, 93 problem, 195 unknotting number, 81, 121, 354, 366, 377 ∞-, 121 BJ-unknotting number, 85

indexfinal

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World Scientific Book - 9in x 6in

485

Index

one, 106, 148 unlink, 48, 107 P -unlink, 341 unlinking gap, 93 operation, 112 unlinking gap, 93, 181 conditions, 98 unlinking number, 81, 121, 302, 366 BJ-unlinking number, 85 one, 106, 148, 149 unsolid, 64 valence, 1 Vassiliev invariant, 35, 211, 219, 220, 382 conjecture, 221 vertex, 1 coloring, 9 connected, 3 connectivity, 4 isolated, 2 negative, 32 positive, 32 sign, 32 single, 2 valence, 1 via, 416 virtual diagram, 50 link, 49 virtual crossing, 23 virtual knot, 22, 49 Vogel’s algorithm, 163 vortex theory, 377 walk, 2 closed, 3 open, 3 Whitney degree, 45 wild knot, 11 winding number, 45 Wirtinger presentation, 201 word, 16 world, 50, 59, 195, 304, 381 arborescent, 64, 229

indexfinal

linear, 69, 227 polyhedral, 241 prismatic, 228 rational, 70, 228 stellar, 228 stratification, 239 subworld, 304 polyhedral-rational, 250 rational-stellar, 235 stellar-rational, 230, 231 stellar-stellar, 238 writhe, 33, 44, 67, 376 average, 383 minimum, 46, 62 zero, 312 zero, 58