Lectures at Knots '96: International Conference Center, Waseda Univ., Tokyo 22 - 31 July 1996

K(AE Series on Knots and Everything — Vol. 15

World Scientifi'

Series on Knots and Everything - Vol. 15

LECTURES AT

KNOTS ' 96 International Conference Center, Waseda Univ., Tokyo 22-31 July1996

SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman Published: Vol. 1: Knots and Physics L. H. Kauffman Vol. 2: How Surfaces Intersect in Space J. S. Carter Vol. 3: Quantum Topology edited by L. H. Kauffman & R. A. Baadhio Vol. 4: Gauge Fields, Knots and Gravity J. Baez & J. P. Muniain Vol. 5: Gems, Computers and Attractors for 3-Manifolds 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 R. E. Miles Vol. 9: Combinatorial Physics T. Bastin & C. W. Kilmister Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics W. M. Honig Vol. 11: History and Science of Knots edited by J. C. Turner & P. van de Griend Vol. 13: Entropic Spacetime Theory J. Armel Vol. 14: Diamond - A Paradox Logic N. S. Hellerstein Vol. 15: Lectures at Knots '96 edited by S. Suzuki

Series on Knots and Everything - Vol. 15

LECTURE S AT

KNOTS '96 International Conference Center, Waseda Univ., Tokyo 22 - 31 July 1996

Editor

S. Suzuki Waseda University, Japan

World Scientific VOW Singapore • NewJersey• London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office : Suite 1B , 1060 Main Street, River Edge, NJ 07661 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.

LECTURES AT KNOTS '96 Copyright m 1997 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in anyform orbyanymeans, 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 981 -02-3094-X

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

PREFACE

This proceedings volume consists of 10 expository or research papers on Knot Theory and Related Topics, which are based on invited lectures delivered at the International Conference and Workshop on Knot Theory (briefly, Knots 96), which was held at Waseda University, Tokyo, Japan from July 22 to July 31, 1996. The order in which the papers appear is the order in which they were received. The other 43 research papers bases on talks delivered at this Conference were included in the other proceedings : Proceedings of Knots 96 , World Scientific Publisher, 1997.

Abstracts of talks that were not published can be found in the Conference/Workshop Report. The Conference/Workshop was the fifth in the series of International Research Institute of the Mathematical Society of Japan. The participant number by country (on the research institution basis) was as follows : Australia 3, Canada 3, France 4, Germany 4, Italy 1, Japan 140, Korea 2, Russia 6, Spain 2, Switzland 1, U.K. 5, and U.S.A. 12. The list of participants is reported in Proceedings of Knots 96. The Conference/Workshop was sponsored also by Waseda University(Subsidy of the International Academic Conference), and was made possible generous grants and gifts from the following foundations, corporations and individuals. I would like to express my deepest appreciation to all of them :

Inoue Foundation of Science, The Kajima Foundation, The Asahi Glass Foundation, Daiwa Anglo-Japanese Foundation, Fuji Xerox Ltd., Hiroshi Noguchi and Akio Kawauchi. This was also supported by a program, represented by S.Suzuki, of Grant-in-Aid for Scientific Reseach(B) #08304008, the Ministry of Education, Science and Culture, Japan. The members of Low dimensional Topology Seminar at Tokyo helped me very much in preparing the Conference/Workshop. I would like to thank all of them, especially, Toshiki Endo, Eiji Nakayama, Miyuki Okamoto, Tomoe Motohashi, Tatsuya Tsukamoto, Makoto Ozawa, Eishin Kawamoto, Yasuhiro Suzuki, Yoichi Sonoda, Ikki Sugiura, Satoshi Taguch and Satosi Miura. A special thanks goes to Toshiki Endo for help in preparing these Lectures.

Vi

Finally, I wish to thank the organizers : K.Kawakubo (Osaka Univ.), A.Kawauchi (Osaka City Univ.), K.Kobayashi (Tokyo Woman's Christian Univ.), Y.Matsumoto (Univ.of Tokyo) and K.Murasugi (Univ.of Toronto) for their support and advice, the other participants for their contributions, the referees for their speedy work, and World Scientific Publishing Co Pte Ltd for publishing this volume.

January 1997 S. Suzuki

vii

TABLE of CONTENTS

Preface ............ .........................................................v

Contents of Proceedings of Knots 96 .......................................viii

Tunnel number and connected sum of knots .................................. 1 K. Morimoto Topological imitations ...................................................... 19 A. Kawauchi Surfaces in 4-space : a view of normal forms and braidings ................... 39 S. Kamada Knot types of satellite knots and twisted knots ...................... ........ 73 K. Motegi Random knots and links and applications to polymer physics ................ 95 T. Deguchi and K. Tsurusaki

Knots and diagrams ....................................................... 123 L. H. Kauffman On spatial graphs .... ...................................................... 195 K. Taniyama Energy and length of knots .............................. .................. 219 G. Buck and J. Simon Chern-Simons perturbative invariants ...................................... 235 T. Kohno Combinatorial methods in Dehn surgery .................................... 263 C. McA. Gordon

vi"

Contents of Proceedings of Knots 96

We here list the titles of papers which are included in Proceedings of Knots 96. • On spatial graphs isotopic to planar embeddings (H. Inaba and T. Soma) • The Conway polynomial of an algebraically split link (J. Levine) • Young diagrams, the Homfly skein of the annulus and unitary invariants (H. R. Morton and A. K. Aiston) • Some new results in the theory of braids and generalised braids (R. Fenn) • The arithmeticity of certain torus bundle cone 3-manifolds and hyperbolic surface bundle 3-manifolds; and an enhanced arithmeticity test. (H. M. Hilden, M-T. Lozano, and J. M. Montesinos-A.) • Equivariant concordance of knots in S3 (S. Naik)

• Note on spatial graphs with good drawings (S. Negami and T. Tsukamoto) • Real representation spaces of 2-bridge knot groups and isometries of the hyperbolic plane (G. Burde) • Two-bridge knots with generalized unknotting number one (Y. Uchida) • Delta-unknotting operation and adaptability of certain graphs (A. Yasuhara) • Planar surfaces in a handlebody and a theorem of Gordon-Reid (K. Morimoto) • Statistics of knots and some relations with random walks on hyperbolic plane (M. Monastyrsky and S. Nechaev) • The fundamental polygons of twist knots and the (-2,3,7) pretzel knot (S. Boyer, T. Mattman and X. Zhang) • On the Tutte polynomial (B. I. Kurpita and K. Murasugi) • Delta unknotting operation and vertex homotopy of graphs in R3 (T. Motohashi and K. Taniyama) • Floer homology for orbifolds and gauge theory knot invariants (0. Collin) • Knots and electricity (L. H. Kauffman) • Uniqueness of essential free tangle decompositions of knots and links (M. Ozawa) • Polynomial invariants of Legendrian links and their fronts (S. Chmutov and V. Goryunov) • Combinatorial analog of the Melvin-Morton conjecture (S. Chmutov)

• A strand passage metric for topoisomerase action (I. D. Darcy and D . W. Sumners) • Algebraic topology based on knots : an introduction (J. H. Przytycki) • Quantum SU(3) invariants derived from the linear skein theory (H. A. Miyazawa and M . Okamoto) • Invariant trace fields and commensurability of hyperbolic 3-manifolds (H. Yoshida)

• Every 2-link with two components is link -homotopic to the trivial 2-link (F. Hosokawa and S. Suzuki) • Alexander invariant and twisting operation (Y. Nakanishi) • Thraev-Viro modules of satellite knots (P. M. Gilmer) • On the invariants of lens knots ( N. Chbili)

• The crossing number of alternating link diagrams on a surface (N. Kamada) • Minimal genus Seifert surfaces for alternating links (M. Hirasawa and M. Sakuma) • Seifert complex for links and 2-variable Alexander matrices (T. Kadokami) • Kauffman polynomials as Vassiliev link invariants (T. Kanenobu)

• Shortest vertical geodesics of manifolds obtained by hyperbolic Dehn surgery on the Whitehead link (H . Akiyoshi, Y. Nakagawa and M. Sakuma) • Energy of knots in a 3-manifold ; the spherical and the hyperbolic cases (J. O'Hara) • A condition for a 3 -manifold to be a knot exterior (T. Sakai) • Conputer programs for knot tabulation (C. Aneziris) • Homeomorphisms of a 3-dimensional handlebody standardly embedded in S3 (S. Hirose) • Framed link diagrams of open 3-manifolds ( J. Hoste) • Open 3-manifolds with infinitely many knot-surgery descriptions (J. Hoste) • Some Seifert 3-manifolds which deconpose S4 as a twisted double (Y. Yamada)

• On simply knotted tori in S4 II (A. Shima) • Hyperbolic three-manifolds and the four-color theorem (S. Yamada) • Representation of mapping class groups via the universal perturbative invariant (J. Murakami)

This page is intentionally left blank

Lectures at Knots 96 edited by Shin'ichi Suzuki ©1997 World Scientific Publishing Co. pp. 1-18

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS KANJI MORIMOTO

1. Introduction Let K be a knot in the 3-sphere S3, and t(K) the tunnel number of K, where t(K) is the minimal number of arcs in S3 whose end points are in K such that the exterior of K and those arcs in S3 is a handlebody (more detailed definition i s given at the end of the present section). Let Kl and K2 be two knots in S3, then we denote the connected sum of Kl and K2 by Kl#K2. In this article, we report on study of the behavior of tunnel numbers of knots under connected sum. By the definition of tunnel number and connected sum of knots, and by taking an arc contained in the decomposing 2-sphere for the connected sum, the following follows immediately.

Fact 1.1.

t(Kl#K2) < t(K1) + t(K2) + 1 for any knots Kl and K2.

By the above inequality, the following two conjectures had been made. Tunnel numbers of knots cannot go down under connected sum, Conjecture A. i.e., the inequality t(K1) + t(K2) < t(Kl#K2) holds for any knots Kl and K2. Tunnel numbers of knots can go up under connected sum, i.e., Conjecture B. there are knots Kl and K2 such that t(Kl#K2) = t(K1) + t(K2) + 1. Concerning the above conjectures, the first result is : Theorem 1.2 ([No, Sc[). Tunnel number one knots are prime, i.e., if t(Kl#K2) = 1 then one of Kl and K2 is a trivial knot (tunnel number zero). The above result was obtained in the first half in 1980's. And this shows that Conjecture A is true if t(Kl#K2) = 1.

In 1991, we studied the case when t(Kl#K2) = 2 and got the following. 1

2 KANJI MORIMOTO

Theorem 1.3 ([Mo2, Theorem]). Let KI and K2 be non-trivial knots in S3. Suppose t(KI#K2) = 2. Then : (1) if neither KI nor K2 is a 2-bridge knot, then t(KI) = t(K2) = 1(2) if one of KI and K2, say K1, is a 2-bridge knot, then t(K2) < 2 and K2 is prime.

And in 1992, we showed that the estimate of the above theorem is best possible by constructing knots K having the property that t(K) = 2 and t(K#K') = 2 for any 2-bridge knot K'. In fact, we got the following. Theorem 1 .4 ([Mo3, Theorem 3]). Let n be a positive integer, and Kn the knot illustrated in Figure 1.1. Then we have : (1) t(Kn) = 2. (2). t(Kn#K) = 2 for any 2-bridge knot K. (3) Kn and Kn, are different types if n # n'.

2n + 1 crossings

Kn

Figure 1.1 The examples in Theorem 1.4 show that Conjecture A is false. On the other hand, in the same year, Moriah and Rubinstein got the following. Theorem 1 . 5 ([MR, Theorem 0.6]). For any positive integers t1 and t2, there are infinitely many pairs of knots KI and K2 such that t (K1) = t1, t(K2) = t2 and t(K1#K2) = t1 + t2 + 1. Theorem 1.5 shows that Conjecture B is true. And the theorem was proved by using argument from hyperbolic geometry, and those examples are corresponding to sufficiently complicated Dehn surgeries along some pretzel knots in S3. Concerning Conjecture B, the author, Sakuma and Yokota proved independently of [MR] by using another method that there are infinitely many pairs of knots KI

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 3

and K2 such that t(K1) = 1, t(K2) = 1 and t(K1#K2) = 3. In fact, we got the following.

Let m be an integer and Km the knot Theorem 1.6 ([MSY, Theorem 2.1]). illustrated in Figure 1.2. Then t(Km) = 1, t(Km') = 1 and t (Km#Km,) = 3 for any integers m and m'.

10m-4 crossings

K.

Figure 1.2 In the following sections, we report on these results and related topics. In section 2, we report on knots whose tunnel numbers go down under connected sum, and in section 3, we report on knots whose tunnel numbers go up under connected sum. Now, let's give more precise definition of tunnel number of knots. Let K be a knot in S3, and put E(K) = cl(S3 - N(K)) be the exterior. Fact 1.7. There is a family of mutually disjoint arcs 71,ry2i • • • ,?'t properly embedded in E(K) such that cl(E(K)-N(71 U72U...Uryt)) is a genus t+1 handlebody, where N(71 U y2 U • • • U -ft) is a regular neighborhood of 'yl U y2 U • • • U •yt in E(K). Let k be a regular diagram of K, and let cl, c2, • • • , ct be the crossing Proof. points in K. Let ry, be the short arc properly embedded in E(K) corresponding to c; (i = 1, 2, • • • , t). Then by the deformation illustrated in Figure 1.3, we see that N(K) U N(-y1 U rye U • • . U ryt) is ambient isotopic to a standard genus t + 1 handlebody in S3. This means that the exterior cl(S3 - (N(K) U N(ryl U rye U ... U ryt))) = cl(E(K) -N(71 U72U • Uryt)) is a genus t + 1 handlebody, and completes the proof of the fact. ❑

4 KANJI MORIMOTO

'Y2

E(K)

Figure 1.3 We call the family of arcs {yl, 12, • • . , yt} in the above fact an unknotting tunnel system for K. In particular , if t = 1 we call it an unknotting tunnel for K. By the above fact , we define t(K) as the minimal number of the arcs among all unknotting tunnel systems for K, and t ( K) is 0 if K is a trivial knot. Then we call t(K) the tunnel number of K.

2. Knots whose tunnel numbers go down under connected sum Outline of the proof of Theorem 1.3. Put K = Kl#K2. Then since t(K) = 2, there is an unknotting tunnel system {yl, y2} for K properly embedded in E(K). Put Vl = N(K) U N(yl U y2), and put V2 = cl(S3 - Vi). Then (VI, V2) is a genus three Heegaard splitting of S3, and Vl contains K as a central curve of a handle of

Vi Let S be the 2-sphere in S3 which gives the connected sum of K = Kl#K2. Then we may assume that S n Vl = Di U D2 U DI U • .. U Dl, where Dj* is a non-separating disk of Vl intersecting K in a single point (i = 1, 2), and D,, is a disk not intersecting K (j = 1, 2, • • • , t). Let W be the genus two handlebody obtained from Vl cut open by D. Then, since D2 is a disk properly embedded in W, according as D2 splits W

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 5

into two solid tori , D2 is a non-separating disk in W and Dz is parallel to a disk in OW, we have the following three cases with respect to Di and D2 (Figure 2.1). Case I : Di U Dz splits V, into two solid tori. Case II Di U Dz does not separate V1. Case III : Di and D2 are mutually parallel. Suppose #(S n V1) = P + 2 is minimal among all 2-spheres which give non-trivial connected sum of K and intersect V, in such disks as above, where #(•) denotes the number of the components . Then by the isotopy of type A argument ([Ja, Oc]), we can show that P = 0 in Case I, and that 8 = 1 in Cases II and III . Moreover, we can show that Dl is a non-separating disk of Vl in Case II such that Di U DZ U Dl splits Vi into a 3-ball and a solid torus and that Dl is a separating disk of Vl in Case III such that Dl splits Vl into a genus two handlebody and a solid torus containing K as a central curve of it (Figure 2.1).

I

II

III

Figure 2.1

6 KANJI MORIMOTO

Then by the argument similar to the proof of [Kol, Theorem], we see that t(K1) = t(K2) = 1 in Case I and that one of Kl and K2 is a 2-bridge knot and the other has tunnel number at most 2 and is prime in Cases 11 and III . This completes the proof of Theorem 1.3. ❑ Outline of the proof of Theorem 1.4. Let S be the 2-sphere in S3 intersecting K„ in four points illustrated in Figure 2.2. Then S splits (S3, K„) into two tangles TI and T., where Tm is a tangle illustrated in Figure 2.3. This shows that (S3, K„) is decomposed into two 2-string non-trivial tangles. Then by [Sc, Theorem 2.3], we have t(KK) > 2.

Figure 2.2

2m+1 crossings

T.

Figure 2.3 On the other hand, by the deformation illustrated in Figure 2.4(1) through Figure 2.4(6), and since the arc p indicated in Figure 2.4(6) is an unknotting tunnel for

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 7

the (4, 3)-torus knot (cf. [BRZ]), we see that the two arcs {ryl,72} is an unknotting tunnel system for K„#K for any 2-bridge knot K, and hence t(K„#K) < 2. Then, together with the inequality t(K„) > 2, we have t(K„) = 2 and t(K,,#K) = 2 for any 2-bridge knot K.

(1)

2-bridge knot

(2)

(3)

(5)

(6) Figure 2.4

8 KANJI MORIMOTO

Moreover, by the uniqueness of the tangle decomposition of K„ into two tangles T1 and T,,, we see that K. and K„ are different types if n # n'. This completes the proof of the theorem. ❑ Theorem 1.4 says that the estimate of Theorem 1.3 is best possible, and the pair of the knot K„ and a 2-bridge knot is a counter example to Conjecture A. Now, let K1 and K2 be two knots such that t(K1) = 1, t(K2) = 2 and t(K1#K2) = 2. Then by Theorem 1.3, K1 is a 2-bridge knot. But what kind of types are the knots K2 ? In order to answer the question, we prepare some terms concerning tangles. Let B be a 3-ball, and let (B, t1 U t2) be a 2-string tangle, where t1 U t2 is a union of mutually disjoint two arcs properly embedded in B. We say that (B, t1 U t2) is a trivial tangle if it is homeomorphic to (D2 X I, {x, y} x I) as pairs, where D2 is a 2-disk and x and y are two points in int(D2), and that it is a non-trivial tangle if it is not a trivial tangle. A component of t1 U t2, say t1, is called unknotted if (B, t1) is homeomorphic to (D2 x I, {x} x I) as pairs. We say that (B, t1 U t2) is a free tangle if cl(B - N(t1 U t2)) is homeomorphic to a genus two handlebody, where N(t1 U t2) is a regular neighborhood of tl U t2 in B. The term " free tangle " is due to T. Kobayashi [Ko3]. Then the answer to the above question is the following. Let K be a knot in S. Then the following Theorem 2 .1 ([Mo4, Theorem 0.4]). three properties are all equivalent to each other. (1) t(K) = 2 and t(K#K') = 2 for some 2-bridge knot K. (2) t(K) = 2 and t(K#K') = 2 for any 2 -bridge knot K'. (3) (S3, K) has a 2 -string tangle decomposition (B, t1 U t2) U (B', t'1 U t2) such that both tangles are non-trivial free tangles and at least one of the two tangles has an unknotted component. Remark 2 .2. The 2-sphere S indicated in Figure 2.2 gives the tangle decomposition of K„ satisfying the property of Theorem 2.1, because the tangle T,,, illustrated in Figure 2.3 is a non-trivial free tangle with an unknotted component. Outline of the proof. First we prove (1) = (3). Suppose t(K) = 2 and t(K#K') = 2 for some 2-bridge knot K'. Put K0 = K#K', and let {y', y2} be an unknotting tunnel system for K0. Put V1 = N(Ko)UN(y1Uy2) and V2 = cl(S3-V1). Then (V1, V2) is a genus three Heegaard splitting of S3 and K0 is a central curve of a handle of V1. Let S be a 2-sphere in S3 which gives the connected sum of K0 = K#K'. Then we may assume that S fl V1 = Di U D2 U D1 U • • • U Di, where D; is a non-separating disk of V1 intersecting K0 in a single point (i = 1, 2), and D3 is a disk not intersecting K0 (j = 1, 2, • • • , 8). Then, since we are in the same situation as the proof of Theorem 1.3, one of the following three cases holds as illustrated in Figure 2.1.

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 9

Case I : S n V1 = Di U D2 and Di U DZ splits V1 into two solid tori. Case II : S n V1 = Di U D2 U D1, Di U D2 does not separate V1 and D1 is a non-separating disk of V1 such that Di U Dz U D1 splits Vi into a 3-ball a solid torus. Case III : S n V1 = Di U Dz U D1, Di and D2 are mutually parallel and D1 is a separating disk of V1 which splits V1 into a genus two handlebody and a solid torus containing K0 as a central curve of it. In Case I, by the proof of Theorem 1.3, we have t (K) = t(K') = 1 . This contradicts that t(K) = 2. In Case III , by more detailed argument than the proof of Theorem 1.3, we have t(K) = 1 and K' is a 2-bridge knot . This contradicts that t(K) = 2. Finally suppose we are in Case II . By deforming S by an isotopy of type A, we may assume that S n V1 = Di U D2 U A and S n V2 = F1 U F2, where A is a nonseparating annulus in Vl as a union of the disk D1 and a band (Figure 2.5(1)) and F; is a non-separating annulus in V2 (i = 1, 2) as a union of a non-separating disk and a band such that those two disks are mutually parallel and those two bands are not mutually parallel (Figure 2 . 5(2)).

V2 F2

(1)

(2)

Figure 2.5 Let X1 and X2 be the closure of the components of V1 - (Di U D2 U A) indicated in Figure 2.5(1), and Y1 and Y2 the closure of the components of V2 - (F1 U F2) indicated in Figure 2.5(2). Then Y1 n 8V2 is identified with X1 n 8V1 and Y2 n 8V1 is identified with X2 n ,9V,. Put B1 = X1 U Yl and B2 = X2 U Y2. Then, since B1 and B2 are the closure of the two components of S3 - S, Bi is a 3-ball (i = 1, 2). Put 6i = Bi n K0 = Xi n K0 (i = 1, 2). Let Bi' be a 3-ball and 6 a trivial are properly embedded in Bi' (i = 1, 2). Put S, = Bi U B; and K. = bi U 6 (i = 1. 2), then Ki is a knot in the 3-sphere S, and K0 = K1#K2. Then by the argument similar to the proof of [Kol, Theorem], we can show that K1 is a 2-bridge knot.

10 KANJI MORIMOTO

Next we show that K2 has the property (3) of Theorem 2.1. We denote the images of Di, D2, A, F1 and F2 in 8X2 and in 8Y2 by the same notations. Let a be the central curve of the annulus A, then a intersects a Claim 2.3. meridian of the solid torus X2 in a single point. Proof of Claim 2.3. Since a is a loop in the 2-sphere S, a splits S into two disks . Then by taking one of the two disks, we get a 2-disk E in cl(S3 - X2) such that E n X2 = OE = a. This shows that a is a prefered longitude of X2 in S3, and ❑ completes the proof of the claim. Let D2 be a 2-disk and x a point in int (D2). Since 62 is a trivial arc in B2, we can regard B2' as D2 x [0, 31 and 62 as {x} x [0, 3]. Put (31 = {x} x [0, 1], ,Q2 = {x} x [1, 2] and ,133 = {x} x [2, 3]. Choose the glueing map f from 8B2 to 8B2 so that f (D2 x {0}) = Di, f (8D2 x [0,1]) = F1, f (8D2 x [1, 2]) = A, f (8D2 x [2,3]) = F2 and f (D2 x {3}) = D. Consider D2 x [0, 3] as a union of D2 x [0,1], D2 x [1, 2] and D2 x [2, 3]. Put W1 = X2Ufj8D2x[1,2] (D2 x [1,2]) and W2 = l2UfIaD2x[o,1]uaD2x[2,3] ((D2 x [0,1])U(D2 x [2, 3])). Then by the above claim, W1 is a 3-ball, and hence W2 = cl(S2 - W1) is a 3-ball too. Moreover, since Wl n K2 = 62 U X32 and W2 n K2 = i31 U,33, we see that (W1, 62 U 02) U (W2i131 U /33) is a 2-string tangle decomposition of (S2, K2) Then by these situations, we see that both tangles above are non-trivial free tangles and 02 is an unknotted component in W1. Then K2 has the property (3) of Theorem 2.1 and so does K. This completes the proof of (1) (3). Next we prove (3) = (2). Suppose K has the property (3) of Theorem 2.1. Then (S3, K) has a 2-string tangle decomposition (B, t1 U t2) U (B', t'1 U t'2) such that both tangles are non-trivial free tangles and t2 is an unknotted component of the tangle (B, t1 U t2). Since tunnel number one knots cannot be decomposed into 2-string non-trivial tangles by [Sc, Theorem 2.3], we have t(K) > 2. On the other hand, let K' be a 2-bridge knot in another 3-sphere S3, and (C, si U s2) U (C, si U s2) a 2-bidge decomposition of (S3, K'), i.e., both tangles are 2- string trivial tangles. For i = 1, 2, put Ni = N(ti; B), N,' = N(tt; B'), Mi = N(si; C) and M'=N(s;;C'). And put Po =cl(S3-(NNUN2UN2))and Qo=cl(S3-(MjUM2U M2)). Since t2 connects t'1 and t'2, and since s2 connects si and s2, both Nl UN2UN2 and Ml U M2 U M2 are 3-balls in S3 and in g3 respectively. Hence both P0 and Qo are 3-balls, and t1 (si resp.) is an arc properly embedded in Po (Qo resp.) (Figure 2.6). Choose the glueing map f from 8Po to 8Qo as follows : f (N1 n 8Po) = Mi n BQo, f(N2n8Po) = M2n8Q0, f (N2n8Po) = M2n8Q0 and f (at1) = as,. Consider the 3-

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 11

sphere Po U f Qo and the knot t1 U f sl in the 3-sphere Po U f Qo. Then by the definition of connected sum, the knot t1 U f sl in the 3-sphere is a connected sum of K and K'. Put W1 = cl(B-N2)Ufc1(C-M2) and W2 = cl(B'-(NfUN2))Ufc1(C'-(MMUM2)). Then by careful observations of these situations, and by the argument in the proof of [Ko3, Theorem], we see that (W1, W2) is a genus three Heegaard splitting of the 3-sphere P0 U f Qo, and K#K' = t1 Uf s1 is a central curve of a handle of W1. This means that t(K#K') < 2, and hence we have t(K) = 2 and t(K#K') = 2. This completes the proof of (3) =^- (2). And since (2) = (1) is trivial, we complete the proof of Theorem 2.1. ❑

' N2 7M2

Figure 2.6 Example 2.4. Let K be the knot 816 in the table of Rolfsen 's book ([Ro]). Then by the deformation illustrated in Figure 2.7, we see that K has a 2-string tangle decomposition such that each tangle is a non-trivial free tangle with an unknotted component . Hence K has the property of Theorem 2.1. And the arcs {ryl, rye} indicated in Figure 2.8 is an unknotting tunnel system for K#K' for any 2-bridge knot K'. Moreover , since prime knots with at most seven crossings are all 2-bridge knots, the minimal crossing number of knots which have the property of Theorem 2.1 is eight. Now, Theorems 1.4 and 2 . 1 show that tunnel numbers can degenerate by one under connected sum. But by extending those examples, Kobayashi showed the following, which shows that tunnel numbers can arbitrarily highly degenerate under connected sum.

12 KANJI MORIMOTO

816

(2)

(1)

(4)

(3)

(5)

(7)

(8) Figure 2.7

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 13

Figure 2.8 Theorem 2 .5 ([Ko3, Theorem)). For any positive integer n, there are infinitely many pairs of knots Kl and K2 such that t(Kl#K2) < t(K1) + t(K2) - n. Outline of the proof.

Before the proof, we prepare a fact without proof.

Fact. Let K be a knot in S3 and E2(K) the 2-fold branched covering space of S3 branched along K, and let g(E2(K)) be the Heegaard genus of E2(K). Then we have g(E2(K)) < 2t(K) + 1.

Now, let K be a knot illustrated in Figure 2.9, and K' a trefoil knot. Put Kl = K#K# • • • #K be the connected sum of 2n copies of K and K2 = K'#K'# • • . #K' be the connected sum of 6n copies of K'.

Figure 2.9

14 KANJI MORIMOTO By [Mol, Theorem 1.1], we have g(E2(K)) = 3. Then by the additivity of Heegaard genus ([Ha]), we have g(E2(K1)) = 3 • 2n = 6n. Then by the above fact, we have 6n < 2t(K1) + 1, and hence t(K1) > 3n. In addition, by calculating directly, we have t(K2) = 6n. On the other hand, by [Ko3, Corollary 3.61, we have t(Kl#K2) < 8n. Thus t(K1) + t(K2) - t(K1#K2) > 3n + 6n - 8n = n, and this completes

the

proof.

O

We close this section by stating two results concerning tunnel numbers and connected sum, one of them is due to Kobayashi and the other is due to Kown. Theorem 2 .6 ([Ko2, Corollary 1]). in S. Then the following holds. n < t(Kl#K2# ... #K3n)•

Let K1i K2, • • • , K3n be 3n non-trivial knots

Theorem 2 .7 ([Kw, Corollary 1]). Let K1, K2,- • • , K.+ 1 be m + 1 non-trivial knots in S3 . Then the following holds. t(K1) + t(K2) + ... + t(Km+1) < 3t(Kl#K2# ... #K3n) + 3m.

3. Knots whose tunnel numbers go up under connected sum In this section, we prove Theorem 1.6. Let K be a knot in an orientable closed 3-manifold M. Then we say that K admits a (g, b)-decomposition if there is a genus g Heegaard splitting (V1, V2) of M such that K intersects V in b-string trivial arc system for i = 1, 2 (cf. [Do, MS]). Let K be a knot in S3 which admits a (g, b)-decomposition, then by taking g central curves of a handlebody of the Heegaard splitting together with b -1 arcs connecting the b-string trivial arcs, we see that K has tunnel number at most g + b - 1 (Figure 3.1). Hence, if a knot Kin S3 admits a (1,1)-decomposition, then we have t(K) < 1.

g=3, b=4

Figure 3.1

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 15

Since tunnel number one knots are prime, together with Fact 1.1, we have the inequality 2 < t(Ki#K2) < 3 for any tunnel number one knots Kl and K2. In [Mo2], we showed : Proposition 3.1 ([Mo2, Corollary 2]). Let Kl and K2 be tunnel number one knots in S. Then t(Kl#K2) = 3 if and only if neither Kl nor K2 admits (1,1)decompositions. By the above proposition, to find tunnel number one knots which we are looking for, it is sufficient to find tunnel number one knots which admit no (1,1)decompositions. Let p and q be coprime integers, and let r be an integer. We denote the knot or the link illustrated in Figure 3.2 by K(p, q; r). Then it is a knot or a 2-component link according as r is even or odd. Let r be the arc indicated in Figure 3.2, then by untying the crossings above the arc through edge slides along the arc, T becomes an unknotting tunnel for the torus knot of type (p, q) (c.f. [BRZJ). Hence we have : Fact 3.2 K(p, q; r) has tunnel number one.

K(p, q; r)

(p, q) torus knot

r crossings

Figure 3.2

In 1991, we made the following. Conjecture 3.3. If p 0 ±1 mod q, q 0 ±1 mod p and n # 0, 1, -1, then the knot K(p, q; 2n) admits no (1,1)-decompositions.

Remark 3.4. By [BRZ], any torus knot of type (p, q) has at most three unknotting tunnels up to isotopy, and the torus knot has exactly three unknotting tunnels

16 KANJI MORIMOTO

up to isotopy if and only if p # ±1 mod q and q 0 ±1 mod p. Then by Proposition 1.5 of [MS], two of the three unknotting tunnels come from (1,1)-decompositions of the torus knot and the third one, which corresponds to the unknotting tunnel T in Figure 3 .2, does not come from any (1,1)-decompositions of the torus knot. This is the basis of Conjecture 3.3. Remark 3.5. Let T be a rational tangle. Then by exchanging the r-crossigs in the diagram of K(p, q; r) for T, we get a knot or a link and denote it by K(p, q; T). Then K(p, q; T) has tunnel number one similarly to that K(p, q; r) has tunnel number one. Consider the case when p = 5 and q = 7. Then by using Montesinos' technique [Mt], we see that E2(K(5,7;T)) is obtained by a Dehn surgery along the pretzel knot of type (3, 3, -3). Hence by Theorem 0.6 of [MR], we see that if K(5, 7; T) is a knot then it admits no (1,1)-decompositions for any sufficiently complicated rational tangle T. Let m be an integer, and let K. be the knot K(7,17; 10m - 4) illustrated in Figure 1.2. Then to prove Theorem 1.6, it is sufficient to prove the following. Theorem 3.6 ([MSY, Theorem 2.1]). decompositions.

For any integer m, Krn admits no (1, 1)-

To prove the above theorem, we use Yokota's result ([Yo2]), which gives a necessary condition for a knot K in a 3-manifold M to admit a (g, b)-decomposition in terms of the quantum SU(2)-invariant ZM,K(s,,(a)) (see [Yo2] for the notations). Then since I ZS3,K( s1(a))I = 2cos

I VK(e1' r-3) , where VK(•) is the Jones polyno-

mial ([Jo]), we have : Proposition 3.7 (a special case of [Yo2, Theorem 5.3]). S3. If K admits a (g, b)-decomposition, then :

Let K be a knot in

6-1 2 sin - -9 2 Cos -

r

r

In particular, if K admits a (1,1)-decomposition, then V

V (e 2x -I) < K 2 sin

r

Proof of Theorem 3.6. Put r = 5 in Proposition 3.7. Then, since 2sin _ 1.902. • • , to show that Km admits no (1,1)-decompositions, it is sufficient to show that the absolute value of the Jones polynomial of the knot at ea s is greater than 1.902••.

TUNNEL NUMBER AND CONNECTED SUM OF KNOTS 17

Let Vm(t) be the Jones polynomial of Km. Then, since Km is obtained by (5m-2)full twists along parallel two strings in the torus knot diagram, and by using the twisting formula of the Jones polynomial due to Yokota ([Yol]), we have : Vm(e& ) = Vo (e 5 )

for any integer m.

Hence, it is sufficient to calculate the Jones polynomial V0(t). Then by using Kauffman's bracket polynomial ([Ka]), we have : Vo(t) = -t46 (t39 - t38 + t37 - t36 - t33 + t32 - 2t31+ 2130-t29 + t28+t22+t20 + t18-t6-t4-t2-1).

Then we have I Vo (e

2, 5

) I = 2.041 • • • . This is greater than 1.902. • • , and com❑

pletes the proof of Theorem 3.6.

Remark 3.7. Let V (t) be the Jones polynomial of a tunnel number one knot.

I

Then by Kohno' s estimate ([Kh] ), we have I V (e a* 5) < 2.148. • .. This shows that the above value 2.041 • • • is a very delicate one.

REFERENCES [BRZ] M. Boileau, M. Rost and H. Zieschang , On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces , Math. Ann. 279 , ( 1988) 553-581. [Do] H. Doll, A generalized bridge number for links in 3-manidolds, Math. Ann. 294 , ( 1992) 701-717. [Ha] W. Haken , Some results on surface in 3-manifolds , Studies in Modern Topology, Math. Assoc. Amer ., Prentice-Hall (1968). [Ja] W. Jaco, Lectures on three manifold topology, CBMS Regional Conf. Ser. in Math. (1980). [Jo] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math . 126, (1987 ) 335-388. [Ka] L. H . Kauffman,

State models and the Jones polynomials , jour Topology 26, (1987)

395-407. [Kol] T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21, (1984) 437-455.

[Ko2] , Structures of full Haken manifolds, Osaka J. Math. 24, (1987) 173-215. [Ko3]

, A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum, J. Knot. Rami. 3, (1994) 179-186.

[Kh] T. Kohno ,

Tunnel number of knots and Jones- Whitten invariants, preprint.

[Kw] H. Z. Kowng, Straightening Tori in Heegaard splittings , preprint.

18 KANJI MORIMOTO [Mol] K . Morimoto, On minimum genus Heegaard splittings of some orientable closed 3manifolds, Tokyo J. Math. 12, (1989) 321-355. ]Mo2]

, On the additivity of tunnel number of knots, Topology Appl. 53, (1993) 37-66.

[Mo3]

, There are knots whose tunnel numbers go down under connected sum, Proc. A. M. S. 123 , ( 1995) 3527-3532. , Charaterization of tunnel number one knots which have the property " 2+1=2 ",

(Mo4]

Topology Appl. 64, (1995) 165-176. [MS] K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 , (1991) 143-167. (MSY] K. Morimoto, M. Sakuma and Y. Yokota,

Examples of tunnel number one knots which

have the property " 1 + 1 = 3 ", Math. Proc. Camb. Phil. Soc. 119, (1996) 113-118. [Mt] J. M. Montesinos , Surgery on links and double branched covers of S3, Ann. Math. Studies 84, (1975 ) 227-259. [MR] Y. Moriah and H . Rubinstein ,

Heegaard structures of negatively curved 3-manifolds,

Preprint. [No] F. H. Norwood, Every two generator knot is prime , Proc. A. M. S. 86 , (1982) 143-147. [Oc] M. Ochiai, On Haken's theorem and its extension, Osaka J. Math. 20, (1983) 461-468. [Rol D. Rolfsen, Knots and links, Math. Lect. Note Series 7. Publish or Perish Inc. (1976). [Sc] M. Scharlemann , Tunnel number one knots satisfy the Poenaru conjecture , Topology Appl. 18, (1984) 235-258. [Yol] Y. Yokota,

Twisting formulae of the Jones polynomial, Math. Proc. Carob. Phil. Soc.

110, (1991) 473-482. , On quantum SU(2) invariants and generalized bridge numbers of knots, Math.

[Yo2]

Proc. Camb. Phil. Soc. 117, (1995) 545-557. DEPARTMENT OF MATHEMATICS, TAKUSHOKU UNIVERSITY, TATEMACHI, HACHIOJI, TOKYO 193, JAPAN. E-mail address : [email protected]

Lectures at Knots 96 edited by Shin'ichi Suzuki

@1997 World Scientific Publishing Co. pp. 19-37

TOPOLOGICAL IMITATIONS AKIO KAWAUCHI

0. Introduction . We consider a (3,1)-manifold pair (or simply a manifold pair) which is a pair (M, L) such that M is a smooth compact connected oriented 3manifold and L is the empty set 0 or a proper (possibly disconnected) oriented smooth 1-submanifold. A topological imitation of a manifold pair (M, L) is a manifold pair (M*, L*) together with a smooth map q : (M*, L*) -- (M, L) with some properties close to a diffeomorphism (cf. [6,7, 8,9,11 , 12,15]). A useful concept in topological imitations is an almost identical imitation, which we call here an AID imitation. It is roughly a topological imitation q : (M*, L*) --> (M, L) of a manifold pair (M, L) with L # 0 which has the following properties (1) and (2) (see §1 for the detail):

(1) (Normalness ). There exist tubular neighborhoods N(L*) and N(L) of L* C M* and L C M, respectively, such that the restriction gI(aM*UN(L*),L*) : (am* U N(L*), L*) -> (aM U N(L), L) is a diffeomorphism and q(M* - intN (L*)) = M intN(L). (2) (Homological Equivalence ). The lifting of the restriction qIM*-intN(L*) : M* intN(L*) -> M - intN(L) to every covering over M - intN(L) is homologically equivalent. (3) (AID Property). The restriction qjM*-intN(L*-a*) : M* - intN(L* - a*) --+ M - intN(L - a) is (boundary- relatively ) homotopic to a diffeomorphism for every pair a* , a of components of L*, L with q(a*) = a, where N(L* - a*) and N(L - a) are the tubular neighborhoods of L* - a* and L - a obtained from N(L*) and N(L) in (1) by removing the components which contain a* and a, respectively.

