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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Fig. 1.24
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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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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
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;