For a trivial link L in the 3-sphere M = S3, the AID imitation q : (M*, L*) -+ (M, L) is closely related to the concept of an almost trivial link (or, equivalently, a link with Brunnian property) in knot theory which is defined as follows: 19

20 AKIO KAWAUCHI

Definition 0.1. A link L in S3 of r (> 2) components K; (i = 1,2,... , r) is almost trivial if the sublink L - K; is trivial in S3 for all i. (In the case r = 2, we further impose that the linking number of the link L is 0.) Trying to strengthen the concept of an almost trivial link, we also have the following concept of an almost trivial link in the strong sense: Definition 0.2. A link L in S3 with r (> 2) components K; (i = 1, 2, ... , r) is almost trivial in the strong sense if L is an almost trivial link and every branchmissing cyclic covering link of L (i.e., the lift of the sublink L - K; to every finite cyclic covering space over S3 branched over K; for every i) is also an almost trivial link. For example, the Whitehead link is almost trivial, but not almost trivial in the strong sense. On the other hand, the Milnor link which is shown in Figure 0.1 (see [20]) is almost trivial in the strong sense.

Figure 0.1 The Milnor link is not any imitation of a trivial link by the homological equivalence of the topological imitation shown in [7 ] ( because this link module is distinct from that of a trivial link), but this concept leads us to the concept of a strongly AID imitation (see [15,18] and Definition 1.4). It is roughly an AID imitation q : (M*, L*) (M, L) of a manifold pair (M, L) with L # 0 together with the following property: (3)* (Strongly AID Property). The lift of the map q : (M`, L*) -+ (M, L ) to every finite regular covering p : (M, L) -+ (M, L) such that k is connected and the branch set is a proper subfamily of the components of L is still an AID imitation after missing the branch set. In this paper, we shall discuss some results around this strongly AID imitation. In §1, several concepts of topological imitations are explained . In §2, the main result (Theorem 2. 4) on the existence of a strongly AID imitation is stated. In §3, an

TOPOLOGICAL IMITATIONS 21

outline of the proof of the main result is shown. In §4, mutative versions of the main result are shown. In §5, we make two applications on the skein (=HOMFLY, LYMPHTOFU) polynomial of links and the 3-manifold invariants, which generalize some results of [10,15,16] . 1. Several concepts of topological imitation . Let I = [-1,1]. The concept of topological imitation arose from an interpretation of reflection, which is stated as follows: Definition 1.1. For a manifold pair (M, L), a smooth involution a on (M, L) x I = (M x I, L x I) is a reflection in (M, L) x I if: (1) a((M, L) x 1) = (M, L) x (-1), and (2) the fixed point set Fix(a, (M, L) x I) of a in (M, L) x I is a manifold pair. A normal covering p : (M, L) -+ (M, L) is a finite regular covering with k connected whose branch set is 0 or a subfamily of the components of L. Let LF denote the branch set of this normal covering . Let L0 = L - LF and L0 = p 1(L0). We note that every reflection a in (M, L) x I lifts uniquely to a reflection a in (M, L0) x I for every normal covering p : (M, L) -+ (M, L) (see [7]). In the following three definitions , we denote by (M, L) a manifold pair contained in a manifold pair (M, L) and by a a reflection in (M, L) x I, and by (M', L') the exterior pair cl(M - M, L - L) which is assumed to be a disjoint union of manifold pairs. Definition 1.2. (1) The reflection a is standard if a(x, t) = (x, -t) for all (x, t) E M X I. (2) The reflection a is normal if a(x, t) = (x, -t) for all (x, t) E 8(M x I) U N(L) x I for a tubular neighborhood N(L) of L in M. (3) The reflection a is isotopically standard if f -'of is standard for a diffeomorphism f of M x I which is isotopic to the identity by an isotopy keeping 8(M x I) U N(L) x I fixed for a tubular neighborhood N(L) of L in M.

(4) The reflection a is isotopically (M', L') -co-standard if a is normal and the reflection in (M U M1', L U Li) x I defined by a and the standard reflection in (Mi, Li) x I is isotopically standard for any (3,1)-manifold pair (M1', Li) such that M1' is a component of M' and L'1 = Mi fl L'. (5) The reflection a is isotopically almost standard if L # 0 and a defines an isotopically standard reflection in (M, L - a) x I for each component a of L. (6) The reflection a is isotopically almost standard in the strong sense if the lift a of a to (M, L0) x I is isotopically almost standard for every normal covering p : (M, L) -+ (M, L) with L0 # 0.

22 AKIO KAWAUCHI

We note that ( 6) means (5) since the identity map is a normal covering map. Definition 1.3.

A reflector of a reflection a in (M, L) x I is a smooth embedding 0. (M', L') -+ (M, L) x I

with 0a,(M', L') = Fix(a, (M, L) x I). Definition 1.4. An imitation of (M, L) is the composite q : (M*, L') ""+ (M, L)

where

X I projection

(M, L)

(M', L*) -+ (M, L) x I is reflector of a reflection a in (M, L) x I.

The manifold pair (M*, L*) is also called an imitation of (M, L) (with imitation map q). In this definition, if the reflection a is normal, then we say that the imitation q is a normal imitation. If a is isotopically (M', L')-co-standard, then we say that the imitation q is an (M', L')-co-identical imitation. If a is isotopically almost standard, then we say that the imitation q is an AID (=almost identical) imitation. If a is isotopically almost standard in the strong sense, then we say that the imitation q is a strongly AID imitation. We note that a normal (or AID, or strongly AID, respectively) imitation of a normal (or AID, or strongly AID, respectively) imitation is a normal (or AID, or strongly AID, respectively) imitation of the original manifold pair (cf. [71). By definition, strongly AID imitations are AID imitations and AID imitations are normal imitations. Every normal imitation q : (M', L') -+ (M, L) induces a homology equivalence and gives the normalness of q (stated in §0) and defines a normal imitation qE : E(L', M*) -+ E(L, M) for the compact exteriors E(L*, M') = M* - intN(L') and E(L, M) = M - intN(L). Every AID imitation q : (M*, L*) -+ (M, L) gives the AID property of q (stated in §0) and in this case we can identify M* with M so that gI8M is the identity on 8M by a choice of the reflector cb used for the definition of q. From this reason, we also denote the AID imitation q : (M', L') -+ (M, L) by q : (M, L') -+ (M, L). A sphere component S of 8M is called an n-pointed boundary sphere for a manifold pair (M, L) if IS n LI = n. If there is a 1-pointed boundary sphere S for (M, L), then the AID imitation map q : (M, L') -+ (M, L) is homotopic to a diffeomorphism by a homotopy relative to 8M U N(L' - a*) for a component a' of L' with IS n a'I = 1 and a tubular neighborhood N(L* - a') of L' - a' in M. If S is a 2-pointed boundary sphere for (M, L), then we can construct a new manifold pair (M+, L+) from (M, L) by a spherical completion, i.e., by adding a cone over (S, S n L) and the AID imitation q : (M, L') -+ (M, L) extends to a unique AID imitation q+ : (M+, L'+) -+ (M+, L+) with E(L'+, M+) = E(L*, M). This means that in order to construct an AID imitation q : (M, L*) -* (M, L) with

TOPOLOGICAL IMITATIONS 23

E(L*, M) 99 E(L, M), we can assume without loss of generality that there are no npointed boundary spheres for (M, L) with 0 < n < 2. Such a manifold pair (M, L) is called a good manifold pair. When L = 0, M is good if and only if 8M has no sphere components. From every manifold pair (M, L), we can obtain a unique good manifold pair which we denote by (M, L)A by taking spherical completions after deleting the arcs of L meeting the 1-pointed boundary spheres. Every normal imitation map q : (M*, L*) -+ (M, L) induces a unique normal imitation q„ : (M*, L*)A -+ (M, L)A which we call the spherical completion of the normal imitation map q. The following property for a normal covering p (M, L) and a normal imitation q : (M*, L*) -+ (M, L) is proved in [7]. Proposition 1.5. In the following pullback diagram, p* : (M*, L*) (M*, L*) is a normal covering and q : (M*, L*) -^ (M, L) is a normal imitation: (1t1, L) P*1

1p

(M*, L*) -° + (M, L). In Proposition 1.5, we call the normal imitation q the p-lift of the normal imitation q, and the normal covering p* the q-lift of the normal covering p. If q : (M*, L*) -+ (M, L) is a strongly AID imitation, then the p-lift q : (M*, L0) -+ (M, L0) and its spherical completion q,, : (M, L0)„ -+ (M, L0),, are AID imitations when L0(= L - LF) # 0. For a good manifold pair (M, L), let Lo be a subfamily of L (possibly, 0) such that Lo D L0 and Lo = p1(Lo). Since (M, L0) has no 1-pointed boundary sphere, we have a unique good manifold pair (M, Lo), by the spherical completions on (Al, Lo) which we simply denote by (M, L). The restricted (in general, branched) covering pl4 : M -+ M extends to a unique covering Tl : M -+ M+ where M+ is a 3-manifold (not necessarily good) obtained from M by some spherical completions. The good manifold pair (M, L) is called a branch-missing manifold pair over (M, L). Let G(MIM) denote the covering transformation group of the normal covering p(M : M --+ M, which naturally extends to an action on (M, L). The group G(MIM) and its order are respectively called the transformation group and the degme of (M, L) over (M, L). We note that there are in general finitely many branch-missing manifold pans (M, L) for each normal covering p : (M, L) -+ (M, L), since there are in general finitely many choices of L0 for each normal covering p : (M, L) -+ (M, L). In particular, the branch-missing manifold pair (M, L) with L0 = L0 (namely, the manifold pair (M, Lo)A)) is denoted by (M, U. We say that a good manifold pair (M', L') has a G-action if it is diffeomorphic to a branch-missing manifold pair (M, L0) = (M, L0),, over a good manifold pair (M, L) with transformation group

24 AKIO KAWAUCHI

G. Then the group G acts faithfully on (M', L') and orientation-preservingly on M' and almost freely on L' (namely, freely on L' missing a finite subset of intL'). We also note that the topological imitation is well fitted to the argument of Dehn surgery. For example, we consider a normal imitation q : (M*, L*) -> (M, L) such that L contains a knot component K and L1 = L - K # 0. Let K* = p - '(K) and L*1 = q-1(L1). Taking the Dehn surgeries of M* and M along K* and K with any surgery slopes related by q, we obtain a normal imitation q' : (M*. Li) -' (M', L'1) induced from q. Further, if q is an AID (or a strongly AID, respectively) imitation, then we see easily that q' is an AID (or a strongly AID, respectively) imitation. 2. The main result on the existence of a strongly AID imitation. We shall make use of hyperbolic invariants such as the hyperbolic volume and the hyperbolic isometry group of a hyperbolic 3-manifold to confirm the existence of non-trivial topological imitations of a good manifold pair (cf. [24, 25,26]).

A (compact connected oriented) 3-manifold M is hyperbolic if we have the following (1 ) or (2): (1) The boundary 8M is 0 or a union of tori and intM has a complete hyperbolic structure (that is, a complete Riemannian structure of constant curvature -1). (2) The double DM of M pasting along the non-torus components of aM has the property of the case (1) when we regard DM as M. A hyperbolic 3-manifold is a good 3-manifold. If M is a hyperbolic 3-manifold in the sense of (2), there is a unique order 2 isometry r of the hyperbolic 3-manifold DM induced from the involution exchanging the two copies of M in DM by Mostow's rigidity theorem, so that the orbit space of DM by r is topologically M (cf. [25]). When M is in the case (1), the volume VolM and the isometry group IsomM of a hyperbolic 3-manifold M are respectively defined to be the hyperbolic volume Vol(intM) and the hyperbolic isometry group Isom(intM). When M is in the case (2), they are respectively defined to be Vol(int(DM))/2 and the quotient group of the group {f E Isom(int D1M) I f r = 7-f } by the subgroup {1, r}(= Z/2Z). The volume VoIM and the isometry group IsomM (up to conjugations in Diff+M) are topological invariants of M by Mostow's rigidity theorem. Definition 2.1. A good manifold pair (M, L) has HCP (= the hyperbolic covering property) if E(L, k) is a hyperbolic 3-manifold for all branch missing manifold pairs (M, L) over (M, L).

TOPOLOGICAL IMITATIONS 25

We note that a good 3-manifold M has HCP if and only if M is hyperbolic. The following concept of a rigid normal imitation is given in [11, 12,15,18]. Definition 2.2. A normal imitation q : (M*, L*) , (M, L) of a good manifold pair (M, L) is rigid if E(L*, M*) is a hyperbolic 3-manifold and IsomE(L*, M*) G(MIM) for the q-lift p* : (M*, L*) --p (M*, L*) of every normal covering p (M, L) -+ (M, L). When L = 0, we note that a normal imitation q : M* -> M is rigid if and only if M* is a hyperbolic 3-manifold with IsomM* = G(M]M) for the q-lift p* : M* -> M* of every normal covering p : M -+ M. The following notion of a J-rigid normal imitation is slightly stronger than that given in [11,12,15] on the condition (2), although we can see (from Lemma 3.4) that the main result of [11] still holds (see [181): Definition 2.3. For a positive integer J, a normal imitation q : (M*, L*) -4 (M, L) of a good manifold pair (M, L) is J-rigid if we have the following (1)-(4): (1) The normal imitation q is rigid. (2) The good manifold pair (M*, L*) has HCP and the q-lift (1tI*, L*) of every branch-missing manifold pair (M, L®) over (M, L) has HCP. (3) There is an isomorphism IsomE (L*, k*) = G for the q-lift (M*, L*) of every branch-missing manifold pair (M, L) over (M, L) with transformation group G of order < J. (4) Every branch- missing manifold pair (M*, L*) over (M*, L*) of order < J is the q-lift of a branch-missing manifold pair (M, L) over (M, L). In this definition, (2) implies that the good manifold pair (M*, L*) has HCP by taking the identity map (which is a normal covering), and (3) is contained in (1) when L = 0, and (4) is equivalent to saying that the map Hom(7rl (E(L, M)), G) --> Hom(irl (E(L*, M*)), G)

induced from the homomorphism (qE)# : 7r1(E(L*,M*)) --4 7r1(E(L,M)) (which is onto by [7]) is bijective for all groups G of order < J. In particular, when L = 0, a normal imitation q : M* -+ M is J-rigid if and only if q is rigid and the map Hom(7rl (M), G) -> Hom(7rl (M*), G) induced from the epimorphism q# : 7r1(M*) -, 7r1 (M) is bijective for all groups G of order < J. By a family of manifold pairs, we mean an infinite family £ of good manifold pairs (Mm, Lm) with m ranging over the set of positive integers N. It is also called a knot (or link) family when Mm = S3 and Lm is a link (or knot) for all m. For a positive

26 AKIO KAWAUCHI

number C, the family £ is said to be C-hyperbolic if E(Lm, Mm) is a hyperbolic 3-manifold for each m with C < VoIE(Lm, Mm) < sup VoIE(Lm, Mm) < oo. mEN

A family S' of normal imitations (Mm, Lm) (m E N) of a good manifold pair (M, L) is said to be regularly C-hyperbolic if the family of the qm-lifts (Mm, Lm) (m E N) of every branch-missing manifold pair (M, L) over (M, L) of any order r is rChyperbolic, where qm denotes the imitation map from (Mm, Lm) to (M, L). We note that every C-hyperbolic family of manifold pairs contains infinitely many members whose exteriors are distinct (up to diffeomorphisms) by Mostow's rigidity theorem (cf. [24]). Our constructing result on strongly AID imitations is stated as follows: Theorem 2 .4. For every good manifold pair (M, L) with L # 0, any positive integer J, and any positive number C, there is a regularly C-hyperbolic family ` of J-rigid strongly AID imitations (M, Lm) (m E N) of (M, L).

3. An outline of the proof of Theorem 2.4. The full proof of Theorem 2.4 is given in [18]. Here we describe an outline of the proof. The following lemma is basic to Theorem 2.4: Lemma 3 .1. For every good manifold pair (M, L) with L yl- 0, there is a strongly AID imitation q : (M, L*) -> (M, L) such that (M, L*) has HCP and the q-lift (M, L0) of every branch-missing manifold pair (M, L0) over (M, L) has HCP. If a hyperbolic 3-manifold M* is a normal imitation of a hyperbolic 3-manifold M, then we have Vo1M* > Vo1M by an argument regarding the hyperbolic volume as the Gromov norm (cf. [24,26]). Since a strongly AID imitation of a strongly AID imitation is a strongly AID imitation of the original manifold pair, the volume estimation for the regularly C-hyperbolic family of Theorem 2.4 follows from the following lemma (if Lemma 3.1 is proved): Lemma 3.2 . For any good manifold pair (M, L) with L # 0, there is a strongly AID imitation q : (M, L*) -4 (M, L) such that (M, L*) has HCP and VoIE(L*, k) > rC for the q-lift (M, L*) of every branch-missing manifold pair (M, L) over (M, L) of any order r.

Proof of Lemma 3.2. For a union L+ of L and a trivial knot 0 in intE(L, M), we take a strongly AID imitation q'+ : (M, L+) -> (M, L+) such that (M, L+) has HCP. Let O' be the component of L' with q+' (0') = O. Then q+ sends a tubular neighborhood N' of O' onto a tubular neighborhood N of 0 homeomorphically. We replace the homeomorphism q+I N' : N' --* N (with correct meridian-longitude systems) by

TOPOLOGICAL IMITATIONS 27

the exterior normal imitation of a normal imitation qo : (S3, 0*) -> (S3, 0) of a trivial knot (S3, 0) such that 0* is a hyperbolic knot and VoIE(O*, S3) > C. The result is a strongly AID imitation q" : (M, L") -* (M, L) such that the exterior E(L", M) is a torus sum of the hyperbolic exteriors E((L+)', M) and E(O*, S3). We take further a strongly AID imitation q* : (M, L*) -* (M, L") such that (M, L*) has HCP. Combining it with the strongly AID imitation q", we obtain a strongly AID imitation q : (M, L*) _ (M, L). Let (M, L*) be the q-lift of every branchmissing manifold pair (M, L) over (M, L) of any order r. Then there is a degree one map (which is a normal imitation map) from the exterior E(L*, M) to the exterior E(L", 1t) of the q"-lift (M, k) of (M, L). Since (M, (L+)') has HCP, we see that the exterior E(L", k) is a torus sum of a hyperbolic 3-manifold E (which is the exterior of a branch-missing manifold pair over (M, (L+)')) and r copies of the hyperbolic 3-manifold E(O*, S3). By regarding the hyperbolic volume as the Gromov norm, we have VoIE(L*, M) > Volt + rVolE(O*, S3) > rC (see Thurston [26] and Soma [23]). This completes the proof of Lemma 3.2. Combining Lemmas 3.1 and 3.2 with the proofs of [9, Main Theorem] and [11,Theorem 1], we can complete the proof of Theorem 2.4. We shall show an outline of the proof of Lemma 3.1. From an argument of Heegaard splittings of a good manifold pair, we see that Lemma 3.1 is a consequence of the following two lemmas (cf. [9]): Lemma 3.3. For every r(> 3)-string trivial tangle (B,T) with B a 3-ball, there is a strongly AID imitation q : (B, T*) --* (B, T) such that (B, T*) has HCP. Let T be an r(> 1)-string trivial tangle in a handlebody V such Lemma 3.4. that genus(V) + r > 3. Let q : (V*,T*) -* (V,T) be a normal imitation such that (V*,T*) has HCP. Then the q-lift (V*,T) of every branch-missing manifold pair (V,Te)) over (V, T) has HCP.

Lemma 3 .4 is seen from [9 , Lemmas 1.4, 1.5] (see [18]). We show Lemma 3.3. Proof of Lemma 3 .3. Let Ti (i = 1, 2, ... , r) be the components of T. We choose mutually disjoint 3-balls Bi(i = 1 , 2, ... , r) in B so that Bi meets T in an (r + 1)trivial tangle with # (Bi n T'3) = 1 + Si, where 6ij denotes Kronecker's delta. Let V = cl (Bi - Uj#iN(Tj)) which is a handlebody of genus r - 1. Let Ti' = Ti n V, C = cl (B - U 1 Vi), and To = T n C. We assume that the component ai of Tin C with 8ai c aV is unknotted in C - (TC - a,). Let h;' C C be a 2-handle on V which surrounds the arc ai and is made disjoint for all i, and h? C V -Tv a 2-handle on C which is a collar of a meridian disk of V. By [11, Lemma 1.2], we have a (U,r=1h;')-coidentical AID imitation qC : (C, (TC)*) - (C, 7) such that (C, (Tc)*) has HCP.

28 AKIO KAWAUCHI

W e consider that the tangles (V, T?') and the 2-handles hT (i = 1, 2, ... , r) are copies of the tangle (V, TV) and the 2-handle h'' shown in Figure 3.1. By assuming that the following lemma is proved, we continue the proof of Lemma 3.3:

Figure 3.1 *--4 (V,Tv) Lemma 3 .5. There is an h''- co-identical AID imitation q'': (V, (Ti')) such that (V, (T'')*) has HCP and the p-lift qV : (V, (T'')*) -, (f/, tv) is an AID imitation for every unbranched normal covering p : (V, Ti') --.. (V, TV). Let q; : (Vi, (T!')*) - (Vi, TV) (i = 1, 2, ... , r) be copies of qv of Lemma 3.5. Then we see from construction that the imitations qC and qV (i = 1, 2, ... , r) define a strongly AID imitation q : (B, T*) -+ (B, T). By the Myers gluing lemma (cf. [8,21]), the tangle (B, T*) has HCP. This completes the proof of Lemma 3.3, assuming Lemma 3.5. Prof of Lemma 3.5. We deform TV isotopically in V to divide (V, TV) into the tangle (W, T') with 2-handles h'', h"' (j = 1, 2, ... , r - 1) and the tangles (Uj, T°) with 2-handles hu (j = 1, 2, ... , r - 1) which are shown in Figure 3.2. We denote the tangle (U„ T') and the 2-handle h° by (U,Tu) and hu (since they are the same one for all j). By [11, Lemma 1.2], we have an (h''Uj:__i h"')-co-identical AID imitation qW : (W, (Tu')*) - (W, Tw) such that (W, (Tw)*) has HCP and an hu-co-identical AID imitation qu : (U, (Tu)*) -* (U,Tu) such that (U, (T")*) has HCP. The imitation qW and the copies qju : (U„ (T°)*) -* (U,, Tu)(j = 1, 2, ... , r *-+ 1) of the imitation qu define an h''-co-identical AID imitation q'' : (V, (Ti')) (VT''). By the Myers gluing lemma (cf. [8,21]), the tangle (V, (T')*) has HCP. Further, from construction we can see that this imitation q'' has a desired property stated in Lemma 3.5. This completes the proof of Lemma 3.5. This completes the outlined proof of Theorem 2.4.

TOPOLOGICAL IMITATIONS 29

hi

(Ui, TU) (U2, T2 U)

(Ur-1,

U Tr

-i)

Figure 3.2 4. Mutative version. We discuss here the concept "mutative imitations "(cf. [11,12,14,15]). This concept is obtained by combining the mutations on closed genus 2 surfaces (see Ruberman [22]) with the topological imitations. An involution p on a closed surface F is called a symmetry of F if the orbit space F/p is a 2sphere. A good manifold pair (M', L') is an e-mutation of a good manifold pair (M, L) if there is a separating closed surface F of genus 2 in intE(L, M) such that (M', L') is obtained from (M, L) by cutting along F and then regluing by a symmetry p of F. Then (M, L) is also an e-mutation of (M', L') and we can say without ambiguity that (M, L) and (M', L') are e-mutative. Two good manifold pairs (M, L) and (M', L') are said to be mutative if there is a finite sequence of good manifold

30 AKIO KAWAUCHI

pairs (M(n), L(n))(n = 0, 1, 2,..., s) with (M(o), L(o)) = (M, L) and (M(,), L(,)) = (M', L') such that (M(n), L(n)) and (M(n+l), L(n+l)) are e-mutative for all n. Two normal imitations q : (M*, L*) -. (M, L) and q' : (M*', L*') -• (M, L) of a good manifold pair (M, L) are said to be related by a trivial e-mutation on (M, L) if there is a handlebody V of genus 2 in an open 3-ball embedded in the compact exterior E(L, M) such that for the preimage V* = q-1 (V), the restrictions qlV* : V* -• V and ql(M* - intV*, L*) : (M* - intV*, L*) -• (M - intV, L) are normal imitations (and in particular, 8V* is a closed genus 2 surface) and q' is obtained from q by taking an e-mutation of (M, L) on 8V and then taking the associated e-mutation of (M*, L*) on OV*. We note that (M, L) is unchanged by an e-mutation on aV (see [11, Lemma 2.2]). Definition 4.1. Two normal imitations q : (M*, L*) -• (M, L) and q: (M*', L*') (M, L) are mutative if there is a finite sequence of normal imitations q(n) : (M(n), L(n)) - (M, L) (n = 1, 2,..., s)

of (M, L) with q(o) = q and q(,) = q' such that q(n) and q(n+1) are related by a trivial e-mutation on (M, L) for all n. We note that if two normal imitations q : (M*, L*) -• (M, L) and q' : (M*', L*') -• (M, L) are mutative, then the good manifold pairs (M*, L*) and (M*', L*') are mutative.

Definition 4.2. Two normal imitations q : (M*, L*) -• (M, L) and q' : (M*', L*') -• (M, L) are properly mutative if q and q' are mutative and E(L*, M*) E(L*', M*') for the q-lift p* : (M*, L*) -• (M*, L*) and the q'-lift p*' (M*', L*') of every normal covering p : (M, L) -. (M, L). Definition 4.3. Two normal imitations q : (M*, L*) -• (M, L) and q: (M*', L*') -. (M, L) are J-properly mutative for a positive integer J if q and q' are properly mutative and E(L*, M*) E(L*', k*) for all branch-missing manifold pairs (M*, L*) and (M*', L*') of the q-lift p* : (M*, L*) -• (M*, L*) and the q'-lift p*' : (M*', L*') -. (M*', L*') of every normal covering p : (M, L) -+ (M, L) of degree < J. When L = 0, the normal imitations q and q' are J-properly mutative if and only if they are properly mutative. Our mutative version of Theorem 2.4 is stated as follows: Theorem 4 .4. For every good manifold pair (M, L) with L y6 0, any positive integers J, N, and any positive number C, there are regularly C-hyperbolic families (n)(n = 1 , 2,..., 2N) of J- rigid strongly AID imitations (M, Lin)) E fi(n)

TOPOLOGICAL IMITATIONS 31

(m E N) of (M, L) such that the imitation maps q(n) : (M, L-)) --+ (M, L) and q(n) : (M, Lin,)) -+ (M, L) are J-properly mutative for all n, n' with n # Ti' and all m E N. Theorem 4.4 can be obtained by an argument parallel to the proof of [11,Theorem 2.4] when we use Theorem 2.4. We have the following theorem which is sharpening the result of [11,Theorem 3.2], but can be similarly proved by using Theorem 4.4. Theorem 4.5. If a good manifold pair (M, L) with L 0 has a G-action, then for any positive integer N and any positive number C, there are C-hyperbolic families (n) (n = 1, 2,..., 2N) of AID imitations (M, L'(„)) E `3`(n) (m E N) of (M, L) such that (1) the AID imitation (M, L^n)) has HCP and IsomE(Ln), M) = G, and

(2) the imitation maps q(n) : (M, Ln)) -+ (M, L) and q(n,) : (M, Lm(,,,)) --+ (M, L) are G-equivariant and properly mutative for all n, n' with n # n' and all mEN. Further, for every proper subgroup H of G, there are C-hyperbolic families ` (n)(H) (n = 1, 2,..., 2N) of AID imitations (M, L(n)H)) E (H) that

(m E N) of (M, L) such

(3) the AID imitation (M, L'(H)) has HCP and IsomE(L'n(H), M) = H, and (4) the imitation maps q'(H) : (M, Ln^") -+ (M, L) and q(n^) : (M, Lin() )) -+ (M, L) are H-equivariant and properly mutative , and are also properly mutative to q(n ) for all n, n' with n # n' and all m E N. 5. Applications. As an application of Theorem 2.4, we consider the coefficient polynomials c,(L; x) (cf. [10,13]) of the skein (=HOMFLY, LYMPHTOFU) polynomial P(L; m, t) of a link L in S3 (cf. [4]). The polynomials cn(L; x)(n = 0, 1, 2.... ) are given by the identity (tm)#L-1PL(f,m) = Enoc,(L;x)(xy)" taking -P2 = x and -m2 = y, where #L denotes the number of components of L. Then there is a non-negative integer nL depending only on the link L such that cn(L;x) = 0 for all n > nL (cf. [13]). Writing cn(L;x) = 0 for n < 0, we have the following properties (which determine the coefficient polynomials uniquely): (1) For a trivial knot cn(O;

x) =

f 0 (n54 0)

E 1 (n=o).

(2) For all n, xc,a(L+; x) - cn(L-; x) = cn-a(Lo; x),

32 AKIO KAWAUCHI

where (L+, L_, Lo ) denotes a skein triple and with S = (#L+ - #Lo + 1)/2 (= 0or 1). We consider a good manifold pair (M, L) such that M is S3 or a compact 3submanifold of S3 obtained by removing some mutually disjoint open 3-balls. We say that such a pair (M, L) is link-admissible if there is a branch- missing manifold pair (M, Lo) over (M, L) which is a link in S3. In this case, this link is called a covering link over (M, L) and denoted by (S3, L). Given a covering link (S3, L) over (M, L), we consider disjoint two 3-balls Di(i = 1, 2) which are trivially lifted to the covering link (S3, L) such that di = Di n L is a trivial arc in Di. Let b C Dl be a trivial band attaching to dl, and let B C D2 be a 3-ball with T = d2 n B a trivial 2tangle in B . A crossing change of T in B is called a trivial crossing change if it does not change the type of (D2id2). The lifts b of b and (B, T) of (B, T) to the covering link (S3, L) over (M, L) are called a trivial lifting band family and a trivial lifting tangle family, respectively. Given a family l. of strongly AID imitations (M, L*) of a link admissible pair (M, L), then the strongly AID imitation q,M : (M, L*) - (M, L) lifts to an AID imitation q : (S3, L*) -4 (S3, L) with (S3, L*) a covering link over (M, L*) for every covering link (S3, L) over (M, L). We say that the family of such covering links (S3, L*) over all (M, L*) E is called the covering link family over associated with the covering link (S3, L) over (M, L) and denoted by the family l J. The following theorem is an equivariant generalization of the result of [10]: Theorem 5.1. For any link-admissible pair (M, L) and any positive integers d, e, J and positive number C, there is a regularly C-hyperbolic family l of J-rigid strongly AID imitations (M, L*) of (M, L) such that the covering link family £ over the family Or associated with every covering link (S3, L) over (M, L) has the following properties:

(1) The AID imitation map q : (S3, L*) --+ (S3, L) sends b* to b and (B*, T*) to (B, T) bijectively for some trivial lifting band families b* and b and some trivial lifting tangle families (.&*, T*) and (B, T) such that the maps q' : (S3 L*') _ (S3, L') and q" : (S3 L*") - (S3 L") obtained from the AID imitation map q by the surgery of L on any one band in b and by a trivial crossing change in any one tangle of (B, T) are homotopic to diffeomorphisms. (2) For the degree r of the covering link (S3, L) over (M, L), we have (i) all the covering links (S3, L*) E `5 have the same n-th coefficient polynomial c„ (L*; x) for every n, (ii) c„(L*; x) = c,,. (L; x) if n < rd, and

(iii) the difference c„ (L*; x) - c„(L; x) is divided by xe - 1 if n > rd.

TOPOLOGICAL IMITATIONS

33

Since the Jones polynomial V(L; t) (cf. [5]) of a link L is identified with P(L; t-1, - (f - t-)), we denote P#(L;

t-1, - (/ - t-1)) by V# (L; t). Then we have V#(L;t- 1) = En cn ( L;t2)tn (t - 1)2n

(cf. [13]) and the statement (2)(i)-(iii ) of Theorem 5.1 implies the following: (3) The (normalized) Jones polynomials V# (L*; t) for all L* E 5 are the same and the difference V#(L*; t) - V#(L; t) is divided by (t - 1)2rd(t2e - 1). Proof of Theorem 5.1. For each component K of L, we choose a 3-ball AK C M which intersects L with only a trivial arc in K, and take one trivial band spanning K and three 3-balls in AK which are indicated in [10 , Fig. 1], and then take arcs attaching to K in these 3-balls as indicated in [10, Fig. 2]. Here we denote these arcs (for all the components K of L) by ak(k = 1, 2, ... , t). Let r = LUi=1ak. Let Hk c F be an H-graph in a 3-ball neighborhood Vk of the arc ak in M which is shown in [10, Fig. 3]. Let Mo = cl(M - Uk=1 Vk). Let (h) , hk) (k = 1, 2, ... , t) be mutually disjoint standard complementary (1, 2)-handle pairs on Vk(k = 1, 2, ..., t) in Mo - P. Let (hk , hk)(k = 1, 2, ... , t) be mutually disjoint standard complementary (1, 2)-handle pairs on Mo in Vk - (P U hk U h2)(k = 1, 2, ... , t). Let Vk = cl(Vk - hk) U hk and M' = cl(Mo - Uk=1hk) Uk-1 hk. Let ((Vk)o, (Hk)o) be a good manifold pair obtained from (Vk, Hk) by removing an open 3-ball neighborhood of each vertex of degree 3. Let (qk)o : ((Vk)o, (Hk)o) - ((Vk)o, (Hk)o) be an hk-co-identical AID imitation such that ((Vk')o, (Hk )o) has HCP (cf. [11, Lemma 1.2]). Let qk : (Vk, Hk) --* (Vk, Hk) be the map obtained from (qk)o by the spherical completions (which is an AID graph imitation in [8]). We replace Hk by a 2-string tangle , say Tk(mk) with mk full twists as shown in [ 10, Fig . 4 (1),(2),(3)]. Then there is a constant c > 0 such that qk induces an hi-co-identical AID imitation qk : (Vk,Tk (Mk)) --> (V,, Tk(mk)) such that (Vk, T; (Mk)) has HCP for all Mk with Fmk > c and all k. For L' = M' fl L, we take by [11, Lemma 1.2] and Theorem 2.4 a (Uk=1 )-co-identical strongly AID imitation hk qm' : (M', (L')*) (M', L') such that the q',,, -lift (M', (L')0) of every branch-missing manifold pair (k,4) over (M', L') has HCP with VolE((L')*, M') > C. For any integers mk(k = 1 ,2,... , t) with 1mkI > c such that (M', L') Uk-1 (Vk, Tk(mk)) _ (M, L) (as indicated in [10]), the imitation maps qk and qM define a strongly AID imitation qM : (M, L*) --+ (M, L) such that the qM-lift (M, L0) of every branchmissing manifold pair (M, L0) over (M, L) has HCP. By Lemma 3.2, we may consider that VoIE(L0, k) > C. From our construction of the graph F, we obtain (1). With a choice of the mk's depending on d and e, the coefficient polynomials of the qM-lift (S3, L*) of every covering link (S3, L) are calculated to have the property (2) by the

34 AKIO KAWAUCHI

same method as [10, Lemmas 2.2, 2.3, 2.4]. This completes the proof of Theorem 5.1. For example, we apply Theorem 5.1 to the cyclic coverings of S3 branched along one component of a Hopf link. Then we obtain the following: Corollary 5.2. For any positive integers d and e and any positive number C, there is a sequence of knots (S3, Kr ) (r = 2,3,4.... ) which are the lifts of a trivial knot by the Z,.- coverings (r = 2,3,4 .... ) of S3 branched along a trivial knot such that (1) the knot (S3, K,) is an AID imitation of a trivial knot with HCP such that VolE(Kr, S3) > rC, (2) the knot (S3, Kr) is an unknotting number one ribbon knot of one fusion, and (3) cn(Kr; x) = 0 (O
1 (n = 0) and c, (K,; x) is divided by xe - 1 for every n > rd. As an application of Theorem 4.5, we discuss a generalization of the result of [15,16] constructing finitely many mutative hyperbolic homology 3-spheres with trivial hyperbolic isometry group and with the same Floer homology, which implies that the following invariants are respectively the same for these homology 3-spheres: the homology, the hyperbolic invariants (hyperbolic volume, hyperbolic isometry, il-invariant, and Chem-Simons invariant), the quantum SU(2)-invariants, and the Floer homology (cf. [14,16,19,22]). The Floer homology, denoted by HF., is a functor from the category whose objects are homology 3-spheres and homology handles and whose morphisms are oriented cobordisms between them to the category of Z/8Z-graded finitely generated abelian groups (see [1, 2, 3]). We have the following: Theorem 5.3. Let M be a homology 3-sphere with an (orientation-preserving faithful) G-action. For the order N = IGI of G and any positive number C, there are normal imitations M(„) (n = 1, 2 ,..., N) of M such that (1) the imitations M(„) for all n are mutually mutative (but mutually distinct) hyperbolic homology 3-spheres with trivial hyperbolic isometry group and with VolM(n) > C, (2) the Floer homology groups HF.(M(„)) for all n are mutually isomorphic and have HF.(M) as a direct summand, and (3) there is a C-hyperbolic family of normal imitations Mm (m E N) of M which have the same Floer homology as M(n) for each n.

TOPOLOGICAL IMITATIONS 35

Proof of Theorem 5.3. We extend the action of G on M to an action on the connected sum Ms = M#n 1S' x Sn so that G acts freely and transitively on the link L = Un 1Ln with L„ = S' x p,,. Let L+ be a G-invariant link in Ms obtained from L by adding one meridian to Ln. By Theorem 4.5, there are AID imitations q+ : (Ms, (L+)*) -* (Ms, L+)

and

(q+)' : (Ms, (L+)*') - (Ms L+)

such that q+ is a G-map and mutative to (q+)' and E((L+)*'; Ms) is a hyperbolic 3-manifold with IsomE((L+)*'; Ms) = {1} and VoIE((L+)*'; Ms) > C. Then by the Dehn surgery on the components of L+ - L with the same surgery coefficient 1/a for any non-zero integer a, q+ and (q+)' induce AID imitations q : (Ms, L*) -. (Ms, L)

and

q' : (Ms L*') _* (Ms, L)

such that q is a G-map and mutative to q'. We choose a G-invariant longitude system of L. Using this longitude system, we can define label functions Pi(i = 0, 1, 2, ... , N) on L so that £1(Ln) = 8i,n for all i and n. We note that the label functions 4- (i = 0, 1, 2, ... , N) specify uniquely the label functions on L* and L*' by the AID imitation maps q and q', which we will write with the same notation. Let M* = X(MS; (L*, to)), M(n) = X(M5; (L*, tn)) and M(n) = X(MS; (L*' en)) (n = 1, 2, ... , N). Since X(M5; (L, ii)) for all i are diffeomorphic to M, the AID imitations q and q' induce normal imitations q*:M*->M, q(n):M(n)- M, q(n):M(n)-+M

(n = 1,2,...,N).

The homology 3-sphere M* has a G-action and, since L* is G-transitive, the homology 3-spheres M(*n) for all n are mutually orientation-preservingly diffeomorphic. Further, using that q( n) and q(n) are mutative, we see that the homology 3-spheres M(n) for all n are mutually mutative . Since (q+)' is an AID imitation such that E((L+)*'; Ms) is a hyperbolic 3-manifold with IsomE((L+)*'; Ms) = {1} and VoIE((L+)*'; Ms) > C, we can see from an argument similar to [16, Lemma 2 . 1] that if a is chosen sufficiently large , then the homology 3-spheres M(n) for all n are mutually non-diffeomorphic, hyperbolic homology 3-spheres with trivial isometry group and Vo1M(n) > C and the knot Kn in M(n) which is the dual knot of (q')-'(Ln) by the Dehn surgery of Ms on (L*', In) is hyperbolic with VoIE(Kn; M(n)) > C. In particular , we have ( 1) at this stage . Since q' is an AID imitation , we see that the zero-surgery manifold X(M(n); (Kn, 0)) is diffeomorphic to the connected sum M#S' x S. Further, by construction, the (+1)-surgery manifold X(M(n); (Kn, +1)) is diffeomorphic to the homology 3-sphere M*. We consider the following part of Floer 's exact triangle (see [1,3]): HF*+1(M#S' X S2) - HF*(M*) -> HF*(M(n)) -+ HF*(M#S' x S2).

36 AKIO KAWAUCHI

Since HF*(M#Sl x S2) = {0} (cf. [1,3]), the natural homomorphism (induced from the surgery trace cobordism) HF*(M*) -+ HF*(M(„)) is an isomorphism for all n. Hence the Floer homology groups HF,(Ml„1)(n = 1, 2,..., N) are mutually isomorphic. From the same calculation , we see also that the Dehn surgery manifolds M(n) = X(M(„); (K,., 1/M)) for all non-zero m have the same Floer homology as M*. Since K„ is a hyperbolic knot in M(„) with Vo1E(Kn; M(„)) > C and M(-n) is still a normal imitation of M, a suitable infinite subfamily of {M(n)jm E N} re-indexed by N forms a desired C-hyperbolic family of (3). For every (normal) imitation M' of a homology 3-sphere M, the Floer homology HF*(M) contains the Floer homology HF,(M) as a direct summand (see [17]). In particular, (2) is proved. This completes the proof of Theorem 5.3. Since every Seifert homology 3-sphere admits a cyclic action of any order, we see that for every integer N > 1 there exit N normal imitations M(n) (n = 1, 2, ... , N) of every Seifert homology 3-sphere M together with the properties (1)-(3) of Theorem 5.3.

REFERENCES 1. P. J. Braam and S. K. Donaldson , Floer's work on instanton homology, knots and surgery, The Floer memorial volume, pp 195 -256, Progress in Math . 133, Birkhauser, 1995. 2. A. Floer An instanton-invariant for 3-manifolds, Commun. Math. Phys., 118 (1988), 215-240. 3. A. Floer, Instanton homology, surgery and knots, Geometry of Low-Dimensional Manifold 1, pp 97-114 (1990), London Math. Soc. Lect. Note Ser., 150, Cambridge Univ. Press. 4. P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett,and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math . Soc. 12 ( 1985), 239-249. 5. V. F. R. Jones, A polynomial invariant for knots via von Neumann algebra, Bull. Amer. Math. Soc. 12 (1985 ), 103-111, 6. A. Kawauchi Imitations of (3,1)- dimensional manifold pairs Sugaku Expositions 2(1989), Amer. Math . Soc., 141-156. 7. A. Kawauchi An imitation theory of manifolds Osaka J. Math. 26 (1989), 447-464. 8. A. Kawauchi , Almost identical imitations of (3,1)- dimensional manifold pairs, Osaka J. Math. 26 (1989), 743-758. 9. A. Kawauchi , Almost identical imitations of (3,1)- dimensional manifold pairs and the branched coverings, Osaka J. Math. 29 (1992 ), 299-327. 10. A. Kawauchi, Almost identical link imitations and the skein polynomial, Knots 90 , pp 465-476, Walter de Gruyter, 1992.

TOPOLOGICAL IMITATIONS 37 11. A. Kawauchi Almost identical imitations of (3,1)- dimensional manifold pairs and the manifold mutation , J. Austral. Math. Soc. (Seri. A) 55 (1993), 100-115. 12. A. Kawauchi, Introduction to topological imitations of (3,1)-dimensional manifold pairs, Topics in Knot Theory, pp 69-83(1993), Kluwer Academic Publishers. 13. A. Kawauchi, On the coefficient polynomials of the skein polynomial of an oriented link, Kobe J. Math. 11 (1994), 49-68. 14. A. Kawauchi , Topological imitation, mutation and the quantum SU(2) invariants, J. Knot Theory Ramifications 3 (1994 ), 25-39. 15. A. Kawauchi A survey of topological imitations of (3,1)-dimensional manifold pairs, The 3rd Korea-Japan School of Knots and Links, pp 43-52 (1994), Proc. Applied Math. Workshop 4, Korea Advanced Institute of Science and Technology. 16. A. Kawauchi, Mutative hyperbolic homology 3-spheres with the same Floer homology, Geom. Dedicata 61 (1996), 205-217. 17. A. Kawauchi, Floer homology of topological imitations of homology 3-spheres, preprint. 18. A. Kawauchi, A stronger concept of almost identical imitation of (3,1)- dimensional manifold pair, preprint. 19. R. Meyerhoff and D. Ruberman, Mutation and the 77-invariant, J. Diff. Geom. 31 (1990), 101-130. 20. J. Milnor, Isotopy of links, Algebraic Geometry and Topology (A symposium in honor of S. Lefschetz), pp 280-306 (1957), Princeton Univ. Press. 21. R. Myers , Homology cobordisms, link concordances, and hyperbolic 3-manifolds , Trans. Amer. Math . Soc. 278 (1983), 271-288. 22. D. Ruberman, Mutation and volumes of knots, Invent. Math. 90 (1987), 189-215. 23. T. Soma, The Gromov invariant of links, Invent. Math. 64 (1981), 445-454. 24. W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry Bull. Amer. Math. Soc. 6 ( 1982), 357-381. 25. W. P. Thurston, Hyperbolic geometry and 3-manifolds, Low-dimensional topology, pp 9-25, London Math. Soc. Lect. Note Ser., 48(1982), Cambridge Univ. Press. 26. W. P. Thurston, The geometry and topology of 3-manifolds, preprint. DEPARTMENT OF MATHEMATICS, OSAKA CITY UNIVERSITY, OSAKA 558, JAPAN

This page is intentionally left blank

Lectures at Knots 96 edited by Shin 'ichi Suzuki

©1997 World Scientific Publishing Co. pp. 39-71

SURFACES IN 4-SPACE: A VIEW OF NORMAL FORMS AND BRAIDINGS SEIICHI KAMADA

In order to describe surfaces in R4 we consider their intersections with parallel hyperplanes R3 [t] (= R3 x {t} c ,R3 x Rl = R4) (t E R1), which are called hyperplane cross sections in R. H. Fox's article [Fol]. He gave several examples of 2-spheres in R4 by this method. Every example of his is a 2-sphere, say. F, in R4 such that F fl R3[0] is a knot and as t goes through a saddle point with increasing absolute value, the number of components increases. Then he asked whether any 2-sphere in R4 is obtainable in this way (p. 134). It is well-known nowadays that the question is solved affirmatively. Moreover a natural analogy holds to any surface in R4; namely, it is defomed into a certain kind of configuration called a normal form.. We shall explain the notion of a normal form and introduce some fundamental results, which might help one to work in 2-knot theory. We shall devote Section 1 to it. Lee Rudolph studied a notion called a braided surface, as a generalization of braids [R1;R2iR3;R4]. On the other hand, Oleg Viro introduced a similar notion called a 2-dimensional braid in [Vi]. (It is a motivation for my reserch on this field.) They are essentially the same notion, but their approaches are different. The former regards classical knot theory and singularity theory, and the latter does rather 2-dimensional knot theory. In Section 2 we treat of 2-dimensional braids and their chart description. A chart is a labelled, directed graph in a 2-disk satisfying a certain condition. Every (simple) 2-dimensional braid is represented by a chart [K2;K3;C-S5]. Viro's theorem [Vi] states that every closed oriented surface embedded in 4-space is equivalent to a closed 2-dimensional braid. In [K4] an alternative proof using normal forms is given, which is convenient to transform a given surface into one in a braid form. In [K5iK6;Kg], Markov's theorem in dimension four is treated. In Section 3, 2-dimensional braids and chart descriptions are generalized to those for immersed surfaces with transverse double points and possibly with chords. (A chord is a simple arc attached to a surface in 4-space intersecting the surface only 39

40 SEIICHI KAMADA

on the end points. It plays an important role for 1-handle surgeries and finger moves, cf. [H-K;Bo;C;Kib2;Krk], etc.) In classical dimension a mapping from a chord diagram (circles with chords) into 3-space connects two generic immersions (that are embeddings, knots or links, related by unknotting operations). Similarly in four dimenion, a mapping from a chord surface diagram (surfaces with chords) into four-space connects two generic immersions (that are immersions whose singularities are transverse double points related by finger moves as unknotting operations). On the other hand, 1-handle surgeries correspond to smoothing operations for knots and links appearing in the skein relation. We propose in the end of the section two combinatorial approaches toward Vassiliev type invariants of 2-knots. We work in the piecewise linear category and all embeddings and immersions are assumed to be locally flat. A surface in R4 means a closed surface embedded in R4 in Section 1, a closed orientable (or oriented) surface embedded in R4 in Section 2 and a closed orientable (or oriented) surface immersed in R4 whose singulrites are transverse double points in Section 3. Two surfaces in R4 are equivalent if they are ambient isotopic. For a subset A of R3 and a subset B of R1, we denote by AB the subset {(x,t)E R3xR1IxEA,teB}ofR3xR1=R.

• oO a 00 minimal

Oa maximal

point

minimal

disk

a00 • point

maximal

disk

saddle point saddle band Figure 1.1

SURFACES IN 4-SPACE 41 1. Normal forms for surfaces in 4-space

1.1. Realizing surface 1.1.1. Any surface F in R4 is deformed, up to equivalence, such that the hyperplane cross sections have a finite number of critical points. A critical point is called a minimal point if a small, unknotted, simple closed polygonal curve appears from it as t increses. A critical point is called a maximal point if a small, unknotted, simple closed polygonal curve shrinks toward it and disappears. A critical point is called a saddle point (or a hyperbolic critical point) if two arcs have a recombination at the point as in Figure 1.1. They are called elementary critical points. We make the neighborhoods of those critical points look like Figure 1.1 by a local isotopy of W. (The deformed surface is of course no longer in general position with respect to the fourth coordinate.) Then they are called a minimal disk , a maximal disk and a saddle band, respectively. 1.1.2. Let t be a link in R3 and B a set of mutually disjoint bands Bi,... , Bm attached to t with attaching arcs {ai, aL} (i = 1, ... , m). Then Cl(e U 0B1 U • . • U 8Bm - (al U ai U . . . U am U an)) is a link in R3. We call it the link obtained from t by the hyperbolic transformations (or surgery) along the bands, and denote it by h(t; B). If #(h(e; B)), the number of components of the link h(t; B), is one (i.e. it is a knot) and if #(t) = m+ 1, then we say that h(t; B) is obtained from t by a complete fusion, and t is obtained from h(t; B) by a complete fission. 1.1.3.

A hyperbolic transformation sequence is a sequence to -+ t1 -> ••• _ to

such that

ti 1

-+ &+1 (i = 0, 1, ..., n - 1) is hyperbolic transformations along bands 1

For an interval [a, b], the realizing surface

F (eo,e1,...,en;Bo,...,Bn-1 is a proper surface F in R3 [a, b] whose hyperplane cross section at t E [a, b] is to for t E [a, to), (to U B°) for t = to, 11 for t E (to,t1), (t1 U B1) for t = t1,

(tn_1 U Bn-1) for t = to-1, to

for t

E (tn_1,b],

where a < to < ti < • < tn_1 < b are real numbers.

42 SEIICHI KAMADA

Using the cellular move lemma (see, Proposition 4.15 in [R-S]), we have the following lemma, which implies that the choice of to , t1,. .. , t._1 is not important. Lemma Two realizing surfaces in R3[a, b] of the same hyperbolic transformation sequence are ambient isotopic in R3 [a, b] rel 9R3 (a, b]. If to and In are trivial links, one can define a closed realizing surface

by (D-)(E)+) in R3 [a, b], where FQ is the realizing surface and D-, D+ are spanning disks for to and Pn in R3 , respectively. 1.2. Normal form 1.2.1. A 2-knot F in R4 is in a normal form if it is a closed realizing surface of a hyperbolic transformation sequence 0_ -> k -+ 0+ such that 0_, 0+ are trivial links and O (k) = 1. (We notice that 0_ -+ k is a complete fusion and k --r 0+ a complete fission.) A closed orientable connected surface F in R4 of genus g is in a normal form if it is a closed realizing surface of a hyperbolic transformation sequence 0_ -+ k_ -+ B -+ k+ -+ 0+ such that 0_, 0+ are trivial links , #(k_) = 0(k+) = 1 and 0(t) = g + 1. (Notice that 0_ -+ k_ and t -+ k+ are complete fusions and k_ -+ t and k+ -' 0+ are complete fissions.) t=3 t=2 t=1 to t=-1 t=-2 t=-3

Figure 1.2 A band B attached to a knot k is noncoherent if h(k; B) is a knot. Mutually disjoint noncoherent bands B1,. .. , Bm attached to a knot k is in regular position

SURFACES IN 4-SPACE 43

to k if there exist m mutually disjoint arcs Ii, ... , I,, on k such that for each i (i = 1, ... , m) the attaching arcs of B; are contained in I. See Figure 1.3.

k

Ii

12

I.

Figure 1.3 A closed non-orientable connected surface F in R4 of non-orientable genus g is in a normal form if it is a closed realizing surface of a hyperbolic transformation sequence O_ -+ k_ -+ k+ --f O+ such that O_, O+ are trivial links, H(k_) = #(k+) = 1 and k_ -+ k+ is hyperbolic transformations along g noncoherent bands in regular position.

Theorem. form.

Any closed connected surface F in R4 is equivalent to one in a normal

A dosed disconnected surface in R4 is in a normal form if each component of which is in a normal form and each sort of critical disks (bands) are in the same critical value. The theorem holds also for F disconnected. 1.2.2. Before giving a proof to Theorem 1.2.1, we have an observation. Let F be a closed realizing surface Fa(QO,A1,...,Bn;0°,...,Bn-1)

of a hyperbolic transformation sequence with respect to real numbers a < to < t1 < < ts_1 < b. Let Bi consist of bands B',,. .. , B. attached to Bi. Put B-1' = {Bi }, 132 = {B,. .. , Bm.} and 2i = h(€€; Bl). Then by the cellular move lemma, F is equivalent to

F.(eo,e1,...,ti+ +Pi+ 1) ...iin;130 ,..., 1ii...13n-1), where the bands of BZ appear at t = ti with ti < t' , < ti+1. Consider an isotopic deformation h, of R3 keeping the link fil setwise fixed and put BB = h1(132). Similarily, put B' = hi(B) (i = j + 1, ... , n -1) and B; = h1(&) (i = j + 1, ... , n). Then the above surface is equivalent to j'a(e0,e1,...,Pj ,Li,!i+1,...,in;130 ,...,B1,O2,...13n-1).

44 SEIICHI KAMADA

If Bi and the bands of 132 are disjoint, then let 133 be their union . Then the surface is equivalent to Fa(co, tl> ... , tj, tj+1, ... , Pn; 13°, ... , B', ... 13n-1).

In this way one can deform 13j quite freely and complicatedly, although the hyperbolic transformation sequence after that changes . We call this modification of 133 a band arrangemant in the upper space. Similarly a band arrangemant in the lower space is defined, although the hyperbolic transformation sequence before 13i changes. 1.2.3. We give a sketch of the proof of Theorem 1.2.1. The basic idea is in FoxMilnor [F-M1], and a concrete proof is given in Kawauchi-Shibuya-Suzuki [K-S-S1] (the outline is also in Suzuki [Su]). For non-orientable surfaces , it is given in Kamada [K1]. For details, refer to [K-S-Si] and [K1]. By an isotopy of R4, deform F such that the hyperplane cross sections have only elementary critical points and that all minimal disks are in R3 [- 3], all maximal disks are in R3 [3 ], and the saddle bands are in distinct hyperplanes R3 [t] with - 3 < t < 3. Using a level-preserving isotopy of R4 (i.e. an isotopy h3 of R4 with he(R3[t]) = R3[t] for t E R1), we deform F such that it is a closed realizing surface in R3[-3,3] of a sequence to --f ti - • • • --> to with tt+1 = h(t;; Be). Here for each i (i = 0, 1, ... , n -1 ) 13' consists of a single band B; attached to t;. There exist mutually disjoint bands B' = {Bo, ... , Bii_1} attached to to such that F is equivalent to r.(to, h(to; B'); B'). [If B1 is disjoint from Bo then by the cellular move lemma, F is equivalent to r (to, t2i ... , tn;13° U {B1}, B2, ... , B.- 1). In case B1 intersects with B o i apply a band arrangement to B1 in the upper space such that it is disjoint to Bo. By induction on n, we have the desired band family B'.] Now we are in a situation that F is a closed realizing surface of a sequence O_ -> 0+ with 0+ = h(O-; B), where O_ and 0+ are trivial links and B is a set of bands attached to 0_.

Since F is connected, we can choose disjoint subsets B_ and B+ of B such that h(O_; B_) and h(O+; B+) are knots, say k_ and k+ respectively. Put Bo = B (B_ U B+). Then F is equivalent to

Fa(0-, k-, k+, 0+; B-, Bo, B+)• Calculating the Euler characteristic, we see that the number of bands of B° is 2g if F is orientable and g if non-orientable. If F is a 2-knot , then Bo is empty and k_ = k+, thus we obtain the conclusion . One might feel that the other cases were also obtained. However in case F is orientable and has non-zero genus , we cannot in general choose g bands from Bo such that the hyperbolic transformations on k_ along them is a complete fission . For example , let k_ and Bo = {B1, ... , B4} be a knot and bands as in Figure 1.4(a), then no two bands make a complete

SURFACES IN 4-SPACE 45

fission. However applying a band arrangement suitably, one can avoid such a bad situation (cf. [K-S-S1]). In case F is non-orientable, l3 does not in general consist of noncoherent bands (for example, see Figure 1.4(b)). By a band arrangement, we may assume that all bands of 8o are noncoherent to k_ and in regular position (cf. [K1]). Thus we have the theorem.

(b)

(a) Figure 1.4 1.3. Standard mistakes 1.3.1. (Standard Mistakes)

(1) Two closed realizing surfaces of the same hyperbolic transformation sequence are not always ambient isotopic in R3 [a, b], but in R4 (i.e. equivalent). (2) The equivalence class of a surface described by motion picture method is not recovered only from the hyperplane cross sections near critical ones. A pair of 2-links which are closed realizing surfaces of the same hyperbolic transformation sequence and not ambient isotopic in R3 [a, b] is given in [K-S-S1]. Thus, if one regards the closed realizing surface as a surface modulo ambient isotopy in R3 [a, b], the maximal and minimal disks must be specified. However the latter assertion of (1) implies that if it is considered up to equivalence, we need no information on them. We shall discuss it later. For (2), consider the spun trefoil and the 1-twist spun trefoil which are nonequivalent 2-knots, but they have motion pictures with the same cross sections near critical ones. 1.3.2. We introduce the Horibe-Yanagawa lemma, which is in Horibe's master thesis [Hr] and a proof written in English is found in [K-S-S1]. Lemma (The Horibe-Yanagawa lemma). Let S1,. .. , S,, be mutually disjoint 2-spheres in 83[0,1] such that for each i (i = 1, ... , n), Si = Di (0] U (aDi)[0,1] U

46 SEIICHI KAMADA

D;[1], where Di and D; are 2-disks in R3 with OD; = BDi. Then there exist n mutually disjoint 3-disks Bl,... , B. in R3[0, 2] with B. = Si. Lemma. Let F and F' be closed realizing surfaces in R3[a, b] of a hyperbolic transformation sequence such that F n R3(a, b) = F' fl R3(a, b). Then they are equivalent by an isotopy of R4 keeping R3[a + c, b - e] fixed for a sufficiently small e > 0. Proof. Let F" be a closed realizing surface in R3 [a + e, b - e'] of the same sequence such that F" fl R3 (a + e', b - e') = F fl R3 (a + e, b - e'), where 0 < e' < E. By the Horibe-Yanagawa lemma and the cellular move lemma, we see that it is equivalent to F and F' by isotopies of R4 whose support are R3 (a - e, a + e) U R3 (b - e, b + e). ❑ This completes the proof. By this lemma we need not specify the spanning disks D- and D+ for a closed realizing surface, when it is considered up to equivalence. 1.3.3. It is known that any 2-knot with four critical points is trivial [Sc] and any projective plane in R4 with three critical points is trivial [B-S]. Question (Problem 4.30 in [Kirl]). standard?

Is any torus with four critical points is

As a consequence of the Horibe-Yanagawa lemma, we have the following, which is not so obvious for the standard mistake (2). Proposition. Let F be a 2-link in R4 each component of which has just two critical points is a trivial 2-link. 1.4. Moving techniques to surfaces in normal forms 1.4.1. Let F be orientable and in a normal form. The moving picture of F is a triple (t, U, L) where £ is the cross section at t = 0 and U (resp. L) the set of upper (resp. lower) bands that are saddle bands appearing at is with t > 0 (resp. t < 0). (If necessary applying a band arrangemant, we may assume that the upper bands are mutually disjoint and so are the lower ones.) Notice that, in case F has positive genus, there is ambiguity to recover the normal form from a moving picture, that is a choice of bands appearing at t = 1 from U and that of bands appearing at t = -1 from L. However this is not important, for the possibility is finite and all of them yield surfaces in normal form equivalent to F. For example, the 3-twist spun knot of the trefoil has a moving picture depicted in Figure 1.5 (cf. [Kn]).

SURFACES IN 4-SPACE 47

Figure 1.5 1.4.2. We list here fundamental moves to moving pictures. Let (e, U, L) be a moving picture of a closed orientable connected surface of genus g (> 0) in R. (1) (Deformation of bands) Deform U (resp. L) by an isotopy of R3 keeping e fixed setwisely, and we have another (e, U', L) (resp. (t, U, L')). (II) (Trading bands) Trade some bands of U and some of L such that the new triple (e, U', L') is a moving picture. (We require that e is a link with g + 1 components again.) (III) (Stabilization) Add a trivial band to U (or L), or do the converse. Conjecture. Two moving pictures of equivalent surfaces in R4 are related by fundamental moves.

This is folklore and believed by some knot theorists in Osaka-Kobe area. Some examples which support the conjecture are given in [H,; M-N; N-T;Y], etc. On the other hand, J. S. Carter and M. Saito [C-S1;C-S2;C-S3;C-S4;C-Ss;C-S6] study projections of surfaces via Reidemeister moves. However the conjecture is still open.

2. Braiding surfaces in 4-space 2.1. 2-dimensional braid

2.1.1. Let p : F --+ D bean m-fold branched covering map from a compact oriented surface F to a 2-disk D. For x E F there is a positive integer q such that p: F -+ D is locally equivalent to z i--+ x4 (z E C) about x. We call it the local degree at x. If it is not one , then x is a singular point and its image is a branch point . The branched covering is simple if Ip 1(y) I = m - 1 for every branch point y E D. 2.1.2. A compact oriented surface F embedded in Dl x D2 (where Di is a 2-disk for i = 1, 2) is called a 2-dimensional m-braid , or 2-dimensional braid of degree m, if the following conditions are satisfied:

48 SEIICHI KAMADA

x D2 -+ D2 (1) The map F - D2 induced from the second factor projection D1 is an m-fold branched covering of D2. (2) The restriction of F to Dl x aD2 is the product of a fixed set X. of m points of Dl with OD2. A 2-dimensional braid is simple if the associated branched covering map F -+ D2 is simple. If two 2-dimensional braids are ambient isotopic by a fiber-perserving isotopy of Dl x D2, as a Dl-bundle over D2, rel Dl x 0D2, then we say that they are equivalent. We usually do not distiguish equivalent ones . Two 2-dimensional braids are braid isotopic if one is isotopic to the other via 2-dimensional braids. Evidently equivalent 2-dimensional m-braids are braid isotopic . However the converse is false. 2.1.3. In order to visualize a 2-dimensional m-braid F, the motion picture method is useful. Identify D2 with the product Il x I2 of the unit intervals and put bt = Ff1D2 x (I x {t}) fort E I2 = [0, 1]. Then bt are (classical) m-braids in D2 x I for all but finite t. For each exceptional t, bt is a singular m-braid, that is like an rn-braid but has intersections of strings . We notice also that bo and bi are trivial m-braids. The one-parameter family { bt}tE (o,l) is called a braid movie of F . See Figure 2.1. Two braid movies of equivalent 2-dimensional m-braids are called equivalent (cf. [C-Ss;C-S6])•

Figure 2.1

2.1.4. In a braid movie, a singular point of bt (for some t) corresponds to a singular point of the 2-dimensional braid F . We often deform the neighborhood of a singular

SURFACES IN 4-SPACE 49

point to make a saddle band which is a half-twisted band attached to the braid. For example, see Figure 2.2.

Figure 2.2 2.2. Chart description 2.2.1. An m- chart is a directed , labelled graph in D2, which may be empty or have closed edges without vertices called hoops, satisfying the following conditions:

( 1) Every vertex has degree one, four or six. (2) The labels of edges are in {1 , 2, ... , m - 1}. (3) For each degree six vertex, three consecutive edges are directed inwardly and the others are outwardly, and these six edges are labelled by i and i + 1 alternately for some i. (4) For each degree four vertex, diagonal edges have the same label and are directed coherently, and the labels i and j of the diagonals satisfy li - jI > 1. We call a vertex of degree one (resp . six) a black (resp . white) vertex. A middle edge of a white vertex W means the middle edge of the three consecutive edges oriented inwardly or that of the other edges.

2.2.2. Associated with an m-chart I', a simple 2-dimensional m-braid a ( P) is defined , refer to [C-Ss;C-S6;K2iK3;K7]. If a simple 2 -dimensional m-braid F is equivalent to .1(I'), then we say that r is a chart description of F, or r represents F. For example , a 4-chart depicted in Figure 2.3 represents the 2 -dimensional 4-braid illustrated in Figure 2.1.

Theorem [K2; K3] ). chart.

Any simple 2-dimensional rn-braid is represented by an m-

50 SEIICHI KAMADA

Figure 2.3 2.2.3. Operations on m-charts listed below are called Cl-, CII- and CIII-moves. A C-move is one of them or its inverse . Two m-charts are C-move equivalent if they are related by a sequence of C-moves (up to ambient isotopy in Dz). (CI) For a 2-disk E on D2 such that r fl E has no black vertices , change r fl E arbitrarily as long as it has no black vertices. (CII) Suppose that an edge a connects a degree four vertex Q and a black vertex B. Remove a and Q, attach B to the edge of Q opposite to a, and connect the other two edges. (CIII) Let a black vertex B and a white vertex W be connected by a non-middle edge a of W. Remove a and W, attach B to the edge of W opposite to a, and connect other four edges in a natural way. It is easily seen that if two m-charts are C-move equivalent, then they represent equivalent 2-dimensional m-braids. The converse is also true. Theorem ([K7]). Two m-charts represent equivalent simple 2-dimensional mbraids if and only if they are C-move equivalent. The 4-chart depicted in Figure 2.3 is simplified as in Figure 2.4 by C-moves. The first step is a CII-move, the second is a CIII-move and the third is a CI-move. The resultant 4-chart yields a braid movie as in Figure 2.5. Thus the braid movies in Figures 2.1 and 2.5 are equivalent. Carter-Saito [C-Ss;C-S6] investigated (14 types of) moves to braid movies in terms of classical braid words, called braid movie moves. It is a good exercise to verify, using braid movie moves, that the braid movies in Figures 2.1 and 2.5 are equivalent.

SURFACES IN 4-SPACE 51

.3

Figure 2.4

u

LI 0

Figure 2.5 2.2.4. Let F and F' be 2-dimensional m-braids in Dl x D2 and Dl x', respectively. Identify a boundary disk sum 2bD2" of D2 and D2' with a 2-disk D. Then the union of F and F' becomes a 2-dimensional m-braid in Dl x D. We call it the product of F and F'. Since we regard equivalent 2-dimensional braids as the same, the product is well-defined. The set of equivalence classes of 2-dimensional m-braids form a commutative semi-group, which we call the 2-dimensional m-braid monoid. (The identity element is the class of the trivial 2-dimensional m-braid Xm x D2.) The subset consisting of the classes of simple ones is called the simple 2-dimensional m-braid monoid and denoted by'3m. Let G„ be the set of ambient isotopy classes of m-charts. Define a product I' • I" of two m-charts I' C D2 and r' C D2' by the union of them in the boundary disk sum DDD2'. Then C.. is a monoid, whose identity element is the class of the empty.

52 SEIICHI KAMADA By Theorems 2.2.2 and 2.2. 3, the map A:Cm -.Bm

is a monoid epimorphism and induces an isomorphism A:CC-. Bm,

where Cm is the quotient monoid of Cm by the C-move equivalence relation. 2.3. Braiding surfaces in 4-space 2.3.1. A 2-dimensional m-braid F is naturally extended to a closed oriented surface F in Di x SZ = D1 X (DZ U DZ) with F fl D1 XT = Xm X D. Identify Di X S2 with a tubular neighborhood of a standard 2-sphere S2 in R4, then P is a surface in R4. We call it the closure of F in R4 and denote it by F. A surface in R4 obtained this way is called a closed 2-dimensional m-braid in R4.

2.3.2. Let {bt} be a braid movie of a 2-dimensional m-braid F. The closure F of F is a surface in R4 such that the hyperplane cross section at t (t E (0, 1)) is the closure of bt in R3[t] and the minimal disks (resp. maximal disks) are in R3[0] (resp. R3[1]), see Figure 2.6.

Figure 2.6 2.3.3. The Alexander theorem [Al] states that any classical knot (or link) is equivalent to a closed braid. For a surface in R4, an analogous result holds. Theorem ([Vi];[K4]). Any closed oriented surface in R4 is equivalent to a closed 2-dimensional braid in R4. Moreover, the closed 2-dimensioal braid may be chosen to be simple.

SURFACES IN 4-SPACE 53

Viro's proof [Vi] seems to follow the Alexander's argument. (If one requires for the closed 2-dimensional braid to be simple, a further argument is needed, for Alexander's argument yields not only simple ones but also non-simple ones. Since there exist non-simple 2-dimensional braids not being braid isotopic to simple ones [K8], the latter assertion of the theorem is not obvious.) An alternative proof using normal forms is given in [K4], which we recall in the next paragraph. 2.3.4. Let us identify R3\z-axis with R+ X Sl in the usual way, where R+ is the half-plane {(xi 0, z) E R3 Ix > 0}. Theorem 2.3.3 is equivalent to the following. Theorem ([K4]). For a closed oriented surface F in R4, there exists a trivial closed m-braid f- in R3 about z-axis and n mutually disjoint bands bl,... , b„ attached to it (for some m > 1 and n > 0) such that (1) each bi(i = 1,...,n) is contained in Di x Ii as in Figure 2.7 for a 2-disk Di C R+ and an interval Ii C S1 with Ii fl I, = 0 for i 54 j,

(2) the link f+ = h(€_; {b1,.. . , b a}) is also a trivial closed rn-braid and (3) F is equivalent to the closed realizing surface of e_ -.1+.

k

'A

M C Figure 2.7 For example, the spun trefoil knot is equivalent to a closed 2-dimensional 3-braid as in Figure 2.8. 2.3.5. In the situation of Theorem 2.3.4, the Euler characteristic of F is given by 2m - n. As an application of the theorem a characterization of 2-knot groups is given in [K4]. F. Gonzalez-Acuiia [Go] gave another characterization of them using a different kind of braid-like form of surfaces in R4. 2.3.6. A ribbon surface in R4 is one obtained from a trivial 2-link by attaching some tubes (1-handle surgeries). For exapmle, every 2-knot obtained by Artin's spinning construction is ribbon. For a ribbon surface, the braiding theorem is strengthened as follows:

54 SEIICHI KAMADA

Figure 2.8 Theorem ([R2]; [K2]). For a ribbon surface F in R4, there exists a hyperboloc transformation sequence of closed m-braids t_ -+ to -+ t+ (for some m) such that (1) t_ --+ to and to -+ t+ are hyperbolic transformations along the same bands satisfying the condition (1) of Theorem 2.3.4 (hence t_ = t+), (2) t_ (= t+) is a trivial closed m -braid and (3) F is equivalent to the closed realizing surface of t_ -+ to --+ t+ The braid movie depicted in Figure 2.8 is equivalent to a closed 2-dimensional 3-braid in Figure 2.9. (Verify it by considering their charts.) 2.3.7. For closed 2-dimensional braids, a result similar to Markov's theorem [Ma] holds: Theorem ([K5; K9] ). If 2-dimensional braids F and F' have equivalent closures in R4, then there exists a sequence F = FO, Fl, ... , F„ = F' of 2- dimensional braids such that for each i(i = 1, . . . , n), F; is obtained from Fi_1 by a braid isotopy, a conjugation, a stabilization or its inverse operation. For the terminologies and a proof, the reader is referred to [KsiKe;K9].

SURFACES IN 4-SPACE 55

Figure 2.9 2.3.8. Recall that any surface in R4 is equivalent to a simple closed 2-dimensional braid. A version of Theorem 2.3.7 for simple ones is as follows: Theorem. If simple 2- dimensional braids F and F' have equivalent closures in R4, then there exists a sequence F = F0, F1, ... , F„ = F' of simple 2-dimensional braids such that for each i (i = 1, . . . , n), F, is obtained from Fi_1 by a braid isotopy, a conjugation, a stabilization or its inverse operation.

The proof will be given elsewhere. The author conjectures that this theorem might be strengthened as follows: Conjecture . If simple 2- dimensional braids F and F' have equivalent closures in R4, then there exists a sequence F = FO, Fl, ... , F. = F of simple 2-dimensional braids such that for each i (i = 1, ... , n), Fi is obtained from Fi_ 1 by an "equivalence", a conjugation, a stabilization or its inverse operation.

2.3.9. We finish this section with questions. Question 1. Do there exist simple 2-dimensional braids that are braid isotopic but inequivalent?

56 SEIICHI KAMADA

Question 2. Consider two simple 2-dimensional braids being braid isotopic. Are they equivalent after applying some conjugations and stabilizations suitably? 3. Immersed surfaces in 4-space 3.1. Immersed surfaces in 4-space 3.1.1. Let F be a closed connected surface generically immersed in R4, i.e. the singularities are transverse double points. (A surface immersed in R4 is connected if it is connected as an abstract surface, not as a subset of R4.) The basic invariant of F is a triple (X(F), e(F), n(F)) where X(F) is the Euler characteristic of F (as an abstract surface), e(F) is the normal Euler number of F and n(F) is the number of double points. (For disconnected F, the basic invariant is defined by a family of those for the components of F. Furthermore in the case that the components are ordered, the family is ordered.) 3.1.2. The surfaces illustrated in Figure 3.1 are "standard" surfaces. A conncted surface in R4 is unknotted if it is a knot sum of standard surfaces . A disconnected surface in R4 is unknotted if it is a split union of unknotted connected surfaces in R4.

Uo 40--0-*Q-* 0 U

4

0'O -* OO ^ C^

U+ To 40404^@

P P+ 0

4

4

-

O^ 0

-+0

O -^ 4

Figure 3.1

4D

SURFACES IN 4-SPACE

Uo 2 0 0

X e n

U_ 2 -2 1

U+ 2 +2 1

To 0 0 0

57

P_ 1 -2 0

P+ 1 +2 0

3.2. Singular 2-dimensional braid 3.2.1. A compact oriented surface F immersed in Di x D2 is called a singular 2-dimensional m-braid if the following conditions (1) and (2) are satisfied: (1) for an immersion f : E -+ Di x D2 of a surface E with f (E) = F, the composition pre o f : E -, D2 is an m-fold branched covering, where pre D1 x D2 -> D2 is the projection. (2) The restriction of F to Di x OD2 is the product of a fixed set Xm of m points of Dl with

eD2.

A point of F is called a singular point if it is a double point of F or the image by f of a singular point of the branched covering pre o f : E --> D2. A fiber pr2-1(y) is called a regular fiber if it has no singular points. A singular 2-dimensional m-braid is simple if Jpr21(y) I = m - 1 or m for every point y E D2. This is equivalent to that the branched covering pre o f is simple and there exists at most one singular point of F in every fiber pr2'(y) (y E D2). If F is embedded, then it is often called an embedded 2-dimensional braid. This is a 2-dimensional braid defined in Chapter 2. The notions of equivalence, a braid isotopy and a product are also defined to singular 2-dimensional braids. 3.2.2. We extend the notion of m-chart. A (singular) m-chart is a directed, labelled graph in D2, which may be empty or have hoops, satisfying the following conditions: (1) Every vertex has degree one, two, four or six. (2) The labels of edges are in {1, 2, ... , m - 1}. (3) For each degree six vertex, three consecutive edges are directed inwardly and the others are outwardly, and these six edges are labelled by i and i + 1 alternately for some i. (4) For each degree four vertex, diagonal edges have the same label and are directed coherently, and the labels i and j of the diagonals satisfy Ii - j I > 1. (5) For each degree two vertex, the two edges are labelled by the same integer and directed noncoherently.

58 SEIICHI KAMADA

We call a vertex of degree two a node. 3.2.3. Associated with an m-chart r, a simple singular 2-dimensional m-braid A(I') is defined such that the nodes correspond to the images of the double points of A(I'). See Figure 3.2. If a simple singular 2-dimensional m-braid F is equivalent to .X(I'), then we call I' a chart description of F, or I' represents F.

i i+l -4 H -4

b

1

r 1

braid movie

chart Figure 3.2 By an argument similar to that in Theorem. m-chart.

[K31,

we have

Any simple singular 2-dimensional m- braid is represented by a singular

3.2.4. In order to treat singular charts with nodes we have to generalize the definition of CI-move and add CIV- and CV-moves to the list of C- moves. (CI) For a 2-disk E on D22 such that r n E has neither black vertices nor nodes, change I' n E arbitrarily as long as it has neither black vertices nor nodes. (CIV) Let a1,a2ia3 be edges, Q a degree four vertex and N a node such that Q E 9a1 f1 8a2 and N E 0a2 f1 8a3. Transmit N across Q, see Figure 3.3. (CV) Let a node N and a white vertex W be connected by a non-middle edge a of W. Transmit N across W (for example, see Figure 3.4).

Q

N

N Figure 3.3

Figure 3.4

Q

SURFACES IN 4-SPACE 59

Theorem. Two singular m-charts represent equivalent simple singular 2-dimensional m-braids if and only if they are C-move equivalent. 3.2.5. In this paragraph, we give a proof of Theorem 3.2.4. The reader may skip over it, since he or she is assumed to be familiar with arguments in [K7]. An m-braid word is a word in the standard generator of the m-braid group. We denote it by x - y if two m-braid words x and y represent the same element of the group . An m-braid word is called a positive/negative symmetric word (resp. a pseudo positive/negative symmetric word) if it is x-1Qjx (resp . x-1v^`x ) for some m-braid word x and e = +1/ - 1. Define an equivalence relation on symmetric words generated by the following transformations (1)-(3);

(1)

x 1 Q j' 5 x +--* y 17 y

where x, y are m-braid words with x - y and e = ±1, (2)

x- 1O'. EO'^O'sx r --- a x 1O'^x

where xis an m-braid word, ji - j1 54 1 ande=f1,S=f1, (3)

xati EO'^O'Tx <--> x-1a^a' o" Ex

where x is an m-braid word, ji- j j = 1 and e = ±1, 5 = ±1. For pseudo symmetric words, define an equivalence relation quite similarly. We say that two symmetric words (resp. pseudo symmetric words ) v and v' are symmetrically equivalent and denote it by v -, -v' if they are equivalent in the sense above. Evidently v -v' implies v - v'. The converse is also true.

Lemma. If two symmetric words (resp . pseudo symmetric words) represent the same element in the braid group then they are symmetrically equivalent. Proof. The case of symmetric words is proved as Theorem 1.4 in [K7]. The proof works well to the case of pseudo symmetric words. (The main difference is to use Lemma 6.2(3) twice in order to obtain L(z) = L(x1 ) - 1 for the case of Ik - it = 1 in the proof of Lemma 6.4 of [K7].) Thus we have the lemma. ❑ Now we prove Theorem 3.2.4. The proof mostly follows that of Theorem 1.1 in [K7]. Since we have the lemma above, our task is to construct a suitable C-move equivalence realizing (in the sense of [K7]) the transformations (2) and (3) from 3.2.5 for quasi symmetric words. This is done as in Figure 3.5 for (2) and as in Figure 3.6 for (3). Then we have the conclusion.

60 SEIICHI KAMADA

(b)

Figure 3.5

Figure 3.6 3.2.6. The singular 2-dimensional m-braid monoid is the monoid consisting of the equivalence classes of singular 2-dimensional m-braids. The submonoid consisting of the classes of simple ones is called the simple singular 2-dimensional m-braid monoid and denoted by B;n"). We denote by Bm") the set of the equivalence classes of simple singular 2-dimensional m-braids with n double points. Then B;,`) is IIB,n" ). Let Cm") be the set of ambient isotopy classes of m-charts with n nodes and put C,(.*) = IIC ). It has a graded monoidal structure by the multiplication as before, which is equivariant under the C-move equivalence relation. Let Cm`)' = IIC$,) be

SURFACES IN 4-SPACE 61

the quotients. By 3.2. 3 and 3.2.4, we have a monoid isomorphism A : C,(n --, 5(t) with .\(C,.„")') =13;,"> for each n > 0. 3.3. Braiding immersed surfaces in 4-space 3.3.1. Let F be an immersed surface in R4 with or without mutually disjoint chords. (A chord attached to F is a simple polygonal arc in R4 whose interior is disjoint from F and the endpoints are regular points of F.) Let D1 x S2 be a tubular neighborhood of a standard 2-sphere in R4. Theorem. In the above situation, there exists an isotopy of R4 carrying F and the chords into D1 x S2 such that F is a simple singular closed 2-dimensional braid and chords are contained in distinct regular fibers.

Proof.

Deform F by an isotopy of R4 such that

(1) all maximal disks are in R3[2], (2) all minimal disks are in R3[-2], (3) all saddle bands, double points, and chords are in R3[0], (4) for t E (-2, 0), F fl R3 [t) = 8_[t] for a link L. in R3, (5) for t E (0, 2), F fl R3[t] = f+[t] for a link e+ in R3. Let N1, ... , N,, be mutually disjoint regular neighborhoods of saddle bands, double points and chords in R3 = R3[0]. Identify R3\z-axis with R+ x [0, 2ir). By an isotopy of R3 = R3 [0] extended to that of R4, we may assume that Ni = B. x I; C R+ x [0, 27r) (i = 1 , ... , n) f o r some 2-disks B 1 , B,, in R+ and mutually disjoint arcs I,, . . . , In in [0, 2ir) and that saddle bands, double points and chords are as in Figure 3.7. By an argument similar to that in [K4], we have the theorem. ❑

Figure 3.7

62 SEIICHI KAMADA

3.3.2. Examples . (1) Uo is represented by the empty 1-chart, namely it is equivalent to a closed 2-dimensional 1-braid. (2) U_ is equivalent to a 2-dimensional 2-braid as in Figure 3.8, which is represented by a 2-chart as in Figure 3.9(1). Since U+ is the mirror image of U_ it has a chart description as in Figure 3.9(2). (3) The standard torus To has a chart description as in Figure 3.9(3). (4) The Montesinos twin [Mo] is a pair of standard 2-spheres having a pair of double points. It has a chart description as in Figure 3.9(4). (5) The Fenn-Rolfsen 2-link [F-R] is a singular 2-link consisting of U_ and U+, which is not homotopically trivial. Its motion picture is depicted in Figure 3.10(1). By brading the middle cross section, we have another motion picture as in Figure 3.10(2). This is not a closed 2-dimensional braid yet, because L and e+ are not trivial closed braids. Introducing saddle bands which are trivial bands attached to B_ and e+ (recall 1.4.2), we transform the motion picure as in Figures 3.11 and 3.12. Do a similar deformation in the lower half space. Then we obtain a closed 2-dimensional 4-braid, whose chart description is as in Figure 3.13. By CIII- and CI-moves, it is simplified as in Figure 3.14.

Figure 3.8

1 1 (2)

1 (3) ~ 1 ^ (4) Figure 3.9

C

SURFACES IN 4-SPACE

63

(1) 19-40

D. 0-1 -4 Q.,

0

t=0

t=-1

hx J

0 0

t=1

mx

0

t=0

t=-1

I^j

I

V,'j) L j__,' I j

ol

00, t=O

0 t=1

Figure 3.10

^, i

Is

r i t=0.5

0 =0 0 t=1

64

SEIICHI KAMADA

t=O

t=0.7

t=0.8

Figure 3.12

Figure 3.13

Figure 3.14

t=1

SURFACES IN 4-SPACE 65

3.3.3. Let F be a simple singular 2-dimensional m-braid and -y a chord attached to F such that y lies in a regular fiber pr2'(y) for some y E D. Without loss of generality we may assume that F fl D x N = X,,, x N, where N is a regular neighborhood of y in D2, and the chord -y is a straight segment as in Figure 3.7. Then, for a chart description r of F with F = A(I'), the chord is described as a vertex of degree zero labelled by an integer in {1, . . . , m - 1} indicating the connection; namely, if the label is i then the chord connects the ith point and the i + 1st point of Xm. Therefore any simple singular 2-dimensional m-braids with chords lying distinct regular fibers has a chart description together with labelled vertices of degree zero. From 3.3.1 we have the following. Theorem. Any immersed surface in R4 (with or without mutually disjoint chords) is represented by a singular chart (with or without degree zero verticies). 3.4. 1-handle surgeries and finger moves 3.4.1. Associated with a chord attached to a surface in R4, two operations are determined, a 1-handle surgery and a finger move. They are schematically illustrated as in Figure 3.15. Refer to [H-K;Bo;C;Kir2iKrk] for the definitions. positive double point

negative double point Figure 3.15 Thanks to Theorem 3.3.1 we assume that the surface is a simple singular closed 2-dimensional m-braid F in Dl X S2 C R4 and the chord is a straight segment y connecting the ith point and the i + 1st point of Xm = pr21(y) for a regular point y E D2 C S2. Let F_, Fx, FO and F+ are surfaces in R4 which differ locally as in Figure 3.16 and the other parts (outside of the figure) are identical such that F is F_ and F shrunk along y is F. F. A 1-handle surgery along y is replacement of F_ by Fo and a finger move is that by F+.

66 SEIICHI KAMADA

F_

Fx

-48 -4 1-48 -41

FO

Figure 3.16 3.4.2. In the above situation, let r be a chart description of F. The chord -y is described by a degree zero vertex with label i. Then m-charts representing F_, F., Fo and F+ are given by I'_, rx, ro and r+ as in Figure 3.17, where they are identical outside the figure and r_ = r. Thus in the chart description, a 1-handle surgery along ry corresponds to insertion of an edge as in ro, called a free edge, and a finger move corresponds to insertion of a cycle as in r+, called an elementary quasi-hoop. Using this fact we have the following, which will be discussed elsewhere.

r0

r+

Figure 3.17 Theorem. A 1-handle surgery (resp. a finger move) is an unknotting operation for surfaces in 4-space and there exists an algorithm for unknotting a given surface.

It is known that a 1-handle surgery (resp. a finger move) is unknotting operations for embedded surfaces in 4-space, [H-K] and [Gi]. A 1-handle surgery is also an unknotting operation for non-orientable embedded surfaces in 4-space, [Ki]. A finger

SURFACES IN 4-SPACE 67

move corresponds to a crossing change in the sense of [Gil. We call it an open crossing change. 3.4.3. As mentioned in the introduction, a 1-handle surgery and a finger move correspond to a smoothing operation and a crossing change (unknotting operation) for classical knots and links, respectively. It is also realized in the following meaning: The four m-charts in Figure 3.17 is equivalent to those in Figure 3.18, whose braid movies are as in Figure 3.19. Then a smoothing operation and a crossing change appear in the middle of the braid movies.

r_

rx

r+

ro Figure 3.18

F_

Fx

FO

F+ Figure 3.19 3.5. Toward Vassiliev type invariants of 2-knots We finish this article by proposing two kinds of combinatorial definitions of Vassiliev type (finite type) invariants of 2-knots. 3.5.1. (Definition 1) By a singular 2-knot we mean a generically immersed 2spheres in R. Let IC- be the set of equivalence classes of singular 2-knots with

68 SEIICHI KAMADA

m chords. (By contracting the chords we often regard a singular 2-knot with m chords as an immersed 2-sphere whose singularites are transverse double points and m non-transverse ones.) Let K be a singular 2-knot with m chords -yl ,... , .1,. Let ( El, ... , E,,,) be an mtuple of signs. For each i (i = I_-, m), if Ei = + 1 then apply a finger move along ryi and if Ei = -1 then eliminate the chord 'yi. Denote by K(E,....,Em) the resultant singular 2-knot. Let V be a singular 2-knot invariant (valued in a ring). It is extended to an invariant for K' via the formula V(K) _ E El ...

EmV(K(E,,.. •, Em))'

If it vanishes on Km+l then we call V a finite type invariant of order m. 3.5.2. (Definition 2) Let G' be the set of equivalence classes of 2-knots with m double loop singularities. A double loop singularity means a singularity like (B3, Ax) x S' where B3 is the unit 3-ball of R3 and Ax is the union of the two arcs on the x-axis and the y-axis. The singularity of Ax is removed by regular homotopies in two ways; one yields positively crossed arcs A+ and the other yields negatively crossed arcs A_. Thus each double loop singularity is removable by regular homotopies along two different directions; one yields (B3, A+) x Sl and the other yields (B3, A_) x S1. We call an operation transforming (B3, A+) x Sl to (B3, A_) x S1, or vice versa, a closed crossing change. This operation does not change the basic invariant.

Let (q,.. . , E,n) be an m-tuple of signs. For K with [K] E G, denote by KBE, „. Em) the 2-knot obtained by removing the double loop singularities according to the signs. Let V be a 2-knot invariant . It is extended to an invariant for G' via the formula V(K) = F El ... Em` • ( K(Ei....,Em))'

If it vanishes on £m+l then we call V a finite type invariant of order m. The author has not settled the following question , except a few classes of 2-knots.

Question.

Is a closed crossing change an unknotting operation?

Problem.

Find unknotting operations preserving basic invariants.

SURFACES IN 4-SPACE 69 REFERENCES

[Al] J. W. Alexander , A lemma on systems of knotted curves, Proc. Nat. Acad. Science USA, 9 (1923), 93-95. [Ar] E. Artin, Zur Isotopie zweidimensionalen Fldchen im R4, Abh . Math . Sem. Univ. Hamburg, 4 (1926), 174-177. [B-L] J. S . Birman and X.-S. Lin, Knot polynomials and Vassiliev 's invariant, Inventh. Math., 11 (1993), 225-270. [B-S) S. Bleiler and M . Scharlemann , A projective planes in R4 with three critical points is standard. Strongly invertible knots have property P, Topology, 27(1988), 519-540. [Bo] J. Boyle, Classifying 1-handles attached to knotted surfaces , Trans. Amer . Math. Soc., 306 (1988), 475-487. [C] A. J. Casson , Three lectures on new-infinite constructions in 4-manifolds, A la Recherche de Is Topologie Perdue , Progress in Math ., 62, Birkhauser ( 1986), 201-214. [C-S1] J. S. Cater and M . Saito, Syzygies among Elementary String Interactions in 2+1 Dimensions, Letters in Math . Physics, 23 (1991 ), 287-300. [C-S2] , Canceling Branch Points on the Projections of Surfaces in 4-space, Proc. Amer. Math . Soc., 116 ( 1992), 229-237. [C-S3] , Reidemeister Moves for Surface Isotopies and Their Interpretations as Moves to Movies, J. Knot Theory Ramifications , 2 (1993), 251-284. [C-S4] , A Diagrammatic Theory of Knotted Surfaces, Quantum Topology, eds. R. A. Baadhio and L. H. Kauffman , Series on knots and everything vol.3, World Scientific, 1993 , 91-115. , Knotted surfaces, bmid movies and beyond, Knots and Quantum Gravity, Oxford [C-S5] Science Publishing , 1994, 191-229. [C-S6] , Knot diagrams and braid theories in dimension 4, Real and complex singularities, Papers from 3rd international workshop held in Sao Carlos 1994 . Pitman Res . Notes Math. Ser. 333 , 1995, 112-147. [F-R] R. Fenn and D . Rolfsen, Spheres may link homotopically in 4-space, J. London Math. Soc., 34 (1986), 177-184. [Fol] R. H . Fox, A quick trip through knot theory, Topology of 3-manifolds and Related Topics (M.K. Fort , Jr., ed .), Prentice-Hall, Englewood Cilffs, New Jersey, 1962 , 120-167. [Fo2] , Rolling, Bull. Amer. Math . Soc., 72 ( 1966), 162-164. [F-Ml] R.H. Fox and J . W. Milnor, Singularities of 2-spheres in 4-space and equivalences of knots, (unpublished ), 1957. [F-M2] , Singularities of 2-spheres in 4-space and cobordism of knots, Osaka J. Math., 3 (1966), 257-267. [Gil C. A. Giller, Towards a classical knot theory for surfaces in R4 , Illinois J. Math., 26 (1982), 591-631. [Go] F. GOnzalez-Acuna, A characterization of 2-knot groups, preprint. [Hr] M. Horibe , On trivial 2-spheres in 4-space (in Japanese), Master Thesis, Kobe Univ., Kobe, 1974. [Hs] F. Hosokawa, On trivial 2-spheres in 4-space, Quart. J. Math . Oxford (2), 19 (1968), 249-256. [H-K] F. Hosokawa and A . Kawauchi , Proposals for unknotted surfaces in four-spaces, Osaka J. Math ., 16 (1979), 233-248. [K1] S. Kamada , Non-orientable surfaces in 4-space , Osaka J. Math., 26 (1989), 367-385. [K2] , Surfaces in R4 of braid index three are ribbon, J. Knot Theory Ramifications, 1 (1992), 137-160.

70 SEIICHI KAMADA , 2-dimensional braids and chart descriptions, Topics in Knot Theory, Proc. NATO ASI, held in Turkey (1992) , 1993, 277-287. , A characterization of groups of closed orientable surfaces in 4-space, Topology, [K4]

[K3]

33 (1994), 113-122. , Alexander's and Markov's theorems in dimension four, Bull. Amer. Math. Soc., 31 (1994), 64-67. , Survey on 2-dimensional braids, Proceedings of the 41st Topology Symposium, [K6] held in Ehime, Japan, 1994, 162-178. [K7] , An observation of surface braids via chart description, J. Knot Theory Ramifications, 5 (1996), 517-529. [K8] , On braid monodromies of non- simple braided surfaces, Math. Proc. Carob. Phil. Soc., 120 (1996), 237-245. , Generalized Alexander's and Markov's theorems in dimension four, preprint. [K9] [Kn] T. Kanenobu, Fox's 2-spheres are twist spun knots, Memoirs Fac. Sci., Kyushu Univ., 37 (1983), 81-86. [K-S-S, ] A. Kawauchi, T. Shibuya and S. Suzuki, Descriptions on surfaces in four-space, I. Normal forms, Math. Sem. Notes Kobe Univ., 10 (1982), 75-125. , Descriptions on surfaces in four-space, II. Singularities and cross-sectional links, [K-S-S2] Math. Sem. Notes Kobe Univ., 11 (1983), 31-69. [Kirl] R. Kirby, Problems in low-dimensional manifold theory, Proc. Symp. Pure Math., 32 (1978), 273-312. [Kirz] , The topology of 4-manifolds, Lecture Notes in Mathematics, 1374, Springer Verlag, 1989. [Krk] P. Kirk, Link maps in the four sphere, Differential Topology, Lect. Notes Math., 1350, Springer Verlag, 1988, 31-43. [Ma] A. A. Markov, Uber die freie Aquivalenz der geschlossner Zopfe, Rec. Soc. Math. Moscou, 1 (1935), 73-78. [M-N] Y. Marumoto and Y. Nakanishi, A note on the Zeeman theorem, Kobe J. math., 8 (1991), 67-71. [Mo] J. M. Montesinos, On twins in the four-sphere I, Quart. J. Math. Oxford, 34 (1983), 171-199. [N-T] Y. Nakanishi and M. Teragaito, 2-knots from a view of moving picture, Kobe J. Math., 8 (1991), 161-172. [K5]

[Rl] L. Rudolph, Algebraic functions and closed braids , Topology, 22 (1983), 191-202. , Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv., 58 (1983), 1-37. , Some topologically locally-flat surfaces in the complex projective plane, Comment. [R3] Math. Helv., 59 (1984), 592-599. , Special positions for surfaces bounded by closed braids, Rev. Mat. Iberoamericana, [R4] 1 (1985), 93-133. [R2]

[R-S] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, SpringerVerlag, 1972. [Sc] M. Scharlemann, Smooth spheres in R4 with four critical points are standard, Invent. Math., 79 (1985), 125-141.

[Su] S. Suzuki, Knotting problems of 2-spheres in the 4-sphere, Math. Sem. Notes Kobe Univ., 4 (1976), 241-371. [Va] V. A. Vassiliev, Cohomology of knot space, Theory of Singularities and Its Applications, Advances in Soviet Math. vol. 1, Amer. Math. Soc., 1990. [Vi] O. Ya. Viro, Lecture given at Osaka City University, September, 1990.

SURFACES IN 4-SPACE

71

[Y] K. Yoshikawa , An enumeration of surfaces in four-space , Osaka J. Math . 31 (1994), 497522. [Z] E. C. Zeeman, Twisting spun knots, Trans . Amer . Math. Soc., 115 (1965 ), 471-495. DEPARTMENT OF MATHEMATICS, OSAKA CITY UNIVERSITY , SUMIYOSHI , OSAKA 558, JAPAN

This page is intentionally left blank

Lectures at Knots 96 edited by Shin'ichi Suzuki ©1997 World Scientific Publishing Co. pp. 73-93

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS KIMIHIKO MOTEGI

Let K be a knot inside a standardly embedded solid torus V in the 3-sphere S3. In the following , for nontriviality, we assume that K cannot be lie in a 3-ball in V. Knotting the solid torus V in the shape of another knot as in Figure 0.1, as the image of K, we obtain a new knot K' in S3. A knot obtained in such a manner is called a satellite knot. On the other hand, twisting the solid torus V several times, we get a new knot K" in S3. The purpose in this article is to give a survey of some aspects of the study of these constructions.

K

Figure 0.1 Supported in part by Grant-in-Aid for Encouragement of Young Scientists 08740074, The Ministry of Education, Science and Culture.

73

74 KIMIHIKO MOTEGI 1. SATELLITE KNOTS OBTAINED FROM A GIVEN PATTERN

Schubert [17] introduced the notion of the product of knots, and afterward generalized this to an operation "taking satellite" [18]. First we recall the construction of satellite knots. Let V be a standardly embedded solid torus in the oriented 3-sphere S3 with the orientation induced from that of S3, and let K be a knot in V, which cannot be contained in a 3-ball in V. Using an orientation preserving embedding f : V -+ S3 such that f (V) is knotted in S3, we can obtain a new knot f (K) in S3. We call the knot f (K) a satellite knot and (V, K) a pattern (Figure 1.1).

(V, K) : pattern f(K) : satellite knot Figure 1.1 The wrapping number (resp.winding number) of K in V is defined to be the minimal geometric intersection number (resp. algebraic intersection number) of K and a meridian disk of V. We denote this number by wrapv(K) (resp. windv(K)). For example, wrapv(K) = 2 and windv(K) = 0 for the pattern (V, K) given by Figure 1.2.

wrapV(K) = 2 windV(K) = 0

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 75

Throughout this section we assume that all knots are oriented and consider two knots K, and K2 to be equivalent if and only if there is an orientation preserving homeomorphism h : S3 -+ S3 which carries Kl onto K2 so that their orientations match. We write Kl = K2 if Kl and K2 are equivalent, and -K denotes the knot obtained from K by inverting its orientation . For an orientation preserving embedding f : V --^ S3, we understand that f (V) and f (K) have orientations induced from that of V and K respectively via the embedding f. The construction of a satellite knot depends on two parameters: the pattern (V, K) and the orientation preserving embedding f : V -* S3. Let us choose a pattern (V, K). Then by changing the embedding as in Figure 1.3, we can obtain other satellite knots. If two embeddings are isotopic, then clearly they define equivalent satellite knots. Conversely, can a satellite knot determine an isotopy class of embeddings of V into S3? Precisely we consider the following problem.

f(K) K f

V

Figure 1.3

76 KIMIHIKO MOTEGI

Problem 1 . 1. Let ( V, K) be a pattern and f : V -+ S3 an orientation preserving embedding such that f (V) is knotted in S3. Determine orientation preserving embeddings g : V -> S3 , up to isotopy, such that g(K) = f (K). When wrapv(K) = 1, the operation "taking satellite" is the same as "taking product" (Figure 1.4).

f

V

satellite 11 product

f(V)

Figure 1.4 In this case, we have the following motivating result due to Schubert [17]. Theorem 1 . 2. (Implication of Schubert's unique factorization theorem [17]) Let (V, K) be a pattern with wrapv(K) = 1 and f, g : V --p S3 be two orientation preserving embeddings. Then f (K) = g(K) if and only if f (Cv) = g(Cv), where Cv denotes an oriented core of V. This is the result which we would like to generalize to any pattern. Before stating a result, we start with some examples. A pattern (V, K) is said to be symmetric if (V, K) admits an orientation preserving homeomorphism zb : V --* V which satisfies ['(Cv)] _ -[Cv] E H1(V) and t,b(K) = K. Let s be the 7r-rotation along the axis L as shown in Figure 1.5. By the definition, for a symmetric pattern (V, K), K is null-homologous in V (or equivalently windv(K) = 0). The pattern (V, K) given by Figure 1.2 is symmetric. In fact, for some homeomorphism isotopic to the identity cp o s gives the symmetry of (V, K). For any symmetric pattern we can observe Example 1.1. Let (V, K) be a symmetric pattern and f : V --+ S3 an orientation preserving embedding such that f (Cv) is a non-invertible knot (i.e. f (Cv) V

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 77

-f (Cv) ). Then for two embeddings f and g = f o s, we have f (K) L g(K) and f (Cv) g( Cv). In particular f and g are not isotopic.

Proof. By the choice of g, g (Cv) = -f (Cv). Hence f (Cv) % g(Cv). Using the symmetry of (V, K) we can verify that f (K) = g(K). ❑

- L

S Figure 1.5 As a concrete example we may take the pattern given by Figure 1.2 and an embedding f : V -p S3 so that f (Cv) is the pretzel knot K (3, 5, 7) (see Figure 1.6), which is known to be non-invertible [22].

f f(C) = k

g(C) = - k Figure 1.6

78 KIMIHIKO MOTEGI

Example 1.2. Let (V, k) be a pattern such that wrapv(k) = 1 and k = K(3, 5, 7) in S3 (Figure 1.7). (For simplicity we assume that [k] = [Cv] E H1(V).)

(V , k) Figure 1.7 Let (V, K) be a pattern such that K is the untwisted double of k. Let f (resp. g) be an orientation preserving embedding from V into S3 so that f (Cv) = (-k)tt(-k) (resp. g(Cv) = kjj(-k)), see Figure 1.8. Then for two embeddings f and g, we have f (K) = g(K). But f (Cv) and g(Cv) cannot be equivalent even in the weakest sense. Proof. We note that since k = K(3, 5, 7) is non-invertible and non-amphicheiral, no two of k, -k, k*, -k* are equivalent, where k* denotes the mirror image of k [22]. In addition k is of genus one and so it is prime. Suppose that there is a homeomorphism of S3 carrying f (Cv) onto eg(Cv) (E = ±1). Then we have (-k)o(-k) = (-ek)O(Ek) or (-k)#(-k) = (-ek*)q(ek*). In any case -k = k or k = k* must hold by Schubert's unique factorization theorem. This is a contradiction. Therefore f (CV) and g(CV) are not equivalent even in the weakest sense. Let us prove f (K) = g(K). First we note that f (K) is the untwisted double of (-k)o(-k)ok and g(K) is the untwisted double of kjj(-k)tjk = -((-k)O(-k)Ok). Using the symmetry of the pattern given by Figure 1.2, we can observe that f (K) ❑ g(K)•

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 79

f

f(C) = (-k) #(-k)

C

g(C) = k # (-k) Figure 1.8 Example 1.2 shows that Theorem in [6] and hence also Corollary in [19] are not true. The theorem below shows that the examples described above are worst that can happen. To state the result we prepare some terminologies.

Let Cv be an oriented core of a solid torus V in S3. Then we choose a preferred meridian-longitude pair (mv, ev) of V so that [2v] = [Cv] E Hl (V) and ek(mv, Cv) = 1, where Pk(a, Q) denotes the linking number of a and Q. Let f : V --> S3 be an orientation preserving embedding, then we adopt f (Cv) as an oriented core of f (V). This determines a preferred meridian-longitude pair (mf(v), ef(v)) of f (V) so that [ef(v)] = [f (Cv)] E Hl (f (V)) and ek(m f(v), f (Cv)) = 1. Then we have an expression [f(Pv)] = [Pf(v)] + n[mf(v)] E Hl(,9f(V)) for some integer n. We define the twist number of the embedding f : V -> S3 to be n and denote it by twist(f ). Note that [f (mv)] = [mf(v)]. An orientation preserving embedding f : V --p S3 is said to be faithful if twist(f) = 0.

80 KIMIHIKO MOTEGI

Theorem 1 . 3 ([81). Let (V, K) be a pattern with wrapv (K) > 2 and f : V --p S3 an orientation preserving embedding such that f (V) is knotted in S3. Let g : V -+ S3 be an orientation preserving embedding which satisfies g(K) = f (K). Then g(Cv) f (Cv), or f (Cv) = K0OK1 and g(Cv) - (-Ko)OK1, where K0 and K1 are knots uniquely determined by the embedding f and the pattern (V, K). Furthermore in any case twist (f) = twist(g)-

We remark that the above decomposition f (Cv) = K0tK1 does not depend on g. To make precise , we explain how we can determine the decomposition of f (Cv) in the above theorem. Let W be a solid torus in V. We say that W has the property (*) if the following conditions are satisfied. • W contains K in its interior, • wrapv(Cw) = 1, where Cw is a core of W, • Cw is not a core of V. If there is no solid torus W in V satisfying the property (*), then we put K0 = f (Cv) and define the decomposition of f (Cv) to be f (Cv) - Ko.

Now let us assume that there is a solid torus W in V satisfying the property (*). We say that the solid torus W(C intV) is (*)-minimal if there is no further solid torus W'(C intW) satisfying the property (*) for W. Then by the uniqueness of the torus decomposition [4] [51, if there is a solid torus satisfying the property (*) in V, then there exists a (*)-minimal solid torus W in V, unique up to isotopy. Let W be a (*)-minimal solid torus in V. We choose an orientation of Cw so that Cw is homologous to Cv. Then we have a description Cw = Cv#k for some nontrivial knot k (see Figure 1.9).

V Figure 1.9

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 81

Let k = kl# ... #k be a prime decomposition of k. First we delete all the invertible factors and all the pairs (kg, k3) with k; = k;. As a result we obtain kl# ... #k,,, (reindexing if necessary ). Then we put K1 = -(k1# ... #k,,,). (Possibly K1 is a trivial knot. ) If f(Cv) has an expression f(Cv) = Ko#K1 for some knot K0, then we define the decomposition of f(Cv) to be f(Cv) = Ko#K1. If f(Cv ) admits no such expression we put K0 = f (Cv) and define the decomposition of f (Cv) to be f (Cv) ?' Ko itself. It should be noted that the decomposition of f (Cv ) depends only on the pattern (V, K) and the embedding f. Example 1.1 corresponds to the case where K1 is trivial in Theorem 1.3. In Example 1.2, the decomposition of f (Cv) is given by f (Cv) = (-k)#(- k) and this example shows that K1 in Theorem 1.3 can be nontrivial. Let f , g : V -> S3 be orientation preserving embeddings . Then f and g are isotopic if and only if f (Cv) = g(Cv) and twist (f) = twist (g). Theorem 1.3, together with this fact , answers the question : How many embeddings (up to isotopy ) can give equivalent satellite knots? Corollary 1.4 ([8] ). Let (V, K) and f be as in Theorem 1.3. Then there is at most one orientation preserving embedding ( up to isotopy) g : V --+ S3 which is not isotopic to f and g(K) = f (K).

If we assume further that windv(K) # 0, then we can improve Theorem 1.3 as follows. Theorem 1 . 5 ([8]). Let (V, K) be a pattern such that wrapv (K) > 2 and K is homologically essential in V, and let f : V -> S3 be an orientation preserving embedding such that f (V) is knotted in S3. If an orientation preserving embedding g : V -+ S3 satisfies g(K) = f (K), then g is isotopic to f . 2. SATELLITE KNOTS OBTAINED FROM A GIVEN EMBEDDING

Recall that to define a satellite knot we need two parameters: a pattern (V, K) and an embedding f : V _4 S3. In the previous section we consider a family of satellite knots obtained from the same pattern , i.e., we take an infinite family of embeddings of V into S3 as parameters . On the contrary, in this section , we consider satellite knots obtained from the same embedding V -> S3, i.e., we take patterns as parameters . In the following we consider unoriented knots in the oriented 3sphere S3 . For two (unoriented) knots K1 and K2, we continue to write K1 = K2 to denote that K1 and K2 are ambient isotopic in S3. For two patterns (V, K1) and (V, K2), if there exists an orientation preserving self-homeomorphism h of V sending preferred-longitude to ±preferred-longitude which satisfies h(K1) = K2, then

82 KIMIHIKO MOTEGI

we write (V, K1) - (V, K2). Furthermore if the homeomorphism h sends preferredlongitude to preferred-longitude, then we write (V, K1) = (V, K2). (V, K1) = (V, K2) if and only if Kl and K2 are ambient isotopic in V. Theorem 2 .1 ([12]). Let (V, K;) (i = 1, 2) be a pattern. Suppose that Kl is unknotted in S3 and windv(K2) # 0. If f (K1) = f (K2) in S3 for some orientation preserving embedding f : V -* S3, then (V, K1) - (V, K2) holds. The next example shows the necessity of the condition "windv (K2) 54 0". Example 2 .1. In Figure 2.1, Kl is unknotted in S3 and K2 is knotted in S3. However windv(K2) = 0. Figure 2.2 indicates an isotopy between f (Kl) and f (K2).

f

(V, K1)

f

(V, K2) Figure 2.1

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 83

slide the twists

^^Ll^Cjt^ ic-rotation

Figure 2.2 If K1 is knotted, even when windv (K2) 34 0, there is an example such that (V, K1) 76 (V, K2) but f (K1) = f (K2) in S3, see Example 2.2.

KIMIHIKO MOTEGI

84

Example 2.2.

f

(V, K1)

f

(V, K2)

Figure 2.3 We then apply Theorem 2.1 to questions: (1) Suppose that Kl is unknotted and K2 is knotted in S. Can f (K1 ) be ambient isotopic to f (K2) in S3 for some embedding f : V -+ S3 ? (2) Suppose that Kl and K2 are both unknotted in S3. How are patterns (V, K,) and (V, K2) related if f (Kl) and f (K2) are ambient isotopic in S3 for some embedding f : V -* S3 ? We can answer the first question by

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 85

Corollary 2.2 ([12]). Let (V, Kt) (i = 1, 2) be a pattern. Suppose that K1 is unknotted and K2 is knotted in S3 and windv (K2) # 0. Then for any embedding f : V - S3, f (K1) Y- f (K2) in S3.

Proof. Assume that f (K1) = f (K2) for some embedding f : V -> S3. Then from Theorem 2 . 1, we have (V, K1) - (V, K2). Extending the orientation preserving homeomorphism of V to that of S3, we see that K1 = K2 in S3, a contradiction. Example 2.1 shows the necessity of the condition "windv(K2) # 0" in Corollary 2.2. As a special case of Theorem 2.1, we have the following which gives an answer to the second question. Corollary 2.3 ([12]). Let (V, K1) be a pattern and K; a trivial knot in S3 (i = 1, 2). Suppose that windv (K1) # 0 or windv (K2) # 0. If f(K1) = f(K2) for some embedding f : V -4 S3, then (V, K1) - (V, K2). Since (V, K1) - (V, K2) implies windv (K1) = windv( K2) and wrapv(K1) _ wrapv ( K2), we have the following.

Corollary 2.4 ([12]). Suppose that Ki is a trivial knot contained in a standardly embedded solid torus V in S3 (i = 1, 2). (1) If windv(K1) 54 windv(K2), then f(K1) f(K2) in S3 for any embedding f:V->S3.

(2) When windv (Ki) = windv(K2) # 0, if wrapv(K1) 54 wrapv (K2), then f (K1) f (K2) in S3 for any embedding f : V -* S3. In the case where windv (K1) = windv ( K2) = 0, the situation is quite different. Theorem 2.5 ([12]). For any faithful embedding f : V -> S3 ( i.e., twist (f) = 0), there exist patterns (V, K1) and (V, K2) which satisfy (1) both K1 and K2 are unknotted in S3, (2) windv (Ki) =windv (K2) = 0, (V, K1) f (V, K2), and (3) f(K1) - f( K2) in S3. Proof. For the given faithful embedding f, actually we can construct required patterns as follows; the construction is due to Makoto Sakuma. First let us consider a 3-components Brunnian link L = k u L1 U L2 depicted in Figure 2.4.

86 KIMIHIKO MOTEGI

k

`' c Figure 2.4 Let (mi, £j) be a preferred meridian-longitude pair of Li (i = 1, 2). Let t be a knot ambient isotopic to f (C), where C denotes a core of V, and (m, e) a preferred meridian-longitude pair of t. Removing a tubular neighborhood N(L1) of Li and gluing the knot exterior E(t) = S3-intN(t) so that mi = e and & = m, we obtain S3 = (S3-intN(Li)) UmA_, E(t) and a new knots K3_i and L3_i as the images of l:-m

k and L3_i respectively, for i = 1, 2. It is easy to see that both K3_i and L3_, are unknotted in S3. Hence by putting V = S3-intN(L3_i)(D K3_i), we have a pattern (V, K3_i) with windv(K3_i) = 0. In this way we obtain two patterns (V, K1) and (V, K2). By the construction, for the faithful embedding f : V -, S3, f (K1) = f (K2) in S3. In fact, roughly speaking, f (Kl) and f (K2) can be described as the knot obtained from k in Figure 2.4 by simultaneously replacing neighborhoods of disks bounded by Ll and L2 by tubes knotted according to the given knot t. We can prove (V, K1) ¢ (V, K2) by showing that wrapv (Kl) = 2 and wrapv (K2) _ 4. For more details, see [12]. ❑ This result can be generalized to Corollary 2 .6 ([12]). For any knot K in S3 and any faithful embedding f : V S3, there exist patterns (V, K1) and (V, K2) which satisfy (1) Ki=K in S3 fori=1,2,

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 87 (2) windv(Ki) =windv(K2) = 0, (V, KI) 96 (V, K2), and (3) f (K1) - f (K2) in S3.

Proof. Let (V, kI) and (V, k2) be the patterns given by Theorem 2.5 depending on the embedding f : V -+ S3. Since each k, is unknotted in S3, we can locally replace an unknotted arc of kt by a knotted arc (with a suitable direction) so that the result K, represents K in S3. Then it follows from the choice of (V, kL) that (V, KI) and (V, K2) are the desired patterns. ❑

3. KNOTS OBTAINED FROM TRIVIAL KNOTS BY TWISTING

In previous sections , we assume that f(V) is knotted in S3. In what follows we consider the case where f ( V) is also unknotted in S. So we may assume that f (V) = V and f : V -+ V is a twisting homeomorphism of V with f (µ) = µ and f (A) _ A + nµ , where (µ, A) is a preferred meridian-longitude pair of V . We denote the image of K by K,,. (See Figure 3.1.)

Kn = f(K)

K

twist

Figure 3.1

88 KIMIHIKO MOTEGI

When the wrapping number of K in V is zero, this operation does not affect the knot types, so we always assume that wrapv(K) > 2 in the following. If the original knot K is a trivial knot in S3, then we call the resulting knot K. a twisted knot. In this section we consider the possibility obtaining knots of special kinds from trivial knots by twisting. 3.1. When can a twisted knot be a trivial knot? In such a special situation as indicated in the title, applying Gabai's result [11, we can deduce the following. Theorem 3.1

([9], [7]).

A twisted knot K. (n# 0) is knotted in S3, except for the

case as in Figure 3.2.

1 - twist

-1 -twist Figure 3.2 This result can be regarded as an answer to a very special case of the following conjecture. Conjecture 3.2. Let K be a knot in a standardly embedded solid torus V in S3. A knot K„ obtained from K by n-twist (n # 0) cannot be ambient isotopic to K, except for the case as in Figire 3.3.

1 - twist

-1 - twist Figure 3.3

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 89 3.2. When can a twisted knot be a composite knot?

We start with a motivating result proved by Scharlemann [15]. Theorem 3 .3 ([15], see also [23]). A crossing change on a trivial knot cannot produce a composite knot (i.e., a knot of unknotting number one is prime). Note that a crossing change on a knot K can be accomplished by a ±1-twist with wrapv(K) = 2 for some V, and Theorem 3.3 was generalized to the following. Theorem 3.4 ([16]). Let V be a standardly embedded solid torus in S3 and K a knot in V with wrapv(K) = 2. Then a twisted knot K„ cannot be a composite knot for any integer n. In connection with this, Mathieu [9] proposed the question: Question 3.5 ([9]). Can we have a composite twisted knot?

Theorem 3.4 shows the impossibility in the case wrapv(K) = 2. On the contrary, we can answer this question in positive by constructing the following concrete example. Example 3.1 ([14]). Let (V, K) be a pattern depicted in Figure 3.4. Then Kl is a product of the (2,3)-torus knot and the (2,5)-torus knot.

Figure 3.4 In this example wrapv(K) = 4. Later Ohyama also foud an interesting and simpler example with wrapv(K) = 3 such that Kl is a prduct of the (2,3)-torus knot and the figure eight knot. We can find other examples of composite twisted knots in [20], [11] and [2]. There is an excellent account of examples of composite twisted knots in [2]. It should be noted that all the examples are ±1-twist, and it was conjectured in [13] that a twisted knot can be a composite knot only for one integer n E {1, -1}. In [21], Teragaito proved that if a twisted knot K. is a composite knot

90 KIMIHIKO MOTEGI

then Inl < 2 applying a combinatorial technique developed by Gordon and Luecke. Later Goodman-Strauss [2], Hayashi and the author [3] independently proved the following. Theorem 3.6 ([2], [3]). If a twisted knot K„ is a composite knot, then n = ±1. Goodman-Strauss [2] shows further that K1 and K_1 cannot both be composite knots. But the following question is still open. Question 3.7. Is the number of prime factors of Kt1 < 2? Compare this with the following well-known question: Is the number of prime factors of a manifold obtained by Dehn surgery on a knot in S3 is less than or equal to 2?

3.3. When can a twisted knot be a torus knot? Let us start with a well-known example. Let K be a (fl, q)-cable of a core of a standardly embedded solid torus V in S3. Then K is a trivial knot in S3 and K. is a torus knot (which is a trivial knot again if q = 2 and n = T-1) for any integer n(# 0). We refer such an example as trivial example (see Figure 3.5).

K

Kn n - twist

(1, q) - torus knot 11 trivial knot

(1+nq, q) - torus knot

Figure 3.5

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 91

As a nontrivial example, we have the following. Example 3.2.

K 1 - twist

trefoil knot

trivial knot Figure 3.6

Theorem 3.8. If a twisted knot K„ (n # 0) is a torus knot, then except for trivial examples n = ±1. This result is an implicit corollary of the joint work with Miyazaki [10, Theorem 1.21 about Seifert fibring surgery on knots in solid tori.

Theorem 3.9 ([10]). Let J be a knot in a solid torus W such that J is not contained in a 3 -ball in W. Suppose that a manifold W(J; y) obtained from W by -y-surgery on J is Seifert fibred. Then one of the following holds. (1) J is a core of W or a cable of a 0-bridge braid in W. (2) y is integral (i.e., a representative of y intersects a meridian of J exactly once). Proof of Theorem 3.8. Let J be a core of the complementary solid torus S3-intV. We note that the twisted knot K„ can be obtained from K by -n-surgery on J. Since K is a trivial knot in S3, W = S3-intN(K) is a solid torus, which contains J in its interior. If K„ is a torus knot, then E(K„) = W(J;-n) is a Seifert fibred manifold over the disk with two exceptional fibres. Applying Theorem 3.9, we have the following possibilities: (1) J is a core of W or a cable of a 0-bridge braid in W. (2) n = ±1.

92 KIMIHIKO MOTEGI

Now we suppose that ( 1) happens. If J is a core of W, then we have wrapv(K) = 1, a contradiction . If J is a cable of a 0-bridge braid in W, then it turns out that K is also a cable of a 0-bridge braid in V. Assume first that the 0-bridge braid is not a core of V. Then since K is unknotted in S3, the 0-bridge braid is a (+1, q)-cable of a core of V (q > 2). Hence K. is a cable of a (±1 + nq, q)-cable of a core of W. The twisted knot K„ can be a torus knot only when n = +1 and q = 2, otherwise K„ has a nontrivial companion. Next assume that the 0-bridge braid is a core of W, then K is a (±1 , q)-cable of a core of V. In this case we have exactly a trivial example. This completes the proof of Theorem 3.8. ❑ Acknowledgement- I would like to thank Masakazu Teragaito for suggesting the application of Theorem 3.9 to Theorem 3.8. I wish to thank Shin 'ichi Suzuki for giving me an opportunity to publish this survey article.

REFERENCES 1. Gabai, D.; Surgery on knots in solid tori, Topology 28 (1989), 1-6. 2. Goodman-Strauss, C .; On composite twisted unknots , to appear in Trans. Amer. Math. Soc.. 3. Hayashi , C. and Motegi, K.; Only single twist on unknots can produce composite knots, to appear in Trans. Amer. Math. Soc.. 4. Jaco, W. and Shalen, P.; Seifert fibered spaces in 3-manifolds , Mem. Amer . Math. Soc. 220, 1979. 5. Johannson , K.; Homotopy equivalences of 3-manifolds with boundaries , Lect. Notes in Math. vol. 761, Springer-Verlag, 1979. 6. Kouno, M.; On knots with companions, Kobe J. Math . 2, (1985), 143-148. 7. Kouno , M., Motegi, K. and Shibuya , T.; Twisting and knot types , J. Math. Soc . Japan 44, ( 1992 ), 199-216. 8. Kouno, M., Motegi, K.; On satellite knots, Math. Proc. Carob . Phil. Soc . 115, (1994), 219-228. 9. Mathieu , Y.; Unknotting , knotting by twists on disks and Property ( P) for knots in S3, Knots 90 (ed Kawauchi , A.), Proc. 1990 Osaka Conf. on Knot Theory and Related Topics, de Gruyter, (1992), 93-102. 10. Miyazaki, K. and Motegi , K.; Seifert fibred manifolds and Dehn surgery III, preprint. 11. Miyazaki , K. and Yasuhara , A.; Knots that cannot be obtained from a trivial knot by twisting, Contemp . Math. 164, (1994 ), 139-150. 12. Motegi , K.; Knotting trivial knots and resulting knot types , Pacific J . Math. 161, (1993), 371-383. 13. Motegi , K.; Primeness of twisted knots, Proc. Amer . Math . Soc. 119 , ( 1993 ), 979-983. 14. Motegi , K. and Shibuya, T.; Are knots obtained from a plain pattern always prime ?, Kobe J. Math. 9, (1992 ), 39-42. 15. Scharlemann , M.; Unknotting number one knots are prime, Invent. Math . 82, (1985), 37-55. 16. Scharlemann , M. and Thompson , A.; Unknotting number, genus, and companion tori, Math. Ann. 280 , ( 1988), 191-205. 17. Schubert , H.; Die eindeutige Zerlegbarkeit eines Knoten in Primknoten , Sitzungsber. Akad. Wiss . Heiderberg, math .-nat. KI.1949 ,3. Abh., 57-104.

18. Schubert , H.; Knoten and Vollringe, Acta Math. 90 (1953), 131-286. 19. Soma, T.; On preimage knots in S3, Proc . Amer. Math. Soc. 100, (1987), 589-592.

KNOT TYPES OF SATELLITE KNOTS AND TWISTED KNOTS 93 20. Teragaito, M.; Composite knots trivialized by twisting, J. Knot Theory and its Ramifications 1, (1992), 467-470. 21. Teragaito, M.; Twisting operations and composite knots, Proc. Amer. Math. Soc. 123, (1995), 1623-1629. 22. Trotter, H.F.; Non-invertible knots exist, Topology 2, (1964), 275-280. 23. Zhang, X.; Unknotting number one knots are prime: a new proof, Proc. Amer. Math. Soc. 113, (1991), 611-612. DEPARTMENT OF MATHEMATICS, COLLEGE OF HUMANITIES & SCIENCES, NIHON UNIVERSITY, SAKURAJOSUI, SETAGAYA-KU 3-25-40, TOKYO 156, JAPAN

E-mail address : [email protected]

This page is intentionally left blank

Lectures at Knots 96 edited by Shin 'ichi Suzuki ©1997 World Scientific Publishing Co. pp. 95-122

RANDOM KNOTS AND LINKS AND APPLICATIONS TO POLYMER PHYSICS TETSUO DEGUCHI AND KYOICHI TSURUSAKI

ABSTRACT. We discuss probabilities of random knotting and linking through numerical simulations using topological invariants of knots and links. We define knotting probability PK (N) by the probability of an N-noded polygon having knot type K. We introduce a universal fitting formula for the knotting probability, and we show that the formula gives good fitting curves to the numerical estimates of knotting probabilities for different models of random polygon. We consider linking of two N-noded random polygons which have fixed knot types Kl and K2, respectively. We define linking probability PKIK-(R; N) by the probability that a given link L is formed when we put a pair of N-noded random polygons of Kl and K2 in distance R. For (L, K1, K2) = (0, 0, 0) and (2i, 0, 0), we numerically evaluate the linking probabilities PLK'K' (R; N), where 0 denotes the trivial knot or the trivial link, 22 the simplest nontrivial link (the Hopf link). We also discuss a formula which approximates the linking probability. Applying the numerical result of the linking probability we calculate the second virial coefficient of a ring polymer solution at the 0 temperature.

1. INTRODUCTION

Knotted ring polymers such as knotted DNA molecules are synthesized in various experiments in chemistry and biology. [5, 31, 49, 29, 30] Furthermore, it is observed that knotted DNA are produced in living cells (in vivo). [31] Thus knotted DNA and knotted molecules exist in reality. From the knot theory the actions of the topological enzymes (topoisomerases) on circular DNAs are studied. [35, 36] From the viewpoint of statistical mechanics and dynamics of polymers, the entanglement effect of long polymer chains gives rise to a number of nontrivial problems. [14, 11, 13] In particular, there is a fundamental question : " what fraction of permissible configurations of a chain of given length will contain a knot ? " [11] Recently, there are two independent approaches to the problem of random knotting and linking; from the rigorous estimates using the Kesten pattern theorem [37, 24, 50, 12, 34], and from the numerical simulations on off-lattice models [46, 4, 3, 19, 2, 17, 8, 9] and on lattice models [27, 18, 40, 41, 23]. In this paper, we study the topological effects of ring polymers via numerical experiments of knotting 95

96 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

and linking of random polygons. We discuss two types of topological problems: random knotting, i.e., self-entanglement of a ring polymer (or a random polygon), and random linking, i.e., mutual-entanglement of two ring polymers (or random polygons). Let us introduce the probability of random knotting. We define random polygon by a 3-dimensional polygon whose vertices are randomly given under certain rules. We denote by N the step number of the random polygon, i.e., the number of the polygonal nodes or vertices. We define (random) knotting probability by the probability that a given random polygon is topologically equivalent to a knot: suppose that we have M random polygons with N vertices and that MK polygons have the same knot type K. Then, the knotting probability PK(N) is given by the ratio MK/M.

The content of the present paper consists of the following. In §2 we introduce a universal formula describing the N-dependence of the knotting probability and show whether or how it fits to the data of our numerical experiments for the Gaussian and the rod-bead random polygons. In §3, we derive a formula for the linking probability, and then we discuss the numerical estimates of the linking probability evaluated by the Gaussian random polygons. In §4, we discuss the second virial coefficient of a ring polymer solution. 2. RANDOM KNOTTING PROBABILITY

In §2 we give a review on a part of the results given in Ref. [9]. We also discuss some related problems and numerical data, which have not been shown in Ref. [9]. 2.1. Knot invariants and random knotting. Let us briefly review on some numerical studies on random knotting related to the present paper. In particular, we consider off-lattice models in this section. We note that there are also important numerical works on lattice polygons. [27, 18, 40, 41, 23] For trivial knot (K = 0) the knotting probability Po(N) has been evaluated for several different models of random polygon with different lengths N less than about 2000. [46, 4, 3, 19, 27, 17] It is discussed numerically that the probablility Po(N) decays exponentially with respect to N for the molecular dynamical model and the rod-bead model. [19, 17] Hereafter we call Po(N) unknotting probability. For nontrivial knots, however, the knotting probabilities had been evaluated only for short polygons with N < 200 in the cases of the off-lattice models. The technical probem is that it will take a long computation time or large memory area for calculating some knot invariants for knot diagrams of long polygons (N large). [6] We should consider the following two points together: (1) computation time, (2) size of memory.

RANDOM KNOTS AND LINKS 97

For the computation-time problem, we note that a straightforward calculation of the Jones polynomial will take an exponential time with respect to the number of crossings . [16, 26, 45] For the memory- size problem, we should consider that if we evaluate a certain knot polynomial of variable t (such as the Alexander polynomial) by putting a real number to t, then the value can be very large or very small; putting t = 2 to the polynomial invariant for a knot diagram with 1000 crossings, then we may have a term such as 2100° or 2-1000. [6, 8] A breakthrough to the problem was given from recent developments in the study of knot invariants. Numerical simulations show that the Vassiliev-type invariants together with the determinant of knot are practical of use. [6] We should remark that there are independent algorithms for calculation of finite-type invariants of some degrees, which may be useful for different situations . [26, 6, 25] For an illustration we give a list of some knot invariants. We denote by AK(t) the Alexander polynomial of variable t for knot K, which is calculated by taking the determinant of the Alexander matrix. The symbol v3(K) denotes the jth coefficient of the quasi-classical expansion of the Jones polynomial. [6, 8]; v2(K) denotes the Vassiliev-type invariant of the second degree. Knot type Knot K

the determinant [4[( -1)I

31(+) 31(-) 41

52(-) 31( +)031 ( +)

3 3 5 5 5 7 7 9

-12 -12 12 -36 -36 -24 -24 -24

60 -36 -12 276 -204 168 - 120 120

-199 -55 31 -1365 -645 -758 -326 -254

31(+)031(-) 31(-)031 (-)

9 9

-24 -24

24 -72

-110 34

31(+)041 31(-)041 41041

15 15 25

0 0 24

48 -48 -24

-312 -168 208

51(+) 51(-) 52(+)

the Vassiliev-type invariants v2(K) v3(K) v4(K)

TABLE I. The values of the determinant of knot OAK(-1)l and those of the j-th coefficient vj(K) in the quasi-classical expansion of Jones polynomial (j = 2, 3, and 4). Symbols 31(+) and 31(-) denote the mirror images of the trefoil. Symbol K10K2 denotes the product of K1 and K2. [6, 8]

2.2. Method of numerical experiment. Let us discuss the method of our numerical experiment. It consists of the following 4 procedures. (1) Making 3-dimensional random configurations of N-noded polygon, (2) Constructing knot diagrams through projections of the 3-dimensional configurations onto a plane,

98 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

(3) Evaluating knot invariants for the knot diagrams, (4) Enumeraing the number of such polygons that have the same set of the knot invariants such as A(-1)K, v2(K), and v3(K). Here we recall that a configuration of a polygon is given by a sequence of the 3dimensional coordinates of the vertices of the polygon. Let us discuss the process 1. We consider the two different types of models of random polygon, the Gaussian random polygon [4] and the rod-bead random polygon [3]. The former is given by ideal chains with no excluded volume, while the latter consists of real chains with the excluded volume. We can study the selfavoiding effect on the knotting probability through simulations on the rod-bead models with different bead radii. [9] We construct Gaussian polygons, step by step, by using the conditional probability distribution Pc(fi; iii, ... ,u";-1)

(2.1) _ (21r) -3/2 exp

(_

2(Nj j) (u' + N

j + 1)2) ,

where XN are the position vectors of the vertices of N-noded Gaussian The conditional polygon, ul, ... , i!N are the jump vectors such that ul = probability is calculated by taking integral of the distribution function P(u"1i... X N) with respect to u3+2, ... , uN where the distribution function is given by (2.2)

P(ul, ... , UN) = Const. x exp(-(ui + • • • + uN)/2)6(u1 + • • • u"N).

We note that 6(X) = 6(x)6(y)6(z) denotes the 3-dim. Dirac's delta function. We now consider the rod-bead polygon. We construct rod-bead polygons first by making linear chains by the dimerization method (chain dimerization) and then by concatenating the linear chains (ring dimerization). [3] Let us explain the chaindimerization. We assume that all the beads have a radius r and the rod has a unit length. First we construct two rod-bead chains with step numbers N1 and N2, respectively. Then we make a chain with step number N1 + N2 by concatenating them and by checking there is no overlap among the beads. If there is an overlap, then we discard the chains N1 + N2, N1 and N2, and we start from the beginning. For an illustration, the binary-tree structure of the chain-dimerization procedure is shown in Fig. 1, where a chain with 100 steps are constructed from 4 chains with 12 steps and 4 chains with 13 steps. Let us consider the ring-dimerization. [3] After constructing 2M chains with step number N/2, we make M polygons with N- (or N + 1-) nodes by picking up a pair of chains out of 2M chains and by concatenating them: if there is an overlap, then we discard the pair and try another pair. We

99

RANDOM KNOTS AND LINKS

should note that in the ring-dimerization process we have to calculate the statistical weights related to the probability of successful concatenation. [3]

Fig.l(a):rod-bead chain consists of beads of radius r and rods with unit length

100 50

25

12

13

25

12

13

50

25

12

13

25

12

13

Fig. l(b):Construction of rod-bead chain;dimerization procedures 2.3. Universal fitting formula. Numerical estimates of random knotting probabilities can be approximated well by a certain theoretical equation. We intoduce the following formula [8] (N - Nini(K)lmixi exp(- - Nini(K)) (2.3) PK(N) = C(K) N(K) ) (- N(K) where m(K), N(K), NN,,i(K) and C(K) are fitting parameters which are determined by the least square method . We shall see later that the formula gives good fitting curves and also that given a model of random polygon, the estimates of N(K) are constant within the error bars: N(K) :: N(O) for several nontrivial knots K. The formula (2.3) is similar to an asymptotic expansion of the partition function or the entropy of a polymer system in statistical mechanics such as random walks or self-avoiding walks . In fact , if we set Ni„i(K) = 0 and take the logarithm of (2.3),

100 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

then it gives the form of an asymptotic expansion.

When the number N of vertices of polygon is small, the knotting probabilities of prime knots can be approximated by linear functions PK(N) ^ C(K) N - Ni i(K) N(K) 1 for where Ni„i(K) are positive numbers of order 0(101). We note that m(K) prime knots, which we shall see in Table II. For the cirular DNA model the knotting probabilities of some prime knots are explicitly described by some linear functions of N, and the results are compared with the biochemical experiments of random knotting. [29] We may conclude that the equation (2.3) is an interpolation formula which connects the large-N behaviour and the small-N behaviour of the knotting probability PK(N). For the large N behaviour, the parameter NN„i(K) is not important. (See also the Table III.) As we shall see later, the estimates of Ni,,i(K) for the Gaussian polygon and the rod-bead polygon are of order 0(10') and are very small when N = 2000 - 5000. 2.4. Numerical estimates of knotting probabilities for Gaussian polygon and rod-bead polygon . Let us discuss the data of our numerical simulations. For a given step number N we constructed 105 polygons to the Gaussian model (M = 105) and 104 polygons to the rod-bead model with a fixed radius (M = 104). [9] In Fig.2, the estimates of the unknotting probability for the Gaussian model and the rod-bead models with r=0.05, 0.10, 0.15, and 0.20 are shown against the step number N. [9] The lines are theoretical curves given by the formula (2.3), whose parameters are shown in Table II. From Fig.2 we may confirm the exponential decay of Po(N) for the rod-bead model shown in Ref. [17] (see also [19]).

Let us explain the method of estimating errors. In Figs.2-6, the error bars denote the standard deviations. For the Gaussian model we assume that the errors are mainly due to the statistical fluctuation of the number MK of polygons of knot K, which are estimated by applying the binomial distribution to MK. For the rodbead models we estimate the variance of the knotting probability PK(N) by taking the sum of the following two contributions: one from the statistical fluctuation of MK and another from the fluctuation of the statistical weights appearing in the ring-dimerization procedure. [9]

RANDOM KNOTS AND LINKS 101

Gaussian and Rod-bead models

1000 2000 3000 4000 5000 Step Number N Fig.2 In Figs.3 and 4, the numerical values of the knotting probability PK(N) of the Gaussian random polygon are plotted against the step number N for five nontrivial knots 31, 31031, 31031031, 41 and 31041 . We recall that the fitting curves are given by the formula (2.3). In Figs .5 and 6 , the numerical estimates of the knotting probability PK( N) of the rod-bead random polygons with r=0.10 and 0.20 are plotted against the step number N for three nontrivial knots 31, 31031i and 31#31031.

Applying the formula (2.3) to the numerical data we see that it gives good fitting curves to the graphs of PK(N) versus N both for the Gaussian random polygon and the rod-bead random polygon . [8, 43, 42 , 9] Here we consider N-noded polygons with N in the region : 50 < N < 5000 , and also consider the cases of several nontrivial knots, for example, some prime knots such as 31i 41, 51, 52 and some composite knots such as 31031 i 31041, 31031031. The least-square estimates of the parameters m(K), C(K), N(K) and N;1,;( K) for the Gaussian and rod-bead models are given in Table II together with the X2 values of the fitting curves. [9] The errors in Table II correspond to 68 . 3 % confidence intervals . The X2 values are consistent with the observation that the fitting curves are good.

102 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

Gaussian Random Polygon

-3_1 --'-- 31 #3_1 - ^- 3-1#3-1#3-1

500

1000 1500 2000 Step Number N

2500

3000

Fig.3

Gaussian Random Polygon

41 --o-- 31 #4_1

500

1000 1500 2000 Step Number N

Fig.4

2500

3000

RANDOM KNOTS AND LINKS

Rod-bead model r=0.10

♦1 Y

0 C

Y

500 1000 1500

Step Number N Fig.5

Rod-bead model r=0.20

-3_1 - 31 #3_1 --^- 3_1#3_1#3_1

1000 2000 3000 4000 5000 Step Number N Fig.6

103

TETSUO DEGUCHI AND KYOICHI TSURUSAKI

104

N(K) x 10 N„,T(K) x 102 X m(K) C(K) Gaussian raondom polygon (22 data points) 3.40 ± 0.04 -0.01 ± 2.84 37 -0.0051 ± 0.0190 1.05 ± 0.87 0 24 0.19 ± 0.02 0.631 ± 0.004 3.50 ± 0.04 0.888 ± 0.024 31 30 0.28 ± 0.04 0.130 ± 0.002 3.49 ± 0.09 0.91 ± 0.05 41 21 0.30 ± 0.06 0.043 ± 0.001 3.31 ± 0.15 1.02 ± 0.09 51 0.28 ± 0.04 32 0.073 ± 0.001 3.31 ± 0.11 1.04 ± 0.07 52 16 0.24 ± 0.04 3.51 ± 0.05 0.198 ± 0.005 1.85 ± 0.05 3103, 26 0.27 ± 0.06 0.078 ± 0.003 3.49 ± 0.08 1.90 ± 0.07 3104, 20 0.23 ± 0.12 0.042 ± 0.005 3.54 ± 0.09 2.80 ± 0.11 31031#31 R.od-bead model with r = 0.05 (21 data points) 16 0.0 ± 136.0 1.0±51.0 2.7 ± 0.2 0.00 ± 0.10 0 16 0.1 ± 0.1 2.7 ± 0.1 0.60 ± 0.02 0.98 ± 0.09 31 14 2.5 ± 0.2 0.2 ± 0.1 0.12 ± 0.01 1.1 ± 0.2 41 13 0.2 ± 0.1 2.8 ± 0.2 0.19 ± 0.02 1.9 ± 0.2 31#31 14 0.0 ± 0.2 2.7 ± 0.3 0.070 ± 0.015 1#41 1-2.1 ± 0 13 0.6 ± 0.3 3.1 ± 0.3 2.4 ± 0.4 0.065 ± 0.019 31031031 Rod-bead model with r = 0.10 (20 data points) 32 -0.1 ± 6.2 1.1±1.6 4.2 ± 0.4 0 -0.08 ± 0.26 14 4.2±0.2 0.2±0.1 0.67±0.02 0.91±0.10 31 17 0.3±0.1 4.4±0.4 0.12±0.01 41 0.8±0.2 16 0.3 ± 0.1 4.4 ± 0.3 0.26 ± 0.02 1.8 ± 0.2 3103, 18 0.3±0.2 4.4±0.4 0.094±0.011 1.8±0.2 31 041 12 0.6 ± 0.3 4.4 ± 0.5 0.079 ± 0.023 2.6 ± 0.3 31#31031 Rod-bead model with r = 0.15 (20 data points) -0.1 ± 19.3 27 8.2 ± 0.5 0.9 ± 2.2 -0.01 ± 0.10 0 23 0.2 ± 0.3 8.5 ± 0.5 0.76 ± 0.03 0.90 ± 0.11 31 6 0.3 ± 0.5 8.5 ± 0.9 0.10 ± 0.01 41 0.9 ± 0.2 10 -0.1 ± 0.4 8.3 ± 0.6 0.30 ± 0.04 2.0 ± 0.2 31 031 18 0.1 ± 0.7 8.5 ± 1.0 0.082 ± 0.017 1.9 ± 0.3 31#41 14 1.4 ± 1.0 8.3 ± 1.1 0.11 ± 0.05 2.7 ± 0.5 31031031 Rod-bead model with r = 0.20 (20 data points) N(K) x 10 N;2,;(K) x 10 x2C(K) m(K) Knot K -0.1 ± 13.1 T52.2 ± 0.6 0.01 ± 0.35 1.0 ± 6.5 0 8 0.0 ± 0.1 2.3 ± 0.5 0.84 ± 0.04 0.9 ± 0.2 31 36 0.0 ± 0.1 2.2 ± 0.7 0.38 ± 0.14 2.1 ± 0.4 31031 21 0.2 ± 0.2 2.1 ± 1.0 0.12 ± 0.12 2.8 ± 0.8 31031031

Knot K

TABLE II . Fitting parameters m(K), C(K), N( K) and N;,,;(K) to Gaussian model and rod-bead models with r =0.05, 0 . 10, 0.15 and 0.20.

RANDOM KNOTS AND LINKS 105

From Table II we see that given a model of random polygon, the parameters N(K) are almost the same value for the different knots: N(K) : N(0). We also find that with respect to the errors, the exponent m(K) of knot K does not change for the Gaussian and the rod-bead models with the four different values of the bead radius: r=0.05, 0.10, 0.15. and 0.20. [9] The numerical result has lead to the conjecture [9] that the exponent m(K) should be universal for each knot type K: for different models of random polygon the knotting probability for knot K is expressed by the formula (2.3) with the same value of the exponent m(K). The exponent m(K) should be determined only by the knot type. [9] Let us give a comment on the estimated values of the exponents m(K). The estimates of m(K) given in Table II are consistent with the conjecture of universality of m(K) within the error bars. However, we should note that the estimated errors for them are not very small. Furthermore, there are other methods for determining the exponent m(K), which may give slightly different estimates. For an illustration we consider the following formula with 3 variables related to the asymptotic form „6(K)

(2.6) PK(N) = C(K) ( N ll N(K))

exp(

N N(K) )

If the step number N is large enough, then the formula (2.6) also fits to the data of the knotting probability. In Table III we give the estimated parameters of the fitting curves to the knotting probability of the Gaussian random polygon. We apply the formula (2.6) to the same data (18 data points) as shown in Figs.2 and 3 except for the cases N=50, 100, 200 and 300. The X2 values are rather small. The estimates of m(K) are a little larger than those in Table H. However, the difference is small with respect to their estimated errors.

m(K) N(K) x 102 X C(K) Gaussian random polygon (18 data points) 0 0.07 ± 0.05 1.012 ± 0.021 3.51 ± 0.0.09 f43.49 ± 0.06 16 0.92 ± 0.04 0.635 ± 0.007 31 1.93 ± 0.05 0.187 ± 0.006 3.47 ± 0.06 12 3103, 31 #31 #31 2.95 ± 0.07 0.0361 ± 0.0024 3.46 ± 0.06 22 Table III. Fitting parameters for the knotting probability of the Gaussian polygon. The data in Figs.2 and 3 for N > 400 are fitted by the 3-parameter formula (2.6) Knot K

Let us discuss a possible connection of the result of off-lattice models to that of lattice models. If the hypothesis of the universality of m(K) should also hold for lattice models, then the simulation of off-lattice models should give the same

106 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

value of m(K) with that of lattice models. Recently the entropic exponents of a few knots are evaluated in a lattice model by using the BFACF algorithm in Ref. [23], which should be equivalent to the exponent m(K) in (2.4). Here we recall that the fitting parameters N(K) and m( K) are equivalent to those of an asymptotic expansion ( 2.4) with respect to 1/N, if we evaluate them by applying the formula (2.3) for large N (See also §2 . 3). It seems that the estimates for the parameters N(K) and m(K) in Ref. [23] have similar properties in commomn with those in Table II [9] and those in the previous papers [8 , 43, 10, 44] with respect to the 1 for prime nontrivial knots K errors; N(K) ^^ N(0) for any knots K, m(K) and m(K10K2 ) .;: m(Kj) + m(K2 ) for composite knots K1OK2. Thus it seems that the result of Ref. [23] is quite favorable to the hypothesis of the universality of the exponent m(K), although the different fitting function is used. We shall investigate this interesting connection more explicitly in later publications. 2.5. Knotting of closed rod-bead chain. Let us consider a different type of random polygon. We define a random polygon by closing the two ends of a random walk (or self-avoiding walk). In this way, any random walk gives a random polygon. It should be noted that the closing edge is much longer than the other edges since its average length is given by the mean end-to-end distance , which is proportional to N°(v=0.50.6).

Fig.7:Polygon obtained by closing the two ends of rod-bead chain. Dashed line shows the closing arc. Let us discuss the motivation. In statistical physics, the shape of polymers is often given by an open chain , not by a closed chain ; it is easier to make linear chains than circular rings. We expect, however, that the topological property of closed linear chains may reflect that of the linear chains ; it will be quite important if we could

RANDOM KNOTS AND LINKS 107 investigate the entanglement effects among linear polymers using knot theory, where the ways of entanglement are not permanent but only temporary and dynamically changing.

Unknotting probability of closed rod-bead chain (r=0.05)

0.01' 0 200 400 600 800 Step Number N

1000 1200

Fig.8 The knottedness of self-avoiding walks has been discussed in Ref. [37]. It seems that we can discuss the knottedness of arcs if tight knots can exist. It should be remarked that some regorous results are also obtained on the knottedness of Gaussian random walks in Ref. [12]. They are quite remarkable since there is no tight knot in the Gaussian chain. Let us show the result of our simulations. We consider the random polygons obtained by closing the two ends of rod-bead chains with the bead radius r = 0.05. For a given step number N we constructed 104 rod-bead chains (M = 10). In Fig.8 the numerical estimates of the unknotting probability of the closed rod-bead chain with r=0.05 are shown against the step number N. In Fig.9 the estimates of the knotting probability of the closed rod-bead chain with r=0.05 are shown against the step number N for K = 31 and 31031. Here the errors are estimated by the standard deviations due to the statistical fluctuation of MK. The theoretical curves are given by the formula (2.3). The fitting parameters and the x2 values are given in Table

IV.

108 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

Knotting probability of closed rod-bead chain ( r=0.05) 31 -^ - 3 #3_1

Y

200 400 600 800 1000 1200 Step Number N Fig.9 Knot K 0 31 31031

m(K) C(K) N(K) x 10 Rod-bead chain (9 data points) 2 .9 ± 0.2 0.07 ± 0.07 1.3 ± 0.3 1.1 ± 0 . 1 0.67 ± 0.01 2.8 ± 0.2 2.3 ± 0.4 2 . 6 ± 0.5 0 . 13 ± 0.05

N;,,;(K) x 10

X

-0 . 2 ± 0.6 0 . 09 ± 0.04 0 . 2 ± 0.2

8 1 28

Table IV There are only 9 data points, which are not sufficient to make good estimates of the fitting parameters. However, the result has many similar properties in common with the Gaussian and rod-bead random polygons. The fitting curves given by the formula (2.3) seems to be good within the estimated errors. The graph of the unknotting probability shows the exponential decay with repsect to N. The parameter N(K) has almost the same value for K = 0 and 31. Thus the numerical simulations suggest that these properties should also hold for such random polygons that are obtained by closing linear chains.

RANDOM KNOTS AND LINKS 109

2.6. Summary of numerical results on random knotting . We observe the following properties from the simulations of the Gaussian random polygon [8, 43, 42, 9], the rod-bead models with 4 different radii [42, 10, 9], and the closed rod-bead chain discussed in §2.5. (1) The formula (2.3) gives good fitting curves to the estimates of the knotting probability for the different models, from the viewpoint of the X2 analysis. (2) Given a model of random polygon, the parameters N(K) for the different knots have the same value with respect to the estimated errors.

(3) Given a knot K, the parameter m(K) gives almost the same value for the different models with respect to the error bars. For the composite knots, m(Ki0K2) m(Ki) + m(K2), with respect to the estimated errors. From the numerical simulations in §2.4 and §2.5, we confirm the following conjectures. [9]

• Conj. 1 The formula (2.3) gives a universal fitting formula; it should give good fitting curves of knotting probability to any model of random polygon.

• Conj. 2 (Characteristic length for random knotting) For any model of random polygon N(K) = N(0). The parameter N(0) can be considered as the characteristic length of randon knotting for the given model. • Conj. 3 (Universality of m(K)) Given a knot K, the exponent m(K) is given by the same value for any model of random polygon.

The conjectures 1, 2, and 3 lead to universality on the N-dependence of random knotting probability. The numerical result of §2.5 suggests that the parameter N(0) can also be considered as the characteric length of self-entanglement of linear chains. Through numerical simulations of other random polygons, we shall investigate how far these conjectures hold. 3. LINKING PROBABILITY

3.1. Definitions of linking probability. Let us disucss the probability of linking. In fact, there are several different definitions of linking probability. In this paper, we define linking probability PL 1K2(R; N) by the probability that a given pair of random polygons with N vertices makes link L where their centers of mass are separated by distance R and the two polygons have the fixed knot types Kl and K2, respectively. For the case of (L, K1, K2) = (0, 0, 0), the symbol PL-K- (R; N)

110 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

denotes the probability that a pair of unknotted N-noded polygons forms the unlink or the trivial link when we put them in distance R; for (L, K1, K2) = (221, 0, 0), Pi1K, (R; N) denotes the probability that a pair of unknotted polygons forms the Hopf link 2' when we put them in distance R. Hereafter we shall call P000 (R; N) unlinking probability. Let us consider a different definition of linking probability. We define averaged linking probability by the following (3.1)

PL" (R; N) - 1 E pL 1K'(R; N)pK1(N)pKK(N) K 1,K2

where the normalization factor Z is given by (3.2) Z = E PK1(N) PK?.(N) = 1. K1,K2 Here we recall that the symbol PK(N) denotes the knotting probability discussed in §2.

In the same way as the linking probability, we can discuss the probability of polygons having a given linking number m. We define linking-number probability P,n 1K' (R; N) by the probability that given two random polygons with N vertices makes a link with linking number m, where their centers of mass are separated by distance R and the two polygons have the fixed knot types Kl and K2, respectively. We also define averaged linking-number probability P,'°° (R; N) by (3.3)

pK1K2 (R; N) pK3 (N) pKa (N).

P,n°o° (R, N) = ZKE

We shall discuss the linking probability as a function of the distance R. We introduce a normalized distance r, which is defined by the distance R divided by the radius of gyration (R9): (3.4) r = R/R9. For the Gaussian polygon, we denote by so the radius of gyration. Thus for the Gaussian polygon, we have r = Rl so. 3.2. Difficulty in the study of linking probability. The question of how the linking probability depends on the separation R has been discussed by several authors through their numerical simulations with the different versions of linking probability. It seems that there are no formula consistent with all the results of the simulations . For example, the r-dependence of unlinking probability is predeicted as exp(-exp(R2)) in Ref. [38] , exp(-R3) in Ref. [47] and exp(-RA) in Ref. [48] where A is a continuous parameter. Furthermore, our numerical simulations using

RANDOM KNOTS AND LINKS 111

the Gaussian polygons show that each of them does not give good fitting curves to the data in the context of the X2 analysis. At first sight the different results of the numerical simulations could be independent from each other since they are on the different versions of linking probability. In fact, none of the authors has discussed the linking probability PL1K2 (R; N) with knot types fixed. In Ref. [38], for example, the averaged linking-number probability P,'°°(R; N) is studied from the viewpoint of the gauge theory. From the results of our numerical simulations, however, it seems that the different versions of linking probability should have a very similar r-dependece. This suggests that the different results of numerical simulations should be related to one another. From the large r behavior of the linking probability we may have some important clues to the problem of random linking. We shall discuss the large r-dependence of linking probability by an intuitive argument leading to the exp(-r3)-dependence, and we shall check the validity of the argument through numerical simulations using some invariants of knots and links. 3.3. Rough derivation of linking probability. Let us now discuss the large rdependence of linking probability P8(r; N). In fact, we can derive it by an intuitive argument in the following. (1) Choose a pair of N-noded polygons, randomly, from an ensemble of random polygons. Put them in a distance r. (2) If they are unlinked, then they should be unlinked when they are placed in a distance r + dr. If they are linked, then they may become unlinked when placed in a distance r + dr. (Assumptions which could be valid only for large r.) (3) The increase dF°(r) in the unlinking pobability should be proportional to the product of the probability (1 - Po (r)) of being linked multiplied by the partial volume 47rr2dr of the configuration space of the two polygons.

dP°(r) - (1 - P°(r)) x 47rr2dr (4) Assume that the coefficient is given by a constant C independent of r, then we have dPo = C(1- P°(r))47rr2 By integrating the equation we obtain

(3.5)

P°(r) = 1 - Aexp(-ar3)

where a = 47rC/3 and A is an integral constant. We can apply the above argument to the different versions of linking probability such as PL °°(r; N), P41K2(r; N) and P,,',0°°(r; N). As far as the large-r behaviors

112 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

are concerened, they should have similar functional dependence on r. Thus, the different versions of linking probability should be connected to one another. Through a similar argument we can discuss the linking probability of a nontrivial link. We propose that the linking probability of the Hopf link can be described by the following fitting formula: (3.6)

P = B1(N) exp(-/3i(N)r3) - B2(N) eXp(-/32(N)r3).

We can improve the above argument for the unlinking probability more precisely. The result is given as follows (3.7) P00'(r; N) = 1 - > A; (N ) exp(-a3 (N)r3).

From the data of our simulations, however, it seems that the formula (3.7) is effective only when r is large (r > 1). A more elaborate study will be given in later publications.

3.4. Method of numerical simulations . Let us explain the method of our numerical experiments . It is essentially the same with that of the numerical simulations of random knotting in §2.2. We construct a large number of random configurations of Gaussian polygon with N vertices. Then we calculate the knot invariant AK(-1) and the Vassiliev invariant v2(K) of the second degree, we select such polygons that have OK(-l)=1 and v2(K)=O. In this way we select only such polygons that may be equivalent to the trivial knot. We repeat this procedure until we have 2M polygons with OK(-1)=l and v2(K)=O. The number M is given by 104 in our simulations of linking probability. We pick up a pair of such configurations, randomly. We put them in distance r, where the distance r is between the two centers of mass. From the projection we make the link diagram, and then calculate the two link invariants: the linking number and the Alexander polynomial evaluated at t = -1. Enumerating the number of such polygons that have the same values of the link invariants, we make estimates of the linking probability.

The two link invariants are the two special values of the Alexander polynomial: AL (S = 1,t = 1) and AL (S = -l,t = -1). Here the symbol AL(s,t) denotes the two variable Alexander polynomial for two-component links. We note that AL(1,1) gives the Gauss linking number. For an illustration we give a list of the values of the Gauss linking number (I = I AL(1,1)1) and COL(-1, -1)J in Table V.

RANDOM KNOTS AND LINKS

L 21 42 52 62 62 62 72

III 1 2 0 3 2 3 1

IoL( -1, -1)1 1 2 4 3 6 5 7

L 72 72 72 72 76 77 78

113

ILL( -1, -1)I

III 1 0 0 2 0 2 0

9 8 8 10 12 2 4

TABLE V. Values of the Gauss linking number (I) and IAL(-1, -1)1The linking number is the simplest but important link invarint . There are some papers related to the Gauss linking integral. [1, 15, 38] The two variable Alexander polynomial for links (IL(s, t)) is much stronger than the Gauss linking number. Unfortunately, there are not many papers which evaluate the linking probabilities numerically using the Alexander polynomial. [47, 20, 22, 44]

( Unlinking probability P000 r ; N) for Gaussian polygon 1

(N=50, 100, 200, 500)

0.810. 6F 0.4+-

8.0

0-50 -F- 0-100 - 0-200 - 0-500 -0-500

1.

0.5

1.0

1.5

1

2.0 2.5 3.0.

normalized distance r Fig.10

3.5. Numerical estimates of linking probability for Ghussiaff polygon. We now show our numerical results. In Fig.10 the hiiile'i"i'cal estiiiates' of the unlinking probability POOO(r; N) are plotted against tie normalized distance r = R/so, for the

114 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

different values of the step number N=50, 100, 200, and 500. We apply the following formula P000(r; N) = 1 - A(N) exp(-a(N)(r - ro(N))3).

(3.8)

This formula is a variant of the equation discussed in §3.3. The error bars denote the standard deviation due to the statistical fluctuation of the enumerated number ML of unlink L=0. The fitting parameters and the X2 values are given in Table VI. It seems that the theoretical curves fit to the data. As a first approximation, the formula (3.8) may be useful. However, the X2 values show that the fitting is not very good from the statistical viewpoint. Gaussian random polygon (31 data points) N A(N) ro(N) a(N) X 50 0.754 ± 0.002 0.229 ± 0.008 -0.182 ± 0.024 213 100 0.827 ± 0.004 0.236 ± 0.008 - 0.008 ± 0.020 141 200 0.878 ± 0.003 0.239 ± 0.007 0.009 ± 0.017 119 500 0.925 ± 0.002 0.266 ± 0.007 0.135 ± 0.015 125 TABLE VI. Fitting parameters for unlinking probability: POOO(r; N) = 1 - A(N) exp(-a(N)(r - ro(N))3)

Unlinking probability of 400 -noded rings P000 (r; N=400) 1 P(0) 0.81-

0.61-

0.41-

0.21

.0 0.5 1.0 1 .5 2.0 2.5 3.0 3.5 Distance between ring polymers : r = R/s0 Fig.11

RANDOM KNOTS AND LINKS 115

In Fig. 11, the data of linking probability POOO (r; N) are plotted against r for N = 400. The theoretical curve is given by the two-variable formula (3.5). The X2 value is 335, which is very large for the case of 31 points of data (A = 0.927 ± 0.002, a = 0.223 ± 0.002). The graph of Fig. 11 suggets that the argument leading to the formula (3.5) may be valid only when r > 1.

Linking probability P200(r; N) for Gaussian polygon (N=50,100,200,500) 0.5 -

II

2_1-50 -2-1 - 100 21 - 200 -2-1-500

0.0

0.5 1.0 1.5 2.0 2.5

3.0

Normalized distance r Fig.12 In Fig.12 the estimates of the linking probability (r; N) is shown against r for N=50,100, 200, and 500. The fitting curves are given by the formula (3.6) given in §3.3. The fitting parameters are listed in Table VII. From the X2 values we can consider that the fitting is good. Gaussian random polygon (31 data points) 2 N B2 31 N Bi(N) X (N) 50 0.503 ± 0 . 008 0 .107 ± 0. 009 0 . 251 ± 0. 004 3.81 ± 0.80 103 100 0 . 536 ± 0 .012 0 . 215 ± 0. 012 0 . 227 ± 0. 004 1 . 85 ± 0.20 62 200 0 . 555 ± 0.016 0 .309 ± 0. 015 0. 207 ± 0.004 1 . 29 ± 0.10 24 500 0 . 558 ± 0. 023 0. 402 ± 0 . 023 0. 199 ± 0 .005 0 .887 ± 0.060 44 TABLE VII. Fitting parameters for the linking probability: P(r; N) = B1( N) exp (-13i(N)r3) - B2(N) exp (-132(N)r3)

116 TETSUO DEGUCHI AND KYOICHI TSURUSAKI

3.6. Discussion on N-dependence of linking probability. Let us discuss how the linking probability depends on the number N of the vertices of polygon. In this subsection we can present only some preliminary results. In Fig.13 the estimates of the unlinking probability Po'(r = 0; N) are plotted against the step number N, where N = 50, 100, 200, 300, 400, and 500. From this plot we may assume a power-law behavior P°(0; N) = 1 - A(N) = B°N-b

(3.9)

where 5 = -0.54 ± 0.02 and B0 = 2.1 ± 0.2.

N-Dependence of Unlinking Probability at r=0: P000(r=0; N) 1

rn C

Y C C

0.01

100 Step Number N of Ring Polmer

1000

Fig.13 Let us consider the N dependence of the parameter a(N) in the two-variable formula (3.5). We evaluated the parameter a(N) by applying the equation (3.5) to the numerical estimates of the unlinking probability P0'0(r; N) for N= 50,100,200, 300, 400, and 500. Here we have 31 data points to each of the step numbers N. In Fig.14 , the estimated a(N) are plotted afainst N. From the graph , we may also assume a power-law behavior (3.10)

a (N) = a°N_µ

RANDOM KNOTS AND LINKS 117

where ao = 0.55±0.03 and p = -0.16±0.01. Here the errors are given by considering only the statistical errors. They may be larger than the estimated ones.

N-dependence of a(N ) (Gaussian polygon) PO00(r; N)=1-A(N )exp(-a(N) r3) 1

0.1

100 Step Number N

1000

Fig. 14 4. APPLICATION TO POLYMER PHYSICS

4.1. Virial expansion of polymer solution. We now discuss the statistical mechanics of a ring polymer solution. Let us consider polymers of a molecular weight Mv, in solution (in a liquid such as water) under the temperature T, where the mass consentration (or density) of the polymers is given by c. It is known that the osmotic pressure 11 of polymer solution can be expressed in terms of the virial expansion (4.1)

II=kBNAT (Mw+A2c2+A3c3+...),

where kBNA denotes the gas constant, which is given by the product of the Boltzmann constant kB and the Avogadro number NA. Here A2 and A3 denote the second and third virial coefficients, respectively. The expansion (4.1) can be compared with the Van der Waals equation of the imperfect gas which describes the relation among the pressure P, the volume V ,

118 TETSUO DEGUCHI AND KYOICHI TSURUSAKI the temperature T and the molar number n

P(V - nb) = nkBNAT P = kB NAT n 1

V 1- nb/V n ( n )2 (4.2) kBNAT V +b +...

(

where b denotes the volume of the molecule of the gas. We note that c corresponds to nM,,,/V. Thus the second virial coefficient A2 corresponds to b/Mw. In general , A2 corresponds to the effective thickness (or excluded volume) of polymers. 4.2. Topological second virial coefficient A2 at 9 temperature. If we synthesize ring polymers in a dilute solution, almost all the ring polymers are not linked one another. Furthermore, the knot type of the ring polymers is given by the trivial knot. The polymer solution has the topological constraint that the ring polymers should not be linked in thermal fluctuations. This topological constraint leads to a repulsive entropic force among the ring polymers. [47, 15, 39] It is known that at a certain temperature, the effective thickness of the polymers in a polymer solution vanishes and consequently we have A2 = 0. Such temperature is called the 9 temperature of the polymer solution. Let us consider the ring- and linear-polymer solutions of polystyrenes in the cyclohexane solvent.[28] Different 0 temperatures have been measured for the ring- and linear polymers. We denote by 01 the 0 temperature for the linear polysthyrene, and by Or that of the ring polysthyrene. In the experiment [28] it is found that 01 is 34.5°C, but Or is 28.0°C. The difference 01 - Or is due to the effective repulsive force from the topological constraint.

The nonzero second virial coefficient A2 of the ring-polymer solution has been measured at 01. [28] For the linear polymers, the excluded volume effect disappears at 01. We may assume that the ring polymers do not have the excluded volume effect at 01i the 0 temperature of the linear polymer solution. Thus the second virial coefficient A2 of the ring polymer solution measured at 01 should be derived from the topological constraint. Let us discuss the second virial coefficient at 01 from some probability of "unlinking". [47, 15, 39] In our viewpoint, it is given by the unlinking probability Po' (r; N). The expression of the virial coefficient at B1 is given by (4.3) A2 =

2M2 w J

47rR2(1- P000 (R; N)) dR,

where the distance R is given by R = sor. We recall that NA is Avogadro's number and M,,, is the molecular weight of the ring polymer. [44]

RANDOM KNOTS AND LINKS 119

In Fig. 15, experimental esitmates and our theoretical evaluations are plotted against N, where we have assumed that the molecular weights MK,,h,, for the Kuhn length is given by 1000. In our numerical evaluation of A2, we can not determine the overall constant. Thus the plot is only valid if we consider the N-dependece of the data.

Second virial coefficient of ring-polymer solution (ring polystyrenes in cyclohexane) 6

I

I

0 N

0 •

•

0 0

• viria12 (Exp) o virial2 (Thory)

0 r ....

I . . . . I . . . . I . . . . .

.. .

0 100 200 300 400 500 Step Number N ( M =1000) Kuhn

600

Fig.15 The plot shows that the N-dependence of the second virial coefficient evaluated thoretically is consistent to that of the experimental data in Ref. [28]. 4.3. Scaling behavior of A2. Let us discuss the scaling behavior of the second virial coefficient. If we assume the formula (3.5) then we have

1

00(1 - P00'(r; N))47rr2dr = A(N)

Assuming the result in §3.6 we have (4.5) a(N) = (1 - BON-a)N"/a0 - Nµ (N -' oo) This gives that A2 - N-0.34 when N is very large.

120 TETSUO DEGUCHI AND KYOICHI TSURUSAKI ACKNOWLEDGEMENT

The authors would like to thank the organizers of KNOTS '96 for hospitality during the conference and encouragement on this work. One of the authors (T.D.) is grateful to I. Dazey, J. O'Hara, J. Simon, D.W. Sumners and K. Taniyama for helpful discussions on interesting and related topics such as knotted DNA, energy of knots and unknotting operation during the conference. REFERENCES 1. M. G. Brereton and S. Shah, J. Phys. A: Math. Gen. 13 (1980) 2751; B. Duplantier, Commun. Math . Phys. 82 (1981) 41. 2. M. Le Bret, Monte Carlo Computation of the Supercoiling Energy, the Sedimentation Constant, and the Radius of Gyration of Unknotted and knotted Circular DNA, Biopolymers, 19 (1980) 619-637. 3. Y.D. Chen, Monte Carlo study of freely jointed ring polymers. I. Generation of ring polymers by dimerization method, J. Chem. Phys. 74 (1981) 2034-2038; II. The writhing number, J. Chem. Phys. 75 (1981) 2447-2453; III. The generation of undistorted perfect ring polymers, J. Chem. Phys. 75 (1981 ) 5160-5163. 4. J. des Cloizeaux and M. L. Mehta , Topological constraints on polymer rings and critical indices, J. Phys. (Paris) 40 (1979) 665-670. 5. F.B. Dean, A.Stasiak, T. Koller and N.R. Cozzarelli, J. Biol. Chem . 260 (1985 ) 4795-4983; S.A. Wasserman, J.M. Duncan and N.R. Cozzarelli, Discovery of a Predicted DNA Knot Substantiates a Model for Site-Specific Recombination , Science 229 (1985 )171-174; Science 232 (1986 ) 951-960. 6. T. Deguchi and K. Tsurusaki, A New Algorithm for Numerical Calculation of Link Invariants, Phys. Lett. A 174 (1993) 29-37. 7. T. Deguchi and K. Tsurusaki, Topology of Closed Random Polygons, J. Phys. Soc. Jpn. 62 ( 1993 ) 1411-1414. 8. T. Deguchi and K. Tsurusaki, A Statitical Study of Random Knotting Using the Vassiliev Invariants , J. Knot Theory and Its Ramifications 3 (1994) 321-354. 9. T. Deguchi and K. Tsurusaki, A Universality of Random Knotting, preprint 1995. 10. T. Deguchi and K. Tsurusaki, Numerical Application of Quantum Invariants to Random Knotting, in Geometry and Physics, Lect. Notes in Pure and Applied Math . Series/ 184, the proceedings of the conference at Aarhus University, July 18 - 27, 1995, Aarhus, Denmark, ed. by J.E. Andersen, J. Dupont, H. Pedersen, and A. Swann, Basel Switzerland, Marcel Dekker Inc., 1997, pp. 557 - 561. 11. M. Delbriick, Knotting Problems in Biology, in Mathematical Problems in the Biological Sciences, ed . R.E. Bellman, Proc. Symp. Appl. Math. 14 (1962) 55-63. 12. Y. Diao, N. Pippenger and D.W. Sumners, On Random Knots, J. Knot Theory and Its Ramifications 3 (1994) 419-429. 13. S.F. Edwards, Statistical mechanics with topological constraints: II, J. Phys. Al (1968) 15-28. 14. H.L. Frisch and E. Wasserman, Chemical Topology, J. Amer. Chem. Soc. 83 (1961), 3789-3795. 15. K. Iwata and T. Kimura, Topological distribution functions and the second virial coefficients of ring polymers, J. Chem. Phys. 74 (1981 ) 2039-2048. 16. F. Jaeger, D.L. Vertigan, and D.J. A. Welsh, On the Computational Complexity of the Jones and Tutte Polynomials, Math. Proc. Camb . Phil. Soc. 108 (1990) 35. 17. K. Koniaris and M. Muthukumar , Knottedness in Ring Polymers, Phys. Rev. Lett. 66 (1991) 2211-2214.

RANDOM KNOTS AND LINKS 121 18. H.A. Lim and E.J. van Rensburg, A numerical simulation of electrophoresis of knotted DNA, preprint of Florida State Univ. FSU-SCRI-91-163, to appear in J. Modelling Sci. Comput., Oxford.

19. J.P.J. Michels and F.W. Wiegel, Probability of knots in a polymer ring, Phys. Lett. 90A (1982) 381-384. 20. J.P.J. Michels and F.W. Wiegel, On the topology of a polymer ring, Proc. R. Soc. Lond. A 403 269. 21. J. O'Hara, Energy of a knot, Topology, 30(1991 ) 241-247. 22. E. Orlandini, E.J. Janse van Rrensburg, M.C. Tesi and S.G. Whittington, Random Linking of Lattice Polygons, preprint (1993). 23. E. Orlandini, M.C. Tesi, E.J. Janse Van Rensburg, and S.G. Whittington, Entropic exponents of lattice polygons with specified knot type, Univ. of Oxford preprint OUTP-96-18S (1996). 24. N. Pippenger, Knots in Random Walks, Discrete Applied Math . 25 (1989 ) 273-278. 25. M. Polyak and 0. Viro, Gauss Diagram Formulas for Vassiliev Invariants, Int. Math. Res. Not. (1994) 445-453. 26. T.M. Przytycka and J.H. Przytycki, Subexponentially Computable Truncations of Jones-type Polynomial, in Graph Structure Theory , eds. N. Robertson and P. Seymour, Contemp. math. AMS 147 (1993) 63-108. 27. E. J. Janse van Rensburg and S. G. Whittington, The knot probability in lattice polygons, J. Phys. A: Math. Gen. 23 (1990) 3573-3590; The dimensions of knotted polygons, 24 (1991) 3935-3948. 28. J. R. Roovers and P. M. Toporowski, Synthesis of high molecular weight ring polystyrenes, Macromolecules 16 (1983) 843-849. 29. V.V. Rybenkov, N.R. Cozzarelli and A.V. Vologodskii, Probability of DNA knotting and the effective diameter of the DNA double helix, Proc. Natl. Acad. Sci. USA 90 (1993) 5307-5311. 30. S.Y. Shaw and J.C. Wang, Knotting of a DNA Chain During Ring Closure, Science 260 (1993) 533-536. 31. K. Shishido, N. Komiyama and S. Ikawa, Increased Production of a Knotted Form of Plasmid pBR322 DNA in Esherichia coli DNA Topoisomerase Mutants, J. Mol, Biol. 195 (1987) 215218. 32. J.K. Simon, Energy functions for polygonal knots, J. Knot Theory and Its Ramifications 3 (1994) 299-320. 33. J.K. Simon, Energy functions for knots: beginning to predict physical behavior, 34. C.E. Soteros, D.W. Sumners and S.G. Whittington, Entanglement complexity of graphs in Z3 Math. Proc. Camb. Phil. Soc. 111 (1992) 75-91. 35. D.W. Sumners, C. Ernst, S.J. Spengler, and N. R. Cozzarelli, Analysis of the mechanism of DNA recombination using tangles, Quarterly Rev. Biophys . 28 (1995 ) pp. 253-313. 36. D.W. Sumners, The topology of DNA I - III, talks in the Workshop of KNOTS '96; see also I. Dazey (with D.W. Sumners), A strand passage metric for topoisomerase action, talk in the Conference of KNOTS '96. 37. D.W. Sumners and S.G. Whittington, Knots in self-avoiding walks, J. Phy. A : Math. Gen. 21 (1988) 1689-1694. 38. F. Tanaka, Gauge Theory of Topological Entanglements I, II , Prog. Theor. Phys. 68 (1982) 148-163, 164-177. 39. F. Tanaka, Osmotic pressure of ring-polymer solutions, J. Chem. Phys. 87 (1987) 4201-4206; See also, K. Iwata, Macromolecules 18 (1985) 115. 40. M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini, and S.G. Whittington, Knot probability for lattice polygons in confined geometries, J. Phys. A: Math. Gen. 27 (1994) 347-360.

122 TETSUO DEGUCHI AND KYOICHI TSURUSAKI 41. M.C. Tesi, E.J. Janse Van Rensburg, E. Orlandini, D.W. Sumners and S.G. Whittington, Knotting and supercoiling in circular DNA: A model incorporating the effect of added salt, Phys. Rev. E 49 (1994) 868-872. 42. K. Tsurusaki, Statistical Study of Random Knotting, Thesis, University of Tokyo, 1995. 43. K. Tsurusaki and T. Deguchi, Fractions of Particular Knots in Gaussian Random Polygons, J. Phys. Soc. Jpn. 64 (1995) 1506-1518. 44. K.Tsurusaki and T. Deguchi, Numerical analysis on topological entanglements of random polygons, the Proceedings of the Satellite meeting of STATPHYS 19, Nankai Institute, Tianjin, China August 8-10, 1995, edited by M. L. Ge and F. Y. Wu, World Scientific, Singapore, 1996, pp. 320-329. 45. K. Tsurusaki, T. Deguchi and M. Wadati, A Topological Study of Random Walks, in Field Theory and Collective Phenomena: In Memory of Prof. H. Umezawa, ed. by S.De Lillo, P. Sodano, F.C. Khanna and G.W. Semenoff, World Scientific, Songapore, 1995, pp. 185-206. 46. A.V. Vologodskii, A.V. Lukashin, M.D. Frank-Kamenetskii , and V.V. Anshelevich, The knot probability in statistical mechanics of polymer chains, Sov. Phys. JETP 39 (1974) 1059-1063. 47. A. V. Vologodskii, A.V. Lukashin, and M.D. Frank-Kamenetskii, Topological interaction be'tween polymer chains, Sov. Phys. JETP 40 (1975) 932-936. 48. K.V. Klenin, A.V. Vologodskii, V.V. Anshelevich, A.M. Dykhne and M.D. Frank-Kamenetskii, Effect of Excluded Volume on Topological Properties of Circular DNA, J. Biomolecular Structure and Dynamics, 5 (1988) 1173-1185. 49. D.M. Walba, Topological stereochemistry, Tetrahedron 41 (1985) 3161-3212. 50. S.G. Whittington, Topology of polymers, in New Scientific Applications of Geometry and Topology, ed. D.W. Sumners, Amer. Math. Soc. PSAM 45 (1992) 73-95. DEPARTMENT OF PHYSICS, FACULTY OF SCIENCE, OCHANOMIZU UNIVERSITY, 2-1-1 OHTSUKA, BUNKYO-KU TOKYO 112, JAPAN

E-mail address: [email protected] MOLECULAR ENGINEERING DIVISION, KANAGAWA INDUSTRIAL TECHNOLOGY RESEARCH INSTITUTE, 705-1 SHIMOIZUMI , EBINA-CITY, KANAGAWA 243-04, JAPAN

E-mail address : [email protected]

Lectures at Knots 96 edited by Shin'ichi Suzuki @1997 World Scientific Publishing Co. pp. 123-194

KNOTS AND DIAGRAMS LOUIS H. KAUFFMAN

ABSTRACT. This paper is a version of the lectures given by the author in Tokyo in July 1996. The theme is knots and diagrams. The topics range from foundations of knot theory to topological quantum field theory.

1. INTRODUCTION

This paper is an exploration of the theme of knot diagrams. I have deliberately focused on basics in a number of interrelated domains. In most cases some fundamentals are done in a new or in a more concise way. Some parts of the paper are expository to fill in the context. This exposition is an outgrowth of the series of lectures that I gave in Tokyo at the Knots 96 conference in the summer of 1996. It gives me great pleasure to thank the organizers of the conference for the stimulating opportunity to deliver those lectures.

A word about proofs. Most proofs given in this paper are sketches, but the author has taken some care to manufacture sketches that the reader should be able to fill in to complete proofs. I hope that this is the case! The paper is divided into five sections. Section 2, on the Reidemeister moves, gives a proof of Reidemeister's basic theorem (that the three Reidemeister moves on diagrams generate ambient isotopy of links in three-space). A discussion on graph embeddings extends Reidemeister's theorem to graphs and proves the appropriate moves for topological and rigid vertices. We hope that this subsection fills in some gaps in the literature. Section 3 discusses Vassiliev invariants and invariants of rigid vertex graphs. This section is expository, with discussions of the four-term relations, Lie algebra weights, relationships with the Witten functional intergal, and combinatorial constructions for some Vassiliev invariants. The discussion raises some well-known problems about Vassiliev invariants. The section on the functional integral introduces a useful abstract tensor notation that helps in understanding how the Lie algebra weight systems are related to the functional integral. Sections 4 and 5 are based on a reformulation of the Reidemeister moves so that 123

124 LOUIS H. KAUFFMAN they work with diagrams arranged generically transverse to a special direction in the plane. We point out how the technique by which we proved Reidemeister's Theorem (it is actually Reidemeister's original technique) generalises to give these moves as well. The moves with respect to a vertical are intimately related to quantum link invariants and to Hopf algebras. Section 4 is a quick exposition of quantum link invariants , their relationship with Vassiliev invariants, classical Yang-Baxter equation and infinitesimal braiding relations. Again, this provides the context to raise many interesting questions. Section 5 is a very concise introduction to the work of the author, David Radford and Steve Sawin on invariants of three-manifolds from finite dimensional Hopf algeras. We touch on the question of the relationship of this work to the Kuperberg invariants. Section 6 is a discussion of the Temperley Lieb algebra. Here we give a neat proof of the relation structure in the Temperley Lieb monoid via piecewise linear diagrams. The last part of this section explains the relationship of the Temperley Lieb monoid to parenthesis structures and shows how this point of view can be used to relate parentheses to the pentagon and the Stasheff polyhedron. This last foray into diagrams gives a taste of joint work of the author with Scott Carter and Masahico Saito. Acknowledgement . It gives the author pleasure to thank the National Science Foundation for support of this research under NSF Grant DMS -2528707.

2. REIDEMEISTER MOVES

Reidemeister [57] discovered a simple set of moves on link diagrams that captures the concept of ambient isotopy of knots in three dimensional space. There are three basic Reidemeister moves. Reidemeister's theorem states that two diagrams represent ambient isotopic knots ' (or links) if and only if there is a sequence of Reidemeister moves taking one diagram to the other. The Reidemeister moves are illustrated in Figure 1.

Reidemeister's three moves are interpreted as performed on a larger diagram in which the small diagram shown is a literal part. Each move is performed without disturbing the rest of the diagram. Note that this means that each move occurs, up to topological deformation, just as it is shown in the diagrams in Figure 1. There are no extra lines in the local diagrams. For example, the equivalence (A) in Figure 2 is not an instance of a single first Reidemeister move. Taken literally,it factors into a move II followed by a move I. Diagrams are always subject to topological deformations in the plane that preserve the structure of the crossings. These deformations could be designated as "Move Zero". See Figure 2.

KNOTS AND DIAGRAMS

Figure 1 - Reidemeister Moves

FQ.-+e r;wt

Aove Zero

Figure 2 - Factorable Move, Move Zero

125

126 LOUIS H. KAUFFMAN A few exercises with the Reidemeister moves are in order . First of all, view the diagram in Figure 3. It is unknotted and you can have a good time finding a sequence of Reidemeister moves that will do the trick . Diagrams of this type are produced by tracing a a curve and always producing an undercrossing at each return crossing. This type of knot is called a standard unknot. Of course we see clearly that a standard unknot is unknotted by just pulling on it, since it has the same structure as a coil of rope that is wound down onto a flat surface.

Can one recognise unknots by simply looking for sequences of Reidemeister moves that undo them? This would be easy if it were not for the case that there are examples of unknots that require some moves that increase the number of crossings before they can be subsequently decreased. Such an demonic example is illustrated in Figure 4.

Figure 3 - Standard Unknot

Figure 4 - A Demon

KNOTS AND DIAGRAMS 127

It is generally not so easy to recognise unknots. However, here is a tip: Look for macro moves of the type shown in Figure 5. In a macro move, we identify an arc that passes entirely under some piece of the diagram (or entirely over) and shift this part of the are, keeping it under (or over) during the shift. Macro moves often allow a reduction in the number of crossings even though the number of crossings will increase during a sequence of Reidemeister moves that generates the macro move.

As shown in Figure 5, the macro-move includes asa special case both the second and the third Reidemeister moves, and it is not hard to verify that a macro move can be generated by a sequence of type II and type III Reidemeister moves. It is easy to see that the type I moves can be left to the end of any deformation. The demon of Figure 4 is easily demolished by macro moves, and from the point of view of macro moves the diagram never gets more complicated. Lets say that an knot can be reduced by a set of moves if it can be transformed by these moves to the unknotted circle diagram through diagrams that never have more crossings than the original diagram. Then we have shown that there are diagrams representing the unknot that cannot be reduced by the Reidemeister moves. On the other hand, unknotted diagrams can not always be reduced by the macro moves in conjunction with the first Reidemeister move. If this were true it would give a combinatorial way to recognise the unknot. (Probably, one needs to at least generalise the macro move to have the arc moving between layers of diagram.)

Figure 5 - Macro Move

128 LOUIS H. KAUFFMAN

2.1. Reidemeister's Theorem. We now indicate how Reidemeister proved his Theorem. An embedding of a knot or link in three dimensional space is said to be piecewise linear if it consists in a collection of straight line segments joined end to end. Reidemeister started with a single move in three dimensional space for piecewise linear knots and links . Consider a point in the complement of the link, and an edge in the link such that the surface of the triangle formed by the end points of that edge and the new point is not pierced by any other edge in the link. Then one can replace the given edge on the link by the other two edges of the triangle, obtaining a new link that is ambient isotopic to the original link . Conversely, one can remove two consecutive edges in the link and replace them by a new edge that goes directly from initial to final points , whenever the triangle spanned by the two consecutive edges is not pierced by any other edge of the link. This triangle replacement constitutes Reidemeister' s three dimensional move. See Figure 6. It can be shown that two piecewise linear knots or links are ambient isotopic in three dimensional space if and only if there is a sequence of Reidemeister triangle moves from one to the other. This will not be proved here. At the time when Reidemeister wrote his book, equivalence via three dimensional triangle moves was taken as the definition of topological equivalence of links.

Figure 6 - Triangle Move

KNOTS AND DIAGRAMS 129

It can also be shown that tame knots and links have piecewise linear representatives in their ambient isotopy class . It is sufficient for our purposes to work with piecewise linear knots and links. Reidemeister's planar moves then follow from an analysis of the shadows projected into the plane by Reidemeister triangle moves in space. Figure 7 gives a hint of this analysis. The result is a reformulation of the three dimensional problems of knot theory to a combinatorial game in the plane.

Figure 7 - Shadows

130 LOUIS H. KAUFFMAN

To go beyond the hint in Figure 7 to a complete proof that Reidemeister's planar moves suffice involves preliminary remarks about subdivision . The simplest subdivision that one wants to be able to perform on a piecewise linear link is the placement of a new vertex at an interior point of an edge - so that that edge becomes two edges in the subdivided link. Figure 8 shows how to accomplish this subdivision via triangle moves.

Figure 8 - Subdivision of an Edge Any triangle move can be factored into a sequence of smaller triangle moves corresponding to a simplicial subdivision of that triangle . This is obvious , since the triangles in the subdivision of the large triangle that is unpierced by the link are themselves unpierced by the link. To understand how the Reidetneister triangle move behaves on diagrams it is sufficient to consider a projection of the link in which the triangle is projected to a non- singular triangle in the plane. Of course , there may be many arcs of the

KNOTS AND DIAGRAMS 131

link also projected upon the interior of the projected triangle. However , by using subdivision , we can assume that the cases of the extra arcs are as shown in Figure 9. In Figure 9 we have also shown how each of these cases can be accomplished by (combinations of) the three Reidemeister moves. This proves that a projection of a single triangle move can be accomplished by a sequence of Reidemeister diagram moves.

Figure 9 - Projections of Triangle Moves

132 LOUIS H. KAUFFMAN

A piecewise linear isotopy consists in a finite sequence of triangle moves. There exists a direction in three dimensional space that makes a non-zero angle with each of theses triangles and is in general position with the link diagram. Projecting to the plane along this direction makes it possible to perform the entire ambient isotopy in the language of projected triangle moves. Now apply the results of the previous paragraph and we conclude R.eidemeister 's Theorem. If two links are piecewise linearly equivalent (ambient isotopic), then there is a sequence of Reidemeister diagram moves taking a projection of one link to a projection of the other. Note that the proof tells us that the two diagrams can be obtained from one spatial projection direction for the entire spatial isotopy. It is obvious that diagrams related by Reidemeister moves represent ambient isotopic links. Reidemeister's Theorem gives a complete combinatorial description of the topology of knots and links in three dimensional space.

2.2. Graph Embeddings. Let G be a (multi-)graph. That is, G is a finite abstract graph with, possibly, a multiplicity of edges between any two of its vertices. Now consider the embeddings of G in Euclidean three space R3. In the category of topological embeddings, any edge of G can acquire local knotting as shown in Figure 10. On top of this there is the possibility of global knotting that results from the structure of the graph as a whole.

Figure 10 - Graph Embedding

KNOTS AND DIAGRAMS 133

Topological or piecewise linear ambient isotopy of graph embeddings is complicated by the fact that arbitrary braiding can be created or destroyed at a vertex, as illustrated in Figure 11.

TOP

Figure 11 - Braiding at a Vertex For this reason , it is useful to consider ways to restrict the allowed movement in the neighborhood of a vertex. One way to accomplish this is to decree that each vertex will come equipped with a specific cyclic order of the edges meeting the vertex. This cyclic order can be instantiated on the boundary of a disk , and the graph replaced by a configuration of disks with cyclic orders of marked points along their boundaries . The edges of the original graph are replaced by edges that go from one disk to another terminating in the marked points. Call such an arrangement a

134 LOUIS H. KAUFFMAN

rigid vertex graph G. If G is a rigid vertex graph, then we consider embeddings of G where the disks are embedded metrically while the (graphical) edges are embedded topologically. A rigid vertex isotopy of one RV (RV will stand for rigid vertex.) embedding G to another G' is a combination of ambient isotopies of the embedded edges of the graph (the strings of the graph) relative to their endpoints on the disks, coupled with affine motions of the disks (carrying along the strings in ambient isotopy). An affine motion of a disk is a combination of parallel translations of the disk along a given direction in three-space and rotations of the disk about an axis through its center. We can think of a given disk as embedded inside a standard three ball with the strings from the disk emanating straight to the boundary of the three ball . Each basic affine motion is assumed to leave the points on the boundary of the containing three -ball fixed . Thus the types of affine motion are as illustrated in Figure 12.

Ro+M.+el

Figure 12 - Rigid Vertex Graphs and Affine Motions

KNOTS AND DIAGRAMS 135

We will give versions of the Reidemeister moves for both topological isotopy and rigid vertex isotopy of embedded graphs. In the topological case the extra moves are illustrated in Figure 13. Here we have indicated the elementary braiding at a vertex and slide moves that take an edge underneath a vertex . The proof that these moves suffice is a generalisaton of our original proof of the Reidemeister moves. That is, we model the graph embeddings by piecewise linear embeddings . This may entail subdividing the edges of the original graph so that those edges can have enough flexibility to sustain a given topological conformation. Thus, when we speak of a piecewise linear embedding of a given graph , we mean a piecewise linear embedding of a graph that is obtained from the given graph by subdividing some of its edges. Piecewise ambient isotopy of graph embeddings is defined exactly as in the case of piecewise linear isotopy for knots and links. The same projection arguments apply and the extra moves are obtained from the three dimensional triangle move as illustrated in Figure 14. This completes the proof of our assertion about the topological Reidemeister moves for graphs.

-roe

Figure 13 - Extra Moves For Topological Isotopy of Graphs

136 LOUIS H. KAUFFMAN

Figure 14 - PL Isotopy Inducing Topological Graphical Moves Consider rigid vertex isotopy of rigid vertex ( RV) graphs . We will assume that the topological moves are performed in the piecewise linear setting. Thus subdivisions of the edges of the graph can be produced . Basic translational affine moves of the embedded disks can have piecewise linear starting and ending states by drawing straight lines from the marked points on the disk boundaries to the corresponding points in the containing balls. Rotatory moves with the center of a disk as axis can also have piecewise linear starting and ending states by taking the braiding that is induced by the rotation and suitably subdividing it. These remarks show that RV isotopy can be held in the PL category. The next point to consider is the result of projection of an RV isotopy on the corresponding diagrams . A sequence of elementary RV isotopies from a graph G to a graph G' has associated with it a direction of projection so that each PL triangle

KNOTS AND DIAGRAMS 137

move has its triangle projected to a non-singular triangle in the plane and each affine move has its disk projected to a non-singular disk in the plane. In the case of the affine moves we can assume that the before to after appearance of the disk and its corresponding containing ball will represent either a topological identity map (albeit an affine shift) or a rotation about the disk axis by it radians. (Higher multiples of it can be regarded as iterates of a it rotation.) Therefore the basic it rotation can be schematized as shown in Figure 15. Figure 15 illustrates the moves that we need to add to the Reidemeister moves to obtain a planar diagram version of RV isotopy. The remaining moves in Figure 15 follow from the same projection arguments that we have used earlier in this section. This completes the construction of the diagrammatic calculus for RV isotopy. Note that the generating moves for rigid vertex graph isotopy are almost the same as the generating moves for topological graph isotopy, except that the braiding at the vertex in the rigid vertex case comes from the twisting the disk as a whole. This circumstance makes the construction of invariants of rigid vertex graphs much easier. We will discuss constructions of such invariants in the next section. In section 4 we will return to the Reidemeister moves and reformulate them once again for the sake of quantum link invariants.

RV

Figure 15 - Diagrammatic Rigid Vertex Isotopy

138 LOUIS H. KAUFFMAN 3. VASSILIEV INVARIANTS AND INVARIANTS OF RIGID VERTEX GRAPHS

If V(K) is a (Laurent polynomial valued, or more generally - commutative ring valued) invariant of knots, then it can be naturally extended to an invariant of rigid vertex graphs by defining the invariant of graphs in terms of the knot invariant via an "unfolding" of the vertex. That is, we can regard the vertex as a "black box" and replace it by any tangle of our choice. Rigid vertex motions of the graph preserve the contents of the black box, and hence implicate ambient isotopies of the link obtained by replacing the black box by its contents. Invariants of knots and links that are evaluated on these replacements are then automatically rigid vertex invariants of the corresponding graphs. If we set up a collection of multiple replacements at the vertices with standard conventions for the insertions of the tangles, then a summation over all possible replacements can lead to a graph invariant with new coefficients corresponding to the different replacements. In this way each invariant of knots and links implicates a large collection of graph invariants. See [32], [33].

v,4 =a, V+bVA C\/ a, =I,6=-1, c-o

V YiCss;Jiev

c -ceyen.ce, Fo'WWII 4,

V-1 - vT^ V c V0

=o

= 0 (de fy,)

=vim, - v^^ vac

Figure 16 - Graphical Vertex Formulas

KNOTS AND DIAGRAMS 139

The simplest tangle replacements for a 4-valent vertex are the two crossings, positive and negative, and the oriented smoothing. Let V(K) be any invariant of knots and links. Extend V to the category of rigid vertex embeddings of 4-valent graphs by the formula (See Figure 16) V(K.) = aV(K+) + bV(K_) + cV(Ko) Here K. indicates an embedding with a transversal 4-valent vertex. This formula means that we define V (G) for an embedded 4-valent graph G by taking thesum V(G) = Eat+(s)b-cs>c0(s)V(S) S with the summation over all knots and links S obtained from G by replacing a node of G with either a crossing of positive or negative type, or with a smoothing (denoted 0). It is not hard to see that if V(K) is an ambient isotopy invariant of knots, then, this extension is an rigid vertex isotopy invariant of graphs . In rigid vertex isotopy the cyclic order at the vertex is preserved, so that the vertex behaves like a rigid disk with flexible strings attached to it at specific points. See the previous section. There is a rich class of graph invariants that can be studied in this manner. The Vassiliev Invariants [69],[7],[4] constitute the important special case of these graph invariants where a = +1, b = -1 and c = 0. Thus V(G) is a Vassiliev invariant if V(K.) = V(K+) - V(K_).

Call this formula the exchange identity for the Vassiliev invariant V. V is said to be of finite type k if V(G) = 0 whenever IGI > k where IGI denotes the number of 4valent nodes in the graph G. The notion of finite type is of extraordinary significance in studying these invariants. One reason for this is the following basic Lemma. Lemma. If a graph G has exactly k nodes, then the value of a Vassiliev invariant vk of type k on G, vk(G), is independent of the embedding of G. Proof. The different embeddings of G can be represented by link diagrams with some of the 4-valent vertices in the diagram corresponding to the nodes of G. It suffices to show that the value of vk (G) is unchanged under switching of a crossing. However, the exchange identity for vk shows that this difference is equal to the evaluation of vk on a graph with k + 1 nodes and hence is equal to zero. This completes the proof.//

The upshot of this Lemma is that Vassiliev invariants of type k are intimately involved with certain abstract evaluations of graphs with k nodes . In fact, there are restrictions (the four-term relations) on these evaluations demanded by the topology (we shall articulate these restrictions shortly) and it follows from results of Kontsevich [4] that such abstract evaluations actually determine the invariants. The

140 LOUIS H. KAUFFMAN

invariants derived from classical Lie algebras are all built from Vassiliev invariants of finite type. All this is directly related to Witten's functional integral [72]. Definition. Let vk be a Vassiliev invariant of type k. The top row of Vk is the set of values that vk assigns to the set of (abstract) 4-valent graphs with k nodes. If we concentrate on Vassiliev invariants of knots, then these graphs are all obtained by marking 2k points on a circle, and choosing a pairing of the 2k points. The pairing can be indicated by drawing a circle and connecting the paired points with arcs. Such a diagram is called a chord diagram. Some examples are indicated in Figure 17.



3

Figure 17 - Chord Diagrams Note that a top row diagram cannot contain any isolated pairings since this would correspond to a difference of local curls on the corresponding knot diagram (and these curls, being isotopic, yield the same Vassiliev invariants. The Four-Term Relation . (Compare [64].) Consider a single embedded graphical node in relation to another embedded arc, as illustrated in Figure 18. The arc underlies the lines incident to the node at four points and can be slid out and isotoped over the top so thatit overlies the four nodes. One can also switch the crossings one-by-one to exchange the arc until it overlies the node. Each of these four switchings gives rise to an equation, and the left-hand sides of these equations will add up to zero, producing a relation corresponding to the right-hand sides. Each term in the right-hand side refers to the value of the Vassiliev invariant on a graph with two nodes that are neighbors to each other. See Figure 18.

KNOTS AND DIAGRAMS 141

=X > A- 6- c+D=^.

Figure 18 - The Four Term Relation There is a corresponding 4-term relation for chord diagrams. This is the 4-term relation for the top row . In chord diagrams the relation takes the form shown at the bottom of Figure 18. Here we have illustrated only those parts of the chord diagram that are relevant to the two nodes in question (indicated by two pairs of points on the circle of the chord diagram ). The form of the relation shows the points on the chord diagram that are immediate neighbors . These are actually neighbors on any chord diagram that realizes this form . Otherwise there can be many other pairings present in the situation.

142 LOUIS H. KAUFFMAN

As an example, consider the possible chord diagrams for a Vassiliev invariant of type 3. There are two possible diagrams as shown in Figure 19. One of these has the projected pattern of the trefoil knot and we shall call it the trefoil graph. These diagrams satisfy the 4-term relation. This shows that one diagram must have twice the evaluation of the other. Hence it suffices to know the evaluation of one of these two diagrams to know the top row of a Vassiliev invariant of type 3. We can take this generator to be the trefoil graph.

Figure 19 - Four Term Relation For Type Three Invariant Now one more exercise: Consider any Vassiliev invariant v and let 's determine its value on the the trefoil graph as in Figure 20.

KNOTS AND DIAGRAMS 143

4j-,^^)D+

VOC-D

_ - it -- -f- #J-7- -'Figure 20 - Trefoil Graph The value of this invariant on the trefoil graph is equal to the difference between its values on the trefoil knot and its mirror image. Therefore any Vassiliev invariant that assigns a non-zero value to the trefoil graph can tell the difference between the trefoil knot and its mirror image. Example. This example shows how the original Jones polynomial is composed of Vassiliev invariants of finite type . Let VK(t) denote the original Jones polynomial

144 LOUIS H. KAUFFMAN

[22]. Recall the oriented state expansion for the Jones polynomial [34] with the basic formulas (6 is the loop value.) VK+ = -t1/2 VKo -

VK_

tVK,O

= -t-1/2VK0 - t-1VK-. 6 = -(tl/2 + t-1/2).

Let t = ex. Then VK+ = -exl2VKo - eXVKOO x/2

x

VK_ _ -e VKo - e VK^. b = -(ex'2 + e-x'2).

Thus VK. = VK+ - VK_ = -2sinh (x/2)VKO - 2sinh (x)VK_.

Thus x divides VK., and therefore xk divides VG whenever G is a graph with at least k nodes . Letting VG(ex) _ >Vk(G)xk, k-0

we see that this condition implies that vk (G) vanishes whenever G has more than k nodes. Hence the coefficients of the powers of x in the expansion of VK (ex) are Vassiliev invariants of finite type! This result was first observed by Birman and Lin [7] by a different argument.

Let's look a little deeper and see the structure of the top row for the Vassiliev invariants related to the Jones polynomial. By our previous remarks the top row evaluations correspond to the leading terms in the power series expansion. Since S = -(ex/2 + e-x/2) = -2 + [higher], /2 -ex/2 + e-x = -x + [higher], -ex + e-x = -2x + [higher],

it follows that the top rows for the Jones polynomial are computed by the recursion formulas v(K„) = -v(Ko) - 2V(KK) v([loop]) = -2. The reader can easily check that this recursion formula for the top rows of the Jones polynomial implies that v3 takes the value 24 on the trefoil graph and hence it is the Vassiliev invariant of type 3 in the Jones polynomial that first detects the difference between the trefoil knot and its mirror image. This example gives a good picture of the general phenomenon of how the Vassiliev invariants become building blocks for other invariants. In the case of the Jones

KNOTS AND DIAGRAMS 145

polynomial, we already know how to construct the invariant and so it is possible to get a lot of information about these particular Vassiliev invariants by looking directly at the Jones polynomial. This, in turn, gives insight into the structure of the Jones polynomial itself. 3.1. Lie Algebra Weights. Consider the diagrammatic relation shown in Figure 21 . Call it ( after Bar-Natan [4]) the STU relation.

11

Figure 21 - The STU Relation Lemma.

STU implies the 4- term relation.

Proof. View Figure 22. STU is the smile of the Cheshire cat. That smile generalizes of the idea of a Lie algebra. Take a Lie algebra with generators T. Then T°7-6 - TbTa = i fabcT` expresses the closure of the Lie algebra under commutators . Translate this equation into diagrams as shown in Figure 23, and see that this translation is STU with Lie algebraic clothing!

146

LOUIS H. KAUFFMAN

Figure 22 - A Diagrammatic Proof

06 J ap b

II 6

Figure 23 - Algebraic Clothing

KNOTS AND DIAGRAMS 147

Here the structure tensor of the Lie algebra has been assumed (for simplicity) to be invariant under cyclic permutation of the indices. This invariance means that our last Lemma applies to this Lie algebraic interpretation of STU. The upshot is that we can manufacture weight systems for graphs that satisfy the 4-term relation by replacing paired points on the chord diagram by an insertion of T° in one point of the pair and a corresponding insertion of T° at the other point in the pair and summing over all a. The result of all such insertions on a given chord diagram is a big sum of specific matrix products along the circle of the diagram, each of which (being a circular product) is interpreted as a trace. Lets say this last matter more precisely: Regard a graph with k nodes as obtained by identifying k pairs of points on a circle. Thus a code such as 1212 taken in cyclic order specifies such a graph by regarding the points 1,2,1,2 as arrayed along a circle with the first and second 1 's and 2's identified to form the graph . Define,for a code ala2 ... am wt(ala2...am ) = trace (T°1T°1T°3...T°-)

where the Einstein summation convention is in place for the double appearances of indices on the right -hand side. This gives the weight system. The weight system described by the above procedure satisfies the 4-term relation, but does not necessarily satisfy the vanishing condition for isolated pairings. This is because the framing compensation for converting an invariant of regular isotopy to ambient isotopy has not yet been introduced . We will show how to do this in the course of the discussion in the next paragraph. The main point to make here is that by starting with the idea of extending an invariant of knots to a Vassiliev invariant of embedded graphs and searching out the conditions on graph evaluation demanded by the topology, we have inevitably entered the domain of relations between Lie algebras and link invariants. Since the STU does not demand Lie algebras for its satisfaction we see that the landscape is wider than the Lie algebra context , but it is not yet undertood how big is the class of link invariants derived from Lie algebras.

In fact , we can line up this weight system with the formalism related to the knot diagram by writing the Lie algebra insertions back on the 4-valent graph . We then get a Casimir insertion at the node. See Figure 24. To get the framing compensation , note that an isolated pairing corresponds to the trace of the Casimir . Let y denote this trace. See Figure 24.

y = tr(>T°T°) a

148 LOUIS H. KAUFFMAN

Y

= A ( ^^T ";r' 7-"Z,) CC.

Figure 24 - Weight System and Casimir Insertion

Let D be the trace of the identity . Then it is easy to see that we must compensate the given weight system by subtracting (-y/D) multiplied by the result of dropping the identification of the two given points . We can diagram this by drawing two crossed arcs without a node drawn to bind them. Then the modified recursion formula becomes as shown in Figure 25. For example , in the case of SU(N) we have D = N, -y = (N2 - 1)/2 so that we get the transformation shown in Figure 25, including the use of the Fierze identity. For N = 2 the final formula of Figure 25 is,up to a multiple, exactly the top row formula that we deduced for the Jones polynomial from its combinatorial structure.

KNOTS AND DIAGRAMS 149

uX N (F; eirz I d Q V4

;a-y)

Figure 25 - Modified Recursion Formula 3.2. Vassiliev Invariants and Witten's Functional Integral. In [72) Edward Witten proposed a formulation of a class of 3-manifold invariants as generalized Feynman integrals taking the form Z(M) where Z(M) = f dAexp[(ik/47r )S(M, A)1.

150 LOUIS H. KAUFFMAN

Here M denotes a 3-manifold without boundary and A is a gauge field (also called a gauge potential or gauge connection ) defined on M. The gauge field is a one-form on a trivial G-bundle over M with values in a representation of the Lie algebra of G. The group G corresponding to this Lie algebra is said to be the gauge group. In this integral the "action" S(M, A) is taken to be the integral over M of the trace of the Chem-Simons three-form CS = AdA + ( 2/3)AAA . (The product is the wedge product of differential forms.) Z(M) integrates over all gauge fields modulo gauge equivalence (See [2] for a discussion of the definition and meaning of gauge equivalence.) The formalism and internal logic of Witten 's integral supports the existence of a large class of topological invariants of 3-manifolds and associated invariants of knots and links in these manifolds. The invariants associated with this integral have been given rigorous combinatorial descriptions [59],[68],[43],[50], [70],[40], but questions and conjectures arising from the integral formulation are still outstanding. (See for example [3 ], [16],[18], [21], [60].) Specific conjectures about this integral take the form of just how it implicates invariants of links and 3-manifolds, and how these invariants behave in certain limits of the coupling constant k in the integral. Many conjectures of this sort can be verified through the combinatorial models. On the other hand, the really outstanding conjecture about the integral is that it exists ! At the present time there is no measure theory or generalization of measure theory that supports it. It is a fascinating exercise to take the speculation seriously, suppose that it does really work like an integral and explore the formal consequences . Here is a formal structure of great beauty. It is also a structure whose consequences can be verified by a remarkable variety of alternative means. Perhaps in the course of the exploration there will appear a hint of the true nature of this form of integration. We now look at the formalism of the Witten integral in more detail and see how it implicates invariants of knots and links corresponding to each classical Lie algebra. In order to accomplish this task, we need to introduce the Wilson loop. The Wilson loop is an exponentiated version of integrating the gauge field along a loop K in three space that we take to be an embedding (knot) or a curve with transversal selfintersections . For this discussion , the Wilson loop will be denoted by the notation WK(A) =< K]A > to denote the dependence on the loop K and the field A. It is usually indicated by the symbolism tr(Pexp( fK A)). Thus WK(A) =< KjA >= tr(Pexp(JK A)). Here the P denotes path ordered integration - we are integrating and exponentiating matrix valued functions, and so must keep track of the order of the operations. The symbol tr denotes the trace of the resulting matrix.

KNOTS AND DIAGRAMS 151

With the help of the Wilson loop functional on knots and links, Witten writes down a functional integral for link invariants in a 3-manifold M: Z(M, K) = f dAexp[ (ik/47r) S(M, A)]tr (Pexp(f A)) = f dAexp[(ik/4ir)S] < KJA > . Here S(M, A) is the Chem-Simons Lagrangian , as in the previous discussion. We abbreviate S(M, A) as S and write < KJA > for the Wilson loop. Unless otherwise mentioned , the manifold M will be the three-dimensional sphere S3 An analysis of the formalism of this functional integral reveals quite a bit about its role in knot theory. This analysis depends upon key facts relating the curvature of the gauge field to both the Wilson loop and the Chern-Simons Lagrangian. The idea for using the curvature in this way is due to Lee Smolin [63] (See also [55]). To this end, let us recall the local coordinate structure of the gauge field A(x), where x is a point in three-space. We can write A(x) = Aa(x) Tadxk where the index a ranges from 1 tom with the Lie algebra basis IT', Tz, T3, ..., T' I. The index k goes from 1 to 3. For each choice of a and k, Aa(x) is a smooth function defined on three-space. In A(x) we sum over the values of repeated indices. The Lie algebra generators Ta are matrices corresponding to a given representation of the Lie algebra of the gauge group G. We assume some properties of these matrices as follows:

1. [Ta, Tb] = i fabcTC where [x, y] = xy - yx , and fabc (the matrix of structure constants) is totally antisymmetric . There is summation over repeated indices. 2. tr(TaTb) = Bab /2 where bab is the Kronecker delta (6ab = 1 if a = b and zero otherwise). We also assume some facts about curvature. (The reader may enjoy comparing with the exposition in [34]. But note the difference of conventions on the use of i in the Wilson loops and curvature definitions.) The first fact is the relation of Wilson loops and curvature for small loops: Fact 1. The result of evaluating a Wilson loop about a very small planar circle around a point x is proportional to the area enclosed by this circle times the corresponding value of the curvature tensor of the gauge field evaluated at x. The curvature tensor is written Fae(x)Tadxrdy3. It is the local coordinate expression of AdA + AA. Application of Fact 1. Consider a given Wilson line < KIS >. Ask how its value will change if it is deformed infinitesimally in the neighborhood of a point x on the line. Approximate the change according to Fact 1, and regard the point x as

152 LOUIS H. KAUFFMAN

the place of curvature evaluation . Let 6 < KJA > denote the change in the value of the line. 6 < KJA > is given by the formula 6 < KIA >= dxrdx,Fae(x)Ta < KJA > . This is the first order approximation to the change in the Wilson line. In this formula it is understood that the Lie algebra matrices Ta are to be inserted into the Wilson line at the point x, and that we are summing over repeated indices. This means that each Ta < K I A > is a new Wilson line obtained from the original line < KIA > by leaving the form of the loop unchanged , but inserting the matrix Ta into that loop at the point x. A Lie algebra generator is diagrammed by a little box with a single index line and two input/output lines which correspond to its role as a matrix (hence as mappings of a vector space to itself ). See Figure 26.

\/- a V <4-> VK

=--T %EK

Figure 26 - Wilson Loop Insertion

^KA)

KNOTS AND DIAGRAMS 153

Remark. In thinking about the Wilson line < KIA >= tr(Pexp(fK A)), it is helpful to recall Euler 's formula for the exponential: e' = lim (1 + x/n)". n-oo

The Wilson line is the limit, over partitions of the loop K, of products of the matrices (1 + A(x)) where x runs over the partition. Thus we can write symbolically, < KIA >= 11 (1 + A(x)) = II (1 + Aa(x)Tadxk). xEK xEK

It is understood that a product of matrices around a closed loop connotes the trace of the product. The ordering is forced by the one dimensional nature of the loop. Insertion of a given matrix into this product at a point on the loop is then a well-defined concept. If T is a given matrix then it is understood that T < KJA > denotes the insertion of T into some point of the loop. In the case above, it is understood from context in the formula

dxrdxaFae(x)Ta < KJA > that the insertion is to be performed at the point x indicated in the argument of the curvature. Remark. The previous remark implies the following formula for the variation of the Wilson loop with respect to the gauge field: S < KIA > /S(Aa(x)) = dxkTa < KIA > . Varying the Wilson loop with respect to the gauge field results in the insertion of an infinitesimal Lie algebra element into the loop.

Proof.

b < KIA > /S(Aa(x)) = b (1 + Aa(y)Tadyk)/S(Aa(x)) yEK

II

(1+A a( y)Tadyk )[Tadxk]

jj

(1 +A" ( Y)T ady.)

y<xEK y>xEK

= dxkTa < KJA > .

Fact 2 . The variation of the Chern -Simons Lagrangian S with respect to the gauge potential at a given point in three-space is related to the values of the curvature tensor at that point by the following formula: he(x)

= Erst5S

/6(Aa(x))•

154 LOUIS H. KAUFFMAN

Here eab, is the epsilon symbol for three indices, i.e. it is +1 for positive permutations of 123 and -1 for negative permutations of 123 and zero if any two indices are repeated. With these facts at hand we are prepared to determine how the Witten integral behaves under a small deformation of the loop K. In accord with the theme of this paper , we shall use a system of abstract tensor diagrams to look at the differential algebra related to the functional integral. The translation to diagrams is accomplished with the aid of Figure 27 and Figure 28.

: Cu,rva.+wre, +eNl.So1^

Z : Gtern,- J ; ^nons Lq. ^^avi^ i a,vl^

A NK ,

T Figure 27 - Notation

KNOTS AND DIAGRAMS

155

Ae

i

Cas i nK;V Vo lume *Of!

n sser + ioL

Figure 28 - Derivation In Figure 27 we give diagrammatic equivalents for the component parts of our machinery. Tensors become labelled boxes. Indices become lines emanating from the boxes. Repeated indices that we intend to sum over become lines from one box to another. (The eye can immediately apprehend the repeated indices and the tensors where they are repeated.) Note that we use a capital D with lines extending from the top and the bottom for the partial derivative with respect to the gauge field, a capital W with a link diagrammatic subscript for the the Wilson loop, a cubic vertex

156 LOUIS H. KAUFFMAN

for the three index epsilon, little triangles with emanating arcs for the differentials of the space variables. The Lie algebra generators are little boxes with single index lines and two input/output lines which correspond to their roles as matrices (hence as mappings of a vector space to itself). The Lie algebra generators are, in all cases of our calculation, inserted into the Wilson line either through the curvature tensor or through insertions related to differentiating the Wilson line. In Figure 28 we give the diagrammatic calculation of the change of the functional integral corresponding to a tiny change in the Wilson loop. The result is a double insertion of Lie Algebra generators into the line, coupled with the presence of a volume form that will vanish if the deformation does not twist in three independent directions. This shows that the functional integral is formally invariant under regular isotopy since the regular isotopy moves are changes in the Wilson line that happen entirely in a plane. One does not expect the integral to be invariant under a Reidemeister move of type one, and it is not. This framing compensation can be determined by the methods that we are discussing [41], but we will not go into the details of those calculations here. In Figure 29 we show the application of the calculation in Figure 28 to the case of switching a crossing. The same formula applies, with a different interpretation, to the case where x is a double point of transversal self intersection of a loop K, and the deformation consists in shifting one of the crossing segments perpendicularly to the plane of intersection so that the self-intersection point disappears. In this case, one T° is inserted into each of the transversal crossing segments so that TaTa < KIA > denotes a Wilson loop with a self intersection at x and insertions of T° at x + el and x + e2 where el and e2 denote small displacements along the two arcs of K that intersect at x. In this case, the volume form is nonzero, with two directions coming from the plane of movement of one arc, and the perpendicular direction is the direction of the other arc. The reason for the insertion into the two lines is a direct consequence of the calculational form of Figure 28: First insertion is in the moving line, due to curvature. The second insertion is the consequence of differentiating the self-touching Wilson line. Since this line can be regarded as a product, the differentiation occurs twice at the point of intersection, and it is the second direction that produces the non-vanishing volume form.

Up to the choice of our conventions for constants, the switching formula is, as shown in Figure 29, Z(K+) - Z(K_) = (47ri/k) f dAexp[(ik/47r)s]TaTa < K„ I A > = (4vri /k)Z(T°TaK..).

KNOTS AND DIAGRAMS 157

The key point is to notice that the Lie algebra insertion for this difference is exactly what we did to make the weight systems for Vassiliev invariants (without the framing compensation). Thus the formalism of the Witten functional integral takes us directly to these weight systems in the case of the classical Lie algebras. The functional integral is central to the structure of the Vassiliev invariants.

nser41iorl.->Ytart-zero Volctwte Solt vY /

Figure 29 - Crossing Switch

158 LOUIS H. KAUFFMAN

3.3. Combinatorial Constructions for Vassiliev Invariants. Perhaps the most remarkable thing about this story of the structure of the Vassiliev invariants is the way that Lie algebras are so naturally implicated in the structure of the weight systems. This shows the remarkably close nature of the combinatorial structure of Lie algebras and the combinatorics of knots and links via the Reidemeister moves. A really complete story about the Vassiliev invariants at this combinatorial level would produce their existence on the basis of the weight systems with entirely elementary arguments. As we have already mentioned, one can prove that a given set of weights for the top row, satisfying the abstract four-term relation does imply that there exists a Vassiliev invariant of finite type n realizing these weights for graphs with n nodes. Proofs of this result either use analysis [4], [1] or non-trivial algebra [9], [4]. There is no known elementary combinatorial proof of the existence of Vassiliev invariants for given top rows. Of course quantum link invariants (See section 4 of these lectures.) do give combinatorial constructions for large classes of link invariants. These constructions rest on solutions to the Yang-Baxter equations, and it is not known how to describe the subset of finite type Vassiliev invariants that are so produced. It is certainly helpful to look at the structure of Vassiliev invariants that arise from already-defined knot invariants. If V(K) is an already defined invariant of knots (and possibly links), then its extension to a Vassiliev invariant is calculated on embedded graphs G by expanding each graphical vertex into a difference by resolving the vertex into a positive crossing and a negative crossing. If we know that V(K) is of finite type n and G has n nodes then we can take any embedding of G that is convenient, and calculate V(G) in terms of all the knots that arise in resolving the nodes of this chosen embedding. This is a finite collection of knots. Since there is a finite collection of 4-valent graphs with n nodes, it follows that the top row evaluation for the invariant V(K) is determined by the values of V(K) on a finite collection of knots. Instead of asking for the values of the Vassiliev invariant on a top row, we can ask for this set of knots and the values of the invariant on this set of knots. A minimal set of knots that can be used to generate a given Vassiliev invariant will be called a knots basis for the invariant. Thus we have shown that the set consisting of the unknot, the right-handed trefoil and the left handed-trefoil is a knots basis for a Vassiliev invariant of type 3. See [51] for more information about this point of view.

A tantalizing combinatorial approach to Vassiliev invariants is due to Michael Polyak and Oleg Viro [54]. They give explicit formulas for the second, third and fourth Vassiliev invariants and conjecture that their method will work for Vassiliev invariants of all orders. The method is as follows.

KNOTS AND DIAGRAMS 159

First one makes a new representation for oriented knots by taking Gauss diagrams. A Gauss diagram is a diagrammatic representation of the classical Gauss code of the knot. The Gauss code is obtained from the oriented knot diagram by first labelling each crossing with a naming label (such as 1,2 ,...) and also indicating the crossing type (+ 1 or -1 ). Then choose a basepoint on the knot diagram and begin walking along the diagram, recording the name of the crossings encountered, their sign and whether the walk takes you over or under that crossing . For example, if you go under crossing 1 whose sign is + then you will record o + 1. Thus the Gauss code of the positive trefoil diagram is ( ol+) (u2+ ) (o3+) (ul +) (o2+) (u3+). for prime knots the Gauss code is sufficient information to reconstruct the knot diagram . See [35] for a sketch of the proof of this result and for other references. To form a Gauss diagram from a Gauss code, take an oriented circle with a basepoint chosen on the circle . Walk along the circle marking it with the labels for the crossings in the order of the Gauss code. Now draw chords between the points on the circle that have the same label . Orient each chord from overcrossing site to undercrossing site. Mark each chord with + 1 or -1 according to the sign of the corresponding crossing in the Gauss code . The resulting labelled and basepointed graph is the Gauss diagram for the knot . See Figure 30 for examples. The Gauss diagram is deliberately formulated to have the structure of a chord diagram (as we have discussed for the weight systems for Vassiliev invariants). If G(K) is the Gauss diagram for a knot K, and D is an oriented (i.e. the chords as well as the circle in the diagram are oriented ) chord diagram , let IG(K) l denote the number of chords in G(K) and IDS denote the number of chords in D. If IDI < JG(K)) then we may consider oriented embeddings of D in G(K). For a given embedding i : D -> G(K) define

< i(D)IG(K) >= sign(i) where sign (i) denotes the product of the signs of the chords in G(K) fl i(D). Now suppose that C is a collection of oriented chord diagrams , each with n chords, and that

eval:C -->R is an evaluation mapping on these diagrams that satisfies the four -term relation at level n. Then we can define < DIK >_ < i(D)IG(K) > i:D-.G(K)

and v(K) < DIK > eval(D). DEC

160 LOUIS H. KAUFFMAN

3

I (c9''...+) (GC 2. +) (e'3 +-) (V- 1-+)

( e'a+) ^ Ck 3 -1-)

(U-I--r) (e" i--) (ct 3 ) Figure 30 - Gauss Diagrams For appropriate oriented chord subsets this definition can produce Vassiliev invariants v(K) of type n. For example, in the case of the Vassiliev invariant of type three taking value 0 on the unknot and value 1 on the right-handed trefoil, -1 on the left-handed trefoil, Polyak and Viro give the specific formula v3(K) =< ASK > +( 1/2) < BIK > where A denotes the trefoil chord diagram as we described it in section 3 and B

KNOTS AND DIAGRAMS

161

413-" W = A= Pa vI

B (A) = e vat I

(B)

e,Aex T

IV3- (K)

Figure 31 - Oriented Chord Diagrams for v3. denotes the three chord diagram consisting of two parallel chords pierced by a third chord. In Figure 31 we show the specific orientations for the chord diagrams A and B . The key to this construction is in the choice of orientations for the chord diagrams in C = {A, B}. It is a nice exercise in translation of the Reidemeister moves to Gauss diagrams to see that v3(K) is indeed a knot invariant. It is possible that all Vassiliev invariants can be constructed by a method similar to the formula v (K) = EDEC < DI K > eval (D). This remains to be seen.

162 LOUIS H. KAUFFMAN

3.4. 817. It is an open problem whether there are Vassiliev invariants that can detect the difference between a knot and its reverse (The reverse of an oriented knot is obtained by flipping the orientation .). The smallest instance of a non-invertible knot is the knot 817 depicted in Figure 32. Thus, at the time of this writing there is no known Vassiliev invariant that can detect the non-invertibility of 817 . On the other hand, the tangle decomposition shown in Figure 32 can be used in conjunction with the results of Siebenmann and Bonahon [61] and the formulations of John Conway [12] to show this non-invertibility . These tangle decomposition methods use higher level information about the diagrams than is easy to encode in Vassiliev invariants. The purpose of this section is to underline this discrepancy between different levels in the combinatorial topology.

Figure 32 - Tangle Decomposition of 817.

KNOTS AND DIAGRAMS 163 4. QUANTUM LINK INVARIANTS

In this section we describe the construction of quantum link invariants from knot and link diagrams that are arranged with respect to a given direction in the plane. This special direction will be called " time". Arrangement with respect to the special direction means that perpendiculars to this direction meet the diagram transversely (at edges or at crossings ) or tangentially (at maxima and minima ). The designation of the special direction as time allows the interpretation of the consequent evaluation of the diagram as a generalized scattering amplitude. In the course of this discussion we find the need to reformulate the Reidemeister moves for knot and link diagrams that are arranged to be transverse (except for a finite collection of standard critical points) to the specific special direction introduced in the previous paragraph. This brings us back to our theme of diagrams and related structures . This particular reformulation of the Reidemeister moves is quite far-reaching . It encompasses the relationship of link invariants with solutions to the Yang-Baxter equation and the relationship with Hopf algebras (to be dealt with in Section 5). 4.1. Knot Amplitudes. Consider first a circle in a spacetime plane with time represented vertically and space horizontally as in Figure 33. The circle represents a vacuum to vacuum process that includes the creation of two "particles" and their subsequent annihilation . We could divide the circle into these two parts (creation "cup" and annihilation "cap") and consider the amplitude < caplcup >. Since the diagram for the creation of the two particles ends in two separate points, it is natural to take a vector space of the form V ® V as the target for the bra and as the domain of the ket . We imagine at least one particle property being catalogued by each factor of the tensor product. For example, a basis of V could enumerate the spins of the created particles. Any non-self-intersecting differentiable curve can be rigidly rotated until it is in general position with respect to the vertical. It will then be seen to be decomposed into an interconnection of minima and maxima. We can evaluate an amplitude for any curve in general position with respect to a vertical direction . Any simple closed curve in the plane is isotopic to a circle , by the Jordan Curve Theorem. If these are topological amplitudes , then the value for any simple closed curve should be equal to the original amplitude for the circle . What condition on creation (cup) and annihilation (cap) will insure topological amplitudes ? The answer derives from the fact that isotopies of the simple closed curves are generated by the cancellation of adjacent maxima and minima as illustrated in Figure 34.

LOUIS H. KAUFFMAN

164


' Mc d

C Lt

la,

b (^ 6 d a,

•

Figure 33 - Spacetime Circle This condition is articulated by taking a matrix representation for the corresponding operators. Specifically , let {e1, e2 ,..., e„} be a basis for V. Let eob = ea®eb denote the elements of the tensor basis for V ® V. Then there are matrices Mob and Mob such that > (1) = E Mobeob

Icup

with the summation taken over all values of a and b from 1 to n . Similarly, < capl is described by < capl(eob) = Mb.

Thus the amplitude for the circle is < caplcup > ( 1) =< cap r Mabeab _ Mab < capl (eab) _ MabMab•

KNOTS AND DIAGRAMS 165

J /y^ I ^Qb ^d

zS AJ b

_46M^d $f/K,4f/^db ^Aab AAcdAac b

Figure 34 - Spacetime Jordan Curve In general , the value of the amplitude on a simple closed curve is obtained by translating it into an "abstract tensor expression" in the Mob and Mab, and then summing over these products for all cases of repeated indices. Returning to the topological conditions we see that they are just that the matrices M°b and Maj are inverses in the sense that M-Wib = bb where bb' denotes the identity matrix. See Figure 24.

One of the simplest choices is to take a 2 x 2 matrix M such that M2 = I where I is the identity matrix . Then the entries of M can be used for both the cup and the cap . The value for a loop is then equal to the sum of the squares of the entries of M: (1)=EMabMab=>MabMab=F_ 4.1-

166 LOUIS H. KAUFFMAN

In particular , consider the following choice for M . It has square equal to the identity matrix and yields a loop value of d = -A2 - A-z, just the right loop value for the bracket polynomial model for the Jones polynomial [29], [28]. iA 0 _ M = [-i A-1 0 Any knot or link can be represented by a picture that is configured with respect to a vertical direction in the plane . The picture will decompose into minima (creations) maxima (annihilations ) and crossings of the two types shown in Figure 35. Here the knots and links are unoriented . These models generalize easily to include orientation.

^( cK fk ab Mai/`tbe^ 1cd

^^k I ^f( L

d6

CL b :

b'

6 46 6 -4b C d ^ cd ^ c ^d' c Figure 35 - Cups, Caps and Crossings Next to each of the crossings we have indicated mappings of V ® V to itself, called R and R-1 respectively. These mappings represent the transitions corresponding to elementary braiding. We now have the vocabulary of cup, cap, R and R-1. Any knot or link can be written as a composition of these fragments , and consequently a choice

KNOTS AND DIAGRAMS 167

of such mappings determines an amplitude for knots and links . In order for such an amplitude to be topological (i.e. an invariant of regular isotopy the equivalence relation generated by the second and third of the classical Reidemeister moves) we want it to be invariant under a list of local moves on the diagrams as shown in Figure 36. These moves are an augmented list of Reidemeister moves , adjusted to take care of the fact that the diagrams are arranged with respect to a given direction in the plane. The proof that these moves generate regular isotopy is composed in exact parallel to the proof that we gave for the classical Reidemeister moves in section 2. In the piecewise linear setting , maxima and minima are replaced by upward and downward pointing angles. The fact that the triangle, in the Reidemeister piecewise linear triangle move , must be projected so that it is generically transverse to the vertical direction in the plane introduces the extra restriction that expands the move set. In this context , the algebraic translation of Move III is the Yang-Baxter equation that occurred for the first time in problems of exactly solved models in statistical mechanics [6].

Figure 36 - Regular Isotopy with respect to a Vertical Direction

168 LOUIS H. KAUFFMAN

All the moves taken together are directly related to the axioms for a quasitriangular Hopf algebra (aka quantum group). Many seeds of the structure of Hopf algebras are prefigured in the patterns of link diagrams and the structure of the category of tangles. The interested reader can consult [59],[73] , [35] and [34] ,[36],[38] and section 5 of this paper for more information on this point. Here is the list of the algebraic versions of the topological moves. Move 0 is the cancellation of maxima and minima. Move II corresponds to the second Reidemeister move. Move III is the Yang-Baxter equation. Move IV expresses the relationship of switching a line across a maximum. (There is a corresponding version of IV where the line is switched across a minimum.) 0. M°'M;b = fib II. Rj6R'c'd = 8c dd III. RiRjc kf R" = RkR°dkRef ii IV. RZMtd = Mbt cd In the case of the Jones polynomial we have all the algebra present to make the model. It is easiest to indicate the model for the bracket polynomial: Let cup and cap be given by the 2 x 2 matrix M, described above so that M;; = MOO. Let R and R-1 be given by the equations

R°a = AM°bMMd + A-16,-Sd, R lab = A-

1M°bMcd

+ ASSbd.

This definition of the R-matrices exactly parallels the diagrammatic expansion of the bracket, and it is not hard to see, either by algebra or diagrams, that all the conditions of the model are met. 4.2. Oriented Amplitudes. Slight but significant modifications are needed to write the oriented version of the models we have discussed in the previous section. See [34], [66], [58], [20]. In this section we sketch the construction of oriented topological amplitudes. The generalization to oriented link diagrams naturally involves the introduction of right and left oriented caps and cups. These are drawn as shown in Figure 37 below.

KNOTS AND DIAGRAMS

169

00

Figure 37 - Right and Left Cups and Caps A right cup cancels with a right cap to produce an upward pointing identity line. A left cup cancels with a left cap to produce a downward pointing identity line. Just as we considered the simplifications that occur in the unoriented model by taking the cup and cap matrices to be identical , lets assume here that right caps are identical with left cups and that consequently left caps are identical with right cups. In fact, let us assume that the right cap and left cup are given by the matrix Mob = A"2aa6

where A is a constant to be determined by the situation , and &j, denotes the Kronecker delta. Then the left cap and right cup are given by the inverse of M: Ma6' _)1-a/2bay.

170 LOUIS H. KAUFFMAN

We assume that along with M we are given a solution R to the Yang-Baxter equation, and that in an oriented diagram the specific choice of R' is governed by the local orientation of the crossing in the diagram. Thus a and b are the labels on the lines going into the crossing and c and d are the labels on the lines emanating from the crossing. Note that with respect to the vertical direction for the amplitude, the crossings can assume the aspects: both lines pointing upward, both lines pointing downward, one line up and one line down (two cases). See Figure 38.

p a^Yct,^^el

-WtT x ecl Figure 38 - Oriented Crossings Call the cases of one line up and one line down the mixed cases and the upward and downward cases the parallel cases . A given mixed crossing can be converted, in two ways, into a combination of a parallel crossing of the same sign plus a cup and a cap . See Figure 39. This leads to an equation that must be satisfied by the R matrix in relation to powers of A (again we use the Einstein summation convention): a°/2S°`RjbA d/25;d = a-°/2b;,R`Xa°1Zdj6.

KNOTS AND DIAGRAMS 171

Figure 39 - Conversion This simplifies to the equation

from which we see that Rdcab is necessarily equal to zero unless b + d = a + c. We say that the R matrix is spin peeserving when it satisfies this condition. Assuming that the R matrix is spin preserving, the model will be invariant under all orientations of the second and third Reidemeister moves just so long as it is invariant under the anti-parallel version of the second Reidemeister move as shown in Figure 40.

172 LOUIS H. KAUFFMAN

'Cd

Figure 40 - Antiparallel Second Move This antiparallel version of the second Reidemeister move places the following demand on the relation between A and R: \(s b)/2,\(t c)/2 9R st

= 6^ fd

Call this the R - A equation. The reader familiar with [22] or with the piecewise linear version as described in [34] will recognise this equation as the requirement for regular homotopy invariance in these models.

KNOTS AND DIAGRAMS 173

4.3. Quantum Link Invariants and Vassiliev Invariants. Vassiliev invariants can be used as building blocks for all the presently known quantum link invariants. It is this result that we can now make clear in the context of the models given in our section on quantum link invariants. Suppose that A is written as a power series in a variable h , say A = exp(h) to be specific. Suppose also, that the R-matrices can be written as power series in h with matrix coefficients so that PR = I+r+h+O(h2) and PR-1 = I + r_ h + O(h2) where P denotes the map of V ® V that interchanges the tensor factors. Let Z(K) denote the value of the oriented amplitude described by this choice of A and R. Then we can write

Z(K) = Zo(K) + Zj(K)h + Z2(K)h2 + ... where each Z„(K) is an invariant of regular isotopy of the link K. Furthermore, we see at once that h divides the series for Z(K+) - Z(K_). By the definition of the Vassiliev invariants this implies that hk divides Z(G) if G is a graph with k nodes. Therefore Z„(G) vanishes if n is less than the number of nodes of G. Therefore Z„ is a Vassiliev invariant of finite type n. Hence the quantum link invariant is built from an infinite sequence of interlocked Vassiliev invariants.

It is an open problem whether the class of finite type Vassiliev invariants is greater than those generated from quantum link invariants. It is also possible that there are quantum link invariants that can not be generated by Vassiliev invariants. 4.4. Vassiliev Invariants and Infinitesimal Braiding. Kontsevich [44], [4] proved that a weight assignment for a Vassiliev top row that satisfies the 4-term relation and the framing condition (that the weights vanish for graphs with isolated double points) actually extends to a Vassiliev invariant defined on all knots. His method is motivated by the perturbative expansion of the Witten integral and by Witten's interpretation of the integral in terms of conformal field theory. This section will give a brief description of the Kontsevich approach and the questions that it raises about the functional integral itself.

The key to this approach is to see that the 4-term relations are a kind of "infinitesimal braid relations". That is, we can re-write the 4-term relations in the form of tangle operators as shown in Figure 41. This shows that the commutator equation [t12 , t13] + [t13 t23] = 0 e

is an algebraic form of the 4-term relation. The 4-term relations translate exactly

174 LOUIS H. KAUFFMAN

B

e>1

R

3

1

3

13

I

w

41 3 tI 3

Figure 41 - Infinitesimal Braiding into these infinitesimal braid relations studied by Kohno [45]. Kohno showed that his version of infinitesimal braid relations corresponded to a flatness condition for a certain connection (the Knizhnik-Zamolodchikov connection) and that this meant that these relations constituted an integrability condition for making representations of the braid group via monodromy. Others have verified that the braid group representations related to the Chern- Simons form and the Witten integral arise in this same way from the Knizhnik-Zamolodchikov equations. In the case of ChernSimons theory the weights in the K-Z equations come from the Casimir of a classical Lie algebra, just as we have discussed. Kontsevich observed that since the arbitrary

KNOTS AND DIAGRAMS 175

4-term relations could also be regarded as an integrability condition it was possible to use them in a generalization of Kohno's ideas to produce braid group representations via iterated integration. He then generalized the process of producing these braid group representations to the production of knot invariants and these become realizations of Vassiliev invariants that have given admissible weight systems for their top rows.

The upshot of the Kontsevich work is a very specific integral formula for the Vassiliev invariants. See [4] for the specifics. It is clear from the nature of the construction that the Kontsevich formula captures the various orders of perturbative terms in the Witten integral. At this writing there is no complete published description of this correspondence. 4.5. Weight Systems and the Classical Yang Baxter Equation.

Lets return momentarily to the series form of the solution to the Yang-Baxter equation, as we had indicated it in the previous subsection. PR = I + r+h + 0(h 2) PR-1 = I + r_h + 0(h 2). Since RR-1 = P, it follows that r_ = -r+ where a' denotes the transpose of a. In the case that R satisfies the R - A equation, it follows that t=r+-r_ = r + r

(letting r denote r+) satisfies the infinitesimal braiding relations [t12 t13] + [t13 t23] = 0. o r J

It is interesting to contemplate this fact, since r, being the coefficient of h in the series for R, necessarily satisfies the classical Yang Baxter Equation [r13 r23] + [r12, r23] + [r12, r13] = 0.

(The classical Yang-Baxter equation for r is a direct consequence of the fact that R is a solution of the (quantum) Yang-Baxter equation.) Via the quantum link invariants, we have provided a special condition (the assumption that r is the coefficient of h in a power series solution of the quantum Yang-baxter equation R, and that R satisfies the R - A equation) ensuring that a solution r of the classical Yang Baxter equation will produce a solution t = r + r' of the infinitesimal braiding relation, whence a weight system for Vassiliev invariants. More work needs to be done to fully understand the relationship between solutions of the classical Yang-Baxter equation and the construction of Vassiliev invariants.

176 LOUIS H. KAUFFMAN 5. HOPF ALGEBRAS AND INVARIANTS OF THREE-MANIFOLDS

This section is a rapid sketch of the relationship between the description of regular isotopy with respect to a vertical direction ( as described in our discussion of quantum link invariants) and the way that this formulation of the Reidemeister moves is related to Hopf algebras and to the construction of link invariants and invariants of three- manifolds via Hopf algebras. More detailed presentations of this material can be found in [20], [35], [36], [37],[38]. Let's begin by recalling the Kirby calculus [42 ]. In the context of link diagrams the Kirby calculus has an elegant formulation in terms of (blackboard) framed links represented by link diagrams up to ribbon equivalence. Ribbon equivalence consists in diagrams up to regular isotopy coupled with the equivalence of a positive (negative) curl of Whitney degree 1 with a positive (negative) curl of Whitney degree - 1. See Figure 42.

rv

f

^

l

f "^'ib^o ^1,

hanjIe

145 blow dow ^'l,

0, C9 ,

blow wP

Figure 42 - Framing and Kirby Calculus

sl ode, 61owivk^ wP a-n.d dowvU

KNOTS AND DIAGRAMS 177

Here we refer informally to the Whitney degree of a plane curve. The Whitney degree is the total turn of the tangent vector If the curve is not closed, then it is assumed that the tangent direction of the initial point is the same as the tangent vector of the endpoint. In Figure 42 we illustrate how curls encode framings and how ribbon equivalent curls correspond to identical framings. A link is said to be framed if it is endowed with a smooth choice of normal vector field. Framing a link is equivalent to specifying an embedded, band(s) of which it is the core. The core of a band is the center curve. Thus Sl x {.5} is the core of S' x [0, 1]. Now introduce two new moves on link diagrams called handle sliding and blowing up and down. These moves are illustrated in Figure 42. Handle sliding consists in duplicating a parallel copy of one link component and then band connect summing it with another component. Blowing up consists in adding an isolated unknotted component with a single curl. Blowing down consists in deleting such a component. These are the basic moves of the Kirby Calculus. Two link diagrams are said to be KC-equivalent if there is a combination of ribbon equivalence, handle-sliding and blowing up and blowing down that takes one diagram to the other. The invariants of three-manifolds described herein are based on the representation of closed three-manifolds via surgery on framed links. Let M3(K) denote the threemanifold obtained by surgery on the blackboard framed link corresponding to the diagram K. In M3 (K) the longitude associated with the diagram, as shown in Figure 43, bounds the meridian disk of the solid torus attached via the surgery. The basic result about Kirby Calculus is that M3(K) is homeomorphic to M3(L) if and only if K and L are KC equivalent. Thus invariants of links that are also invariant under Kirby moves will produce invariants of three-manifolds. It is the purpose of this section to sketch one on the approaches to constructing such invariants.

The ideas behind this approach are quite simple. We are given a finite dimensional quasitriangular Hopf algebra A. We associate to A a tensor category Cat(A). The objects in this category are the base field k of the Hopf algebra , and tensor powers of a formal object V. It is assumed that the tensor powers of V are canonically associative and that the tensor product of V with k on either side is canonically isomorphic to V. The morphisms in Cat(A) are represented by Hopf algebra decorated immersed curves arranged with respect to a vertical direction. An immersed curve diagram is a link diagram where there is no distinction between undercrossings and overcrossings . Segments of the diagram can cross one another transversely as in a standard link diagram , and we can arrange such a diagram with respect to a vertical direction just as we did for link diagrams . A vertical place on such a diagram is a point that is not critical with respect to the vertical direction, and is not a crossing . A decoration of such an immersed curve diagram consists in a

178 LOUIS H. KAUFFMAN

Figure 43 - Surgery on a Blackboard Framed Link subset of vertical places labelled by elements of the Hopf algebra A. The diagrams can have endpoints and these are either at the bottom of the diagram or at the top (with respect to the vertical). The simplest decorated diagram is a vertical line segment with a label a (corresponding to a element a of the Hopf algebra) in its interior. In the category Cat(A) this segment is regarded as a morphism [a] : V -+ V where V is the formal object alluded to above. Composition of these morphisms corresponds to multiplication in the algebra: [a] [b] = [ab]. By convention, we take the order of multiplication from bottom to top with respect to the vertical direction.

A tensor product a ® b in A ® A is represented by two parallel segments, one decorated by a, the other decorated by b. It is our custom to place the decorations for a and for b at the same level in the diagram. In the Hopf algebra we have the

KNOTS AND DIAGRAMS

179

V 01

V Cad [6]

Ca] ®[b] =[Q®b]. ,Z) ) b

a ^Q = Q 1 ^^, 4

b

n a^ =

-

= ([b] ®[2]

O

F

n'^) kt) = U -V (.t.)

Figure 44 - Morphisms in Cat(A) coproduct A : A --> A ® A. We shall write A(a) = Eal ®a2 where it is understood that this means that the coproduct of a is a sum over elements of the form ai ® a2. It is also useful to use a version of the Einstein summation convention and just write

A (a) = al ® a2 where it is understood that the right hand side is a summation . In diagrams, application of the antipode makes parallel lines with doubled decorations according to the two factors of the coproduct . See Figure 44.

180 LOUIS H. KAUFFMAN

A crossing of two undecorated segments is regarded as a morphism P : V ® V ---> V ®V. Since the lines interchange , we expect P to behave as the permutation of the two tensor factors. That is, we take the following formula to be axiomatic: Po([a](9 [b])=([b]®[a])oP. A cap (see Figure 44) is regarded as a morphism from V ® V to k, while a cup is regarded as a morphism form k to V ® V. As in the case of the crossing the relevance of these morphisms to the category is entirely encoded in their properties. The basic property of the cup and the cap is that if you "slide" a decoration across the maximum or minimum in a counterclockwise turn, then the antipode S of the Hopf algebra is applied to the decoration. In categorical terms this property says Cup o ([a] (9 1) = Cup o (10 [Sal) and ([Sa] ®1) o Cap = (1(9 [a]) o Cap. These properties and some other naturality properties of the cups and the caps are illustrated in Figure 44. These naturality properties of the flat diagrams include regular homotopy of immersions, as illustrated in Figure 44. In Figure 45 we see how this property of the cups and the caps leads to a diagrammatic interpretation of the antipode. This, in turn, leads to the interpretation of the flat curl as a grouplike element G in A such that S2(a) = GaG-1 for all a in A. G is a flat curl diagram interpreted as a morphism in the category. We see that formally it is natural to interpret G as an element of A and that A(G) = G ® G is a direct consequence of the diagrams for Cat(A). In a so-called ribbon Hopf algebra there is such a grouplike already in the algebra. In the general case it is natural to extend the algebra to contain such an element. We are now in a position to describe a functor F from the tangle category T to Cat(A). (The tangle category is defined for link diagrams without decorations. It has the same objects as Cat(A). The morphisms in the tangle category have relations corresponding to the augmented Reidemeister moves described in the section on quantum link invariants.) F simply decorates each positive (with respect to the vertical - see Figure 45) crossing of the tangle with the Yang-Baxter element (given by the quasi-triangular Hopf algebra A) p = Ee ®e and each negative crossing (with respect to the vertical) with p-1 = ES(e) ®e . The form of the decoration is indicated in Figure 46. The key point about this category is that because Hopf algebra elements can be moved around the diagram, we can concentrate all the algebra in one place. Because the flat curls are identified with either G or G-1, we can use regular homotopy

KNOTS AND DIAGRAMS 181

of immersions to bring each component of a link diagram to the form of a circle with a single concentrated decoration (involving a sum over many products). An example is shown in Figure 46. Let us denote by \(a) : k -p k the morphism that corresponds to decorating the right hand side of a standard circle with a. That is .(a) = Cap o (1® [a]) o Cup. We can regard A as a linear functional defined on A as a vector space over k.

s 'CQ)

o(6) =gy(p) _ y Figure 45 - Diagrammatics of the Antipode

182 LOUIS H. KAUFFMAN

see =1 L&j

uJ Figure 46 - The Functor F : T -p Cat(A). We wish to find out what properties of A will be appropriate for constructing invariants of three-manifolds. View Figure 47. Handle sliding is accomplished by doubling a component and then band summing . The doubling corresponds to applying the antipode. As a result , we have that in order for A to be invariant under handle-sliding it is sufficient that it have the property A(x)1 = EA( xl)x2. This is the formal defining property of a right integral on the Hopf algebra A. Finite dimensional Hopf algebras have such functionals and suitable normalizations lead to well -defined three-manifold invariants . For more information see the references cited at the beginning of this section . This completes our capsule summary of Hopf algebras and invariants of three-manifolds.

KNOTS AND DIAGRAMS 183

There are a number of problems related to this formulation of invariants of threemanifolds . First of all, while it is the case that the invariants that come from integrals can be different from invariants defined through representations of Hopf algebras as in [59] it is quite difficult to compute them and consequently little is known. Another beautiful problem is related to the work of Greg Kuperberg [46], [47]. Kuperberg defines invariants of three-manifolds associated via Hopf diagrams associated with a Heegard splitting of the three-manifold. Does our invariant on the Drinfeld double of a Hopf algebra H give the same result as Kuperberg 's invariant for H? This conjecture is verified in the (easy) case where H is the group ring of a finite group . Finally, it should be mentioned that the way in which handle -sliding invariance is proven for the universal three-manifold invariant of finite type of Le and Murakami [48] is directly analogous to our method of relating handle sliding, coproduct and right integral . It remains to be seen what is the relationship between three-manifold invariants of finite type and the formulations discussed here.

Figure 47 - Handle Sliding and Right Integral

184 LOUIS H. KAUFFMAN 6. TEMPERLEY LIEB ALGEBRA

This final section is devoted to the structure of the Temperley Lieb algebra as revealed by its diagrammatic interpretation. We begin with a combinatorial description of this algebra. It is customary, in referring to the Temperley Lieb algebra to refer to a certain free algebra over an appropriate ring. This freealgebra is the analog of the group ring of the symmetric group S„ on n letters. It is natural therefore to first describe that multiplicative structure that is analogous to S,,. We shall call this structure the Temperley Lieb Monoid Mn. We shall describe the Temperley Lieb algebra itself after first defining this monoid. There is one Temperley Lieb monoid, M, for each natural number n. The connection elements of Mn consist in diagrams in the plane that make connections involving two rows of n points. These rows will be referred to as the top and bottom rows. Each point in a row is paired with a unique point different from itself in either the top or the bottom row (it can be paired with a point in its own row). These pairings are made by arcs drawn in the space between the two rows. No two arcs are allowed to intersect one another. Such a connection element will be denoted by U, with subscripts to indicate specific elements. If the top row is the set Top = {1, 2, 3, ..., n} and the bottom row is Bot = 11',2 .. ..., n'}, then we can regard U as a function from Top U Bot to itself such that U(x) is never equal to x, U(U(x)) = x for all x, and satisfying the planar non-intersection property described above. In topological terms U is an n-tangle with no crossings, taken up to regular isotopy of tangles in the plane. If U and V are two elements of M„ as described above, then their product UV is the tangle product obtained by attaching the bottom row of U to the top row of V. Note that the result of tatting such a product will produce a new connection structure plus some loops in the plane. Each loop is regarded as an instance of the loop element 6 of the Temperley Lieb monoid M,,. The loop element commutes with all other elements of the monoid and has no other relations with these elements. Thus UV = 6kW for some non-negative integer k, and some connection element W of the monoid.

The Temperley Lieb algebra T„ is the free additive module on Mn modulo the identification 6=-A2-A 2,

over the ring Z[A, A-1] of Laurent polynomials in the variable A. Products are defined on the connection elements and extended linearly to the algebra. The reason for this loop identification is the application of the Temperley Lieb algebra for the bracket polynomial and for representations of the braid group [22], [28], [40]. The Temperley Lieb monoid Mn is generated by the elements 1, U1, U2, ..., U„_1 where the identity element 1 connects each i in the top row with its corresponding

KNOTS AND DIAGRAMS 185

member i' in the bottom row. Here Uk connects i to i' for i not equal to k, k + 1, k', (k + V. Uk connects k to k + 1 and k' to (k + 1)'. It is easy to see that

2 = Wk,

U

UkUk±lUk = Uk, U1Uj=U;U;,

li-il>1.

See Figure 48.

U

Figure 48 - Relations in the Temperley Lieb Monoid

LOUIS H. KAUFFMAN

186

We shall prove that the Temperley Lieb Monoid is the universal monoid on G„ = {1, U1, U2, ..., U„_1} modulo these relations. In order to accomplish this end we give a direct diagrammatic method for writing any connection element of the monoid as a certain canonical product of elements of G,,. This method is illustrated in Figure 49. I 0. 3 'f- S 6 7 $

N

;3

1/ ! a' 'f ` 31

(

// U3UR(ti) (c

61 ,/ u3)

S67 8 9 U

H

71

gg

Ii

(6666. u.IZ / '

l 1 ) (6(6

) (V7) (`1g)

Figure 49 - Canonical Factorization in the Temperley Lieb Monoid As shown in Figure 49, we represent the connection diagram with vertical and horizontal straight arcs such that except for the height of the straight arcs, the form of the connection between any two points is unique - consisting in two vertical arcs

KNOTS AND DIAGRAMS 187

and one horizontal arc. The horizontal arc has as its endpoints on the vertical lines that go through the row points that are being connected. (Diagram is drawn so that each pair in the set { (i, i') : 1 < n} determines such a vertical line. The vertical arcs in the connection are chosen as segments from these vertical lines. All connections are chosen so that the connections do not intersect. It is from this diagram that we shall read out a factorization into a product of elements of Gn. The factorization is achieved via a decoration of the straight arc diagram by dotted vertical arcs, as shown in Figure 49. Each dotted arc connects midpoints of the restrictions of horizontal arcs to the columns of the diagram, where a column of the diagram is the space between two consecutive vertical lines (vertical lines described as in the previous paragraph). The index of a column is the row number associated to the left vertical boundary of the column. The dotted lines in a given column are uniquely determined by starting at the bottom of the column and pairing up the horizontal arcs in that column in vertical succession. Each dotted are is labelled by the index of the column in which it stands. In a given diagram a sequence of dotted arcs is a maximal set of dotted arcs (with consecutive indices) that are interconnected by horizontal segments such that one can begin at the top of the dotted arc (in that sequence) of highest index, go down the arc and left to the top of the next arc along a horizontal segment, continuing in this manner until the whole sequence is traversed. It is clear from the construction of the diagram that the dotted segments in the diagram collect into a disjoint union of sequences {s1, s2, ..., sk} where each s` denotes the corresponding descending sequence of of consecutive indices:

s' = (mi, mi - 1, mi - 2,..., ni + 1, ni). These indices satisfy the inequalties: m1 < m2 < m3 < ... < Mk

and nl
The sequences { s1, s2, ..., sk} occur in that order on the diagram read from left to right. Of course the descent of each sequence goes from right to left. If D is a diagram with sequence structure s(D) as we have just described, let U(s(D)) be the following product of generators of the Temperley Lieb monoid: U(D) = U(s1)U(s2)U(s3)...U(sk)

where U(s') = U,m,Umi_l...Uni+iUni.

By looking carefully at the combinatorics of these diagrams , as illustrated in Figure 49 , one sees that D and U(D) represent the same connection structure in the

188 LOUIS H. KAUFFMAN

Temperley Lieb monoid. Furthermore, any sequence structure s satisfying the inequalities given above (call these the canonical inequalities) will produce a standard diagram from the product U(s). Thus the sequence structure of a Temperley Lieb diagram completely classifies this diagram as a connection structure in the monoid. We must now prove that any product of elements of G„ = {1, U1, U2, ..., U„_1} can be written , up to a loop factor , as U(s ( D)) for some diagram D, or equivalently as U(s) for a sequence structure satisfying the canonical inequalities . This is a simple exercise in using the relations we have already given for the products of elements of G,,. We leave the details to the reader . This completes the proof that the relations in Figure 48 are a complete set of relations for the Temperley Lieb monoid. 6.1. Parentheses. Elements of the Temperley Lieb monoid M. are in one to one correspondence with well-formed parenthesis expressions using n pairs of parentheses . The proof of this statement is shown in Figure 50.

22

Figure 50 - Creating Parentheses

KNOTS AND DIAGRAMS 189

One result of this reformulation of the Temperley Lieb monoid is that one can rewrite the product structure in terms of operations on parentheses, getting an interesting formal algebra that encodes properties of the topology of plane curves [39],[30]. In this section we will take the relationship with parentheses as an excuse to indicate a further relationship with non-associative products. First consider the abstract structure of non-associative products. They are usually written in the forms: (a * b) * c a * (b * c) ((a*b)*c)*d (a*(b*c))*d a*((b*c)*d) a* (b* (c*d)) (a*b)*(c*d) In writing products in this standard manner, one sees the same structure of parentheses occurring in different products, as in ((a*b)*c)*d and (a*(b*c))*d. In each of these cases if we eliminated the algebraic literals and the operation symbol *, we would be left with the parenthetical: (()). There is another way to write the products so that different products correspond to different arrangements of parentheses. To do this, rewrite the products in the operator notation shown below:

a*b=a(b)^() (a * b) * c = a (b)(c) ()() a * (b * c) = a(b(c)) (()) ((a * b) * c) * d = a(b)(c)(d) ()()() (a * (b * c)) * d = a(b(c))(d) (())() a * ((b * c) * d) = a(b(c)(d)) (()()) a * (b * (c * d)) = a(b(c(d))) ((())) (a * b) * (c * d) = a(b)(c(d)) --, ()(()) In the operator notation, each product of n + 1 terms is uniquely associated with an expression using n pairs of parentheses. Of course we can replace the expressions in parentheses in terms of nested caps as we did in Figure 50. Once this is done, we notice the very interesting fact that re-associations can be visualized in terms of "handle-sliding" at the cap level. The meaning of this remark should be apparent to the reader from Figure 51.

190 LOUIS H. KAUFFMAN

a*b H nb1 (a-*-b) * c H a (6) n

-

nn a-*(b*c)E^i ct ('b-) <-> ^^1^

Handle S l i d iY6q

1 Q,r.EYL, I'.hi,ses

^=Q6 a bn ^a'=4!. * (b*c))*J

car-)

a b =a-x- (b*(c--* 4

Figure 51 - Pentagon In Figure 51 we illustrate the pentagon of re-associations of a product of four terms in terms of sliding caps. Notice that we can do this sliding without writing any algebraic literals by labelling the "active" right feet of the caps that do the sliding. In Figure 52 we show the structure of the Stasheff polyhedron with reassociations of five literals in the cap sliding formalism.

KNOTS AND DIAGRAMS 191

Cap sliding is a new formulation of a continuous background for these re-association moves. There is a related continuous background in terms of recoupling formalisms for trees. This formalism is intimately related to many topics in topological quantum field theory. The interested reader will find more about these points of view in [30],[14], [ 15], in the author's lecture notes with Scott Carter and Masahico Saito [ 11] and in his book with Sostenes Lins [40].

nn (1 V 3 (n^ ^1 Figure 52 - The Stasheff Polyhedron

192 LOUIS H. KAUFFMAN REFERENCES 1. Daniel Altshuler and Laurent Riedel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, preprint ( 1996). 2. M. F. Atiyah, Geometry of Yang-Mills Fields, Accademia Nazionale dei Lincei Scuola Superiore Lezioni Fermiare , Pisa, 1979. 3. M. F. Atiyah, The Geometry and Physics of Knots, Cambridge University Press, 1990. 4. D. Bar-Natan, On the Vassiliev knot invariants, (to appear in Topology). 5. Dror Bar-Natan, Perturbative Aspects of the Chern-Simons Topological Quantum field Theory, Ph. D. Thesis, Princeton University, June 1991. 6. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, 1982. 7. J. Birman and X. S. Lin, Knot polynomials and Vassiliev's invariants, (to appear in Invent. Math.).

8. R. Bott and C. Taubes, On the self-linking of knots, Jour. Math. Phys. 35 (1994), pp. 52475287. 9. P. Cartier, Construction combinatoire des invariants de Vassiliev - Kontsevich des noeuds, C. R. Acad. Sci. Paris 316, Serie I, (1993), pp. 1205-1210. 10. J. Scott Carter, D. E. Flath, M. Saito, The Classical and Quantum 6j-Symbols, Mathematical Notes 43, Princeton University Press (1995). 11. J. Scott Carter, L. H. Kauffman, M. Saito, Diagrammatics, Singularities and Their Algebraic Interpretations, (To appear in Conference Proceedings of the Brasilian Mathematical Society.) 12. John H. Conway, (private conversation) 13. J. H. Conway, An enumeration of knots and links and some of their algebraic properties, in Computational Problems in Abstract Algebra, Pergammon Press, N.Y.,1970, pp. 329-358. 14. L. Crane and I. Frenkel, Four dimensional topological quantum field theory, Hopf categories and canonical bases, J. Math. Physics, No. 35 (1994), pp. 5136-5154. 15. L. Crane, L. H. Kauffman, D. Yetter, State sum invariants of 4-manifolds, Journal of Knot Theory and Its Ramifications, (to appear ( 1996)). 16. S. Garoufalidis, Applications of TQFT to invariants in low dimensional topology, (preprint 1993). 17. E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons model and new relations between the Homfly coefficients, Physics Letters B, Vol. 238, No. 4, Sept. 28 (1989), pp. 489-494. 18. D. S. Freed and R. E. Gompf, Computer calculation of Witten's three-manifold invariant, Commun. Math. Phys., No. 141 (1991), pp. 79-117. 19. B. Hasslacher and M. J. Perry, Spin networks are simplicial quantum gravity, Phys. Lett., No. 103 B (1981). 20. M. A. Hennings , Hopf algebras and regular isotopy invariants for link diagrams, Math. Proc. Camb. Phil. Soc., Vol. 109 (1991), pp. 59-77. 21. L. C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys., No. 147, (1992), pp. 563-604. 22. V. F. R. Jones, A polynomial invariant of links via von Neumann algebras, Bull. Amer. Math. Soc., 1985, No. 129, pp. 103-112. 23. V. F. R.Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., Vol.126, 1987, pp. 335-338. 24. V. F. R.Jones, On knot invariants related to some statistical mechanics models, Pacific J. Math., Vol. 137, no. 2,1989, pp. 311-334. 25. L. H. Kauffman, The Conway polynomial, Topology, 20 (1980), pp. 101-108. 26. L. H. Kauffman , Formal Knot Theory, Princeton University Press, Lecture Notes Series 30 (1983). 27. L. H. Kauffman, On Knots, Annals Study No. 115, Princeton University Press (1987).

KNOTS AND DIAGRAMS 193 28. L. H. Kauffman, State Models and the Jones Polynomial , Topology, Vol. 26, 1987, pp. 395-407. 29. L. H. Kauffman, Statistical mechanics and the Jones polynomial, AMS Contemp. Math. Series, Vol. 78, 1989, pp. 263-297. 30. L. H. Kauffman, Map coloring, q-deformed spin networks, and Turaev-Viro invariants for 3manifolds, Int. J. of Modern Phys. B, Vol. 6, Nos. 11, 12 (1992), pp. 1765-1794. 31. L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., Vol. 318. No. 2 , 1990, pp. 417-471. 32. L. H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly, Vo1.95, No.3, March 1988, pp. 195-242. 33. L. H. Kauffman and P. Vogel, Link polynomials and a graphical calculus, Journal of Knot Theory and Its Ramifications, Vol. 1, No. 1, March 1992, pp. 59-100. .

34. L. H. Kauffman, Knots and Physics, World Scientific Pub.,1991 and 1993. 35. L. H. Kauffman, Gauss codes, quantum groups and ribbon Hopf algebras, Reviews in Mathematical Physics, Vol.5, No. 4.,1993, pp. 735-773. 36. L. H. Kauffman and D. Radford, Invariants of 3-Manifolds derived from finite dimensional Hopf algebras, Journal of Knot Theory and Its Ramifications . Vol. 4, No. 1 (1995), 131-162. 37. L. H. Kauffman, D. Radford and S. Sawin, Centrality and the KRH Invariant, ( to appear). 38. L. H. Kauffman, Hopf algebras and 3-Manifold invariants, Journal of Pure and Applied Algebra. Vol. 100 (1995), 73-92. 39. L. H. Kauffman, Spin Networks, Topology and Discrete Physics, in second edition of Braid Group, Knot Theory and Statistical Mechanics, ed. by Yang and Ge, World Sci. Pub. (1994) 40. L. H. Kauffman and S. L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3Manifolds, Annals of Mathematics Study 114, Princeton Univ. Press,1994. 41. L. H. Kauffman, Functional Integration and the theory of knots, J. Math. Physics, Vol. 36 (5), May 1995, pp. 2402-2429. 42. R. Kirby, A calculus for framed links in S3, Invent. Math., 45, (1978), pp. 35-56. 43. R. Kirby and P. Melvin, On the 3-manifold invariants of Reshetikhin- Turaev for sl (2, C), Invent. Math. 105, 473-545, 1991. 44. M. Kontsevich, Graphs, homotopical algebra and low dimensional topology, (preprint 1992). 45. T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemporary Mathematics, Vol. 78, Amer. Math. Soc., 1988, pp. 339-364. 46. G. Kuperberg, Involutory Hopf algebras and 3-manifold invariants, Int. J. Math., 1991, pp. 41-66. 47. G. Kuperberg, Non-involutory Hopf algebras and 3-manifold invariants , (preprint 1994.) 48. T. Q. T. Le and J . Murakami , Universal finite type invariants of 3-manifolds, (preprint 1995.) 49. R. Lawrence, Asymptotic expansions of Witten-Reshetikhin-T uraev invariants for some simple 3-manifolds. J. Math. Phys., No. 36(11), November 1995, pp. 6106-6129. 50. W. B. R. Lickorish, The Temperley Lieb Algebra and 3-manifold invariants, Journal of Knot Theory and Its Ramifications , Vol. 2,1993, pp. 171-194. 51. J. Mathias, Ph.D. Thesis, University of Illinois at Chicago, 1996. 52. R. Penrose, Angular momentum: An approach to Combinatorial Spacetime, In Quantum Theory and Beyond, edited by T. Bastin, Cambridge University Press ( 1969). 53. R. Penrose, Applications of negative dimensional tensors, In Combinatorial Mathematics and Its Applications, edited by D. J. A. Welsh, Academic Press (1971). 54. Michael Polyak and Oleg Viro, Gauss diagram formulas for Vassiliev invariants, Intl. Math. Res. Notices, No. 11, (1994) pp. 445-453. 55. P. Cotta-Ramusino, E. Guadagnini, M. Martellini, M. Mintchev, Quantum field theory and link invariants, (preprint 1990)

194 LOUIS H. KAUFFMAN 56. G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients, In Spectroscopic and Group Theoretical Methods in Theoretical Physics, (1968) North Holland, Amsterdam. 57. K. Reidemeister, Knotentheorie, Chelsea Pub. Co., New York, 1948, Copyright 1932, Julius Springer, Berlin. 58. N. Y. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I and II, LOMI reprints E-4-87 and E-17-87, Steklov Institute, Leningrad, USSR. 59. N. Y. Reshetikhin and V. Tllraev, Invariants of Three Manifolds via link polynomials and quantum groups, Invent. Math., Vo1.103, 1991, pp. 547-597. 60. L. Rozansky, Witten's invariant of 3-dimensional manifolds: loop expansion and surgery calculus, In Knots and Applications, edited by L. Kauffman, (1995), World Scientific Pub. Co. 61. L. Siebenmann and F. Bonahon, (Unpublished Manuscript) 62. Lee Smolin, Link polynomials and critical points of the Chern-Simons path integrals, Mod. Phys. Lett. A, Vol. 4,No. 12, 1989, pp. 1091-1112. 63. Lee Smolin, The geometry of quantum spin networks, (preprint 1996), Center for Gravitational Physics and Geometry, Penn. State University, University Park, PA. 64. T. Stanford, Finite-type invariants of knots, links and graphs, (preprint 1992). 65. H. 'hotter, Non-invertible knots exist, Topology, Vol.2, 1964, pp. 275-280. 66. V. G. T uraev, The Yang-Baxter equations and invariants of links, LOMI preprint E-3-87, Steklov Institute, Leningrad, USSR., Inventiones Math., Vol. 92, Fasc.3, pp, 527-553. 67. V. G. Thraev and 0. Y. Viro, State sum invariants of 3-manifolds and quantum 6j symbols, Topology 31, (1992), pp. 865-902. 68. V. G. T raev and H. Wenzl, Quantum invariants of 3-manifolds associated with classical simple Lie algebras, International J. of Math., Vol. 4, No. 2,1993, pp. 323-358. 69. V. Vassiliev, Cohomology of knot spaces, In Theory of Singularities and Its Applications, V. I. Arnold, ed., Amer. Math. Soc., 1990, pp. 23-69. 70. K. Walker, On Witten's 3-Manifold Invariants, (preprint 1991). 71. J. H. White, Self-linking and the Gauss integral in higher dimensions, Amer. J. Math., 91 ( 1969), pp. 693-728. 72. Edward Witten, Quantum field Theory and the Jones Polynomial, Commun. Math. Phys., vol. 121, 1989 , pp. 351-399. 73. D. Yetter, Quantum groups and representations on monoidal categories, Math. Proc. Camb. Phil. Soc., Vol. 108 (1990), pp. 197-229. DEPARTMENT OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE, UNIVERSITY OF ILLINOIS AT CHICAGO, 851 SOUTH MORGAN STREET, CHICAGO, IL, 60607-7045

Lectures at Knots 96 edited by Shin 'ichi Suzuki @1997 World Scientific Publishing Co. pp. 195-217

ON SPATIAL GRAPHS KOUKI TANIYAMA

Preface

Fig. 0.1 Fig. 0.1 means that CAT=PL. Namely we will work in the piecewise linear category throughout this survey article. The article is devided into the following two parts. Part 1. Equivalence relations on spatial graphs. Part 2. Knots in a spatial graph.

Part 1 is a research done by the author some years ago. Part 2 is a research with Y. Ohyama and A. Yasuhara that is now in progress. Acknowledgement The author is very appreciate to the organizers of KNOTS '96, especially to Professor Shin'ichi Suzuki, for giving him an opportunity to introduce his recent reserches in KNOTS '96. The author is grateful to Professor Toshiki Endo, the general manager of KNOTS '96, for his continuous kindness for the author (and for all other participants) throughout the period of KNOTS '96. Finally the 195

196

KOUKI TANIYAMA

author would like to express his hearty gratitude to Professor Jonathan Simon for his lecture in Tokyo in 1990, which inspired the author very much and the author could achieve the results in Part 1.

Part 1. Equivalence relations on spatial graphs Let G be a finite graph without loops and multiple edges. (We ocasionally treat graphs with loops and multiple edges. This is just a convenience. Since any graph has a subdivision without loops and multiple edges, we may suppose without loss of generality that a graph has no loops and multiple edges.) Let SE(G) be the set of all embeddings of G into the three-dimensional Euclidean space R. We call an element of SE(G) a spatial embedding of a graph, or simply a spatial graph. We will define seven equivalence relations (1) ambient isotopy, (2) cobordism, (3) isotopy, (4) Iequivalence, (5) edge-homotopy, (6) vertex-homotopy and (7) homology on SE(G) so that they satisfy the following relations which the author call the fundamental theorem of spatial graph theory.

Theorem 1 .1 [38]. (2) ( 1) (4) - (5) -+ (6 ) -> (7). (3) In order to define these equivalence relations , we prepare the following notions. Let I = [0, 1] be the unit closed interval and f, g elements of SE(G). We say that a continuous map 1 : G x I -> R3 X I is (a) between f and g if there is a real number e > 0 such that 4)(x, t) = (f (x), t) forallxEG, 0 R3 for each t E I such that (D(x, t) = (Ot(x), t) for all x c G, t E I. (c) locally flat if each point of the image of 4) has a neighbourhood N such that the pair (N, N fl 4D(G x I)) is homeomorphic to the standard disk pair (D4, D2) or (D3 x I, X„ x I) for some non-negative integer n where the pair (D3, Xn) is shown in Fig. 1.1.

ON SPATIAL GRAPHS 197

n segments

Fig. 1.1

We say that f and g are (1) ambient isotopic if there is a level preserving locally flat embedding 4) : G x I --+ R3 x I between f and g. (2) cobordant if there is a locally flat embedding 4t : G x I -> R3 x I between f and g. (3) isotopic if there is a level preserving embedding 4i : G X I -+ R3 x I between f and g.

(4) I-equivalent if there is an embedding
We note that the definition of ambient isotopy is equivalent to the usual definition using isotopic deformation of R3. The first four arrows in Theorem 1.1 directly follow from the definitions. The equivalence relations (2) (3) and (4) are not main objects of this article . However we introduce some results on (2) and (3) here. An example of cobordism of spatial graphs is illustrated in Fig. 1.2. The spatial theta curve of t = 0 in Fig. 1.2 is a "vertex connected sum" (see [45]) of a spatial theta curve and its "reflected inverse". Therefore this example implies the existence of "cobordism inverse" and the cobordism classes of spatial theta curves form a group under the vertex-connected sum. See [36]. Miyazaki showed , using the CassonGordon invariant, that this group is not abelian [20].

198 KOUKI TANIYAMA

^ !5(!!5 OCT Fig. 1.2 Fig. 1 . 3 is an example of isotopy that is an application of Alexander's trick. Here we have a non-locally flat point in the level t = 4.

X88 t =0

t =1/4 t=1/2 t=3/4 t=1 Fig. 1.3

An application of an Alexander 's trick is called a blowing- down and the reverse operation is called a blowing-up. It is easy to see that an isotopy is generated by blowing-ups, blowing-downs and ambient isotopies , cf. [27] . Moreover Soma showed the following remarkable rearrangement theorem. Theorem 1.2. [32] Let G be a finite graph. Let f, g E SE( G) be isotopic embeddings. Then there is an embedding h E SE(G) such that h is obtained from f by a sequence of blowing- downs and ambient isotopies , and g is obtained from h by a sequence of blowing-ups and ambient isotopies. Concerning above theorem, Soma showed many interesting results involving isotopy, cobordism, minimally knotted embeddings ([30] [48]) and generalized bouquets ([38]). We refer the reader to [31 ], [11] and [32] for the results. We also note here that an isotopy invariant of spatial graphs coming from branched covering spaces is defined and studied in [5]. Now we continue the definitions . We say that f and g are

(5) edge-homotopic if g is obtained from f by a series of "self-crossing changes" and

ON SPATIAL GRAPHS

199

ambient isotopy. Here a self-crossing change is a change of a crossing whose over-path and underpath belong to the same edge of G. See Fig. 1.4.

same edge Fig. 1.4 (6) vertex-homotopic if g is obtained from f by a series of "crossing changes between adjacent edges"and ambient isotopy.

Here a crossing change between adjacent edges is a change of a crossing whose over-path and under-path belong to two edges that has a common vertex. See Fig. 1.5.

Fig. 1.5 (7) homologous if there is a locally flat embedding 4i : (G x 1)# U° 1 Si R3 x I between f and g where n is a natural number, Si is a closed orientable surface and # means the connected sum. More precisely, there is an edge e of G for each i such that Si is attached to the open disk int(e x I) by

200 KOUKI TANIYAMA

the usual connected sum of surfaces. See Fig. 1.6 and Fig. 1.7 for examples.

XJIS11\ Z I G G xI (G xI) # 6 Si 1=1

Fig. 1.6

CQ Fig. 1.7 Roughly speaking, a handle ( a connected sum of a torus) corresponds to an application of Whiteney 's trick in higher dimension . Thus homology abelianizes everything and admits a nice algebraic classification as in the following Theorem 1.6. The above seven equivalence relations on spatial graphs are natural generalizations of various equivalence relations on links (link cobordism , link homotopy, link homology etc., see [6 ], [18], [19], [22] etc. ). The proof of the nontrivial part (4) -* ( 5) of Theorem 1.1, that is shown for links in [3], [7] and [8], is similar to that in [8]. See [38] for the detail. Before stating the homology classification , we state the unknotting theorems of spatial graphs under the seven equivalence relations . Let X and Y be topological

ON SPATIAL GRAPHS 201

spaces. Suppose that X is embeddable into Y. The first question is whether or not there are different embeddings of X into Y. Many unknotting theorems, like those of Jordan-Schonflies, Brown, Zeeman etc., are known as basic theorems of topology. In our case, if G is a forest (a disjoint union of trees) then it is clear that any two embeddings of G into R3 are ambient isotopic. Conversely if G is not a forest then G contains a cycle and hence there are many different embeddings of G up to ambient isotopy because the cycle can be knotted in various ways. In order to generalize this observation we make the following definition. Let (i) be one of (1)-(7). A graph G is called unique up to (i)-equivalence if any two embeddings of G into R3 are (i)-equivalent. In [38] the author proved the following results. Theorem 1.3.

For a graph G, the following conditions are mutually equivalent:

(i) G is unique up to ambient isotopy. (ii) G is unique up to cobordism. (iii) G is a forest.

(iv) G does not contain any subdivision of the loop (Fig. 1.8 (a)). Theorem 1.4. For a graph G, the following conditions are mutually equivalent: (v) G is unique up to isotopy. (vi) G is unique up to I-equivalence. (vii) G is unique up to edge-homotopy. (viii) There is a vertex v of G such that the maximal subgraph of G that does not contain the vertex v is a forest. (ix) G does not contain any subdivision of the graphs (b), (c) and (d) of Fig. 1.8.

Theorem 1.5. For a graph G, the following conditions are mutually equivalent: (x) G is unique up to vertex-homotopy. (xi) G is unique up to homology. (xii) G is a planar graph which does not contain disjoint cycles. (xiii) G does not contain any subdivision of the graphs (b), (e) and (f) of Fig. 1.8.

202 KOUKI TANIYAMA

0 00 .^. a

(e)

(fl

Fig. 1.8 These results show how intrinsic properties of a graph affect extrinsic properties of the spatial embeddings of the graph. Next we state the homology classification of spatial graphs that is a first step towards the ambient isotopy classification of spatial graphs. Theorem 1.6. [40]

Two elements f and g of SE(G) are homologous if and only

if £(f) = G(g) • Now we explain the Wu invariant .£(f ). In 1965 W. T. Wu defined an invariant of embeddings of a complex into a Euclidean space in a general dimensional setting [46], see also [47]. However, as far as the author know , little was known in our dimensional setting (1,3). Let X be a topological space and C2 (X) = {(x,y) E X x X I x # y}. Namely C2(X) is the configuration space of ordered two points on X. Let a : C2(X) -+ C2(X) be an involution defined by o(x,y) = (y , x). Suppose that an elment f of SE(G) is given . Then we have a map (f x f) : C2(G) --* C2(R3) that maps (x, y) to (f (x), f (y)). We note that the injectivity off and x # y assure f (x) # f (y). Then (f x f) induces a homomorphism (f x f )g : H2(C2( R3), a) -i H2(C2(G), a) where H2(C2(X), a) is the second cohomology group of the subchain complex Ai+i(C2(X ), a) a-+' Ai(C2(X ), a) 6'' Ai-i(C2(X ), a) .. . of the singular chain complex of the space C2(X) ... -+ Ai+i(C2(X)) '+-'+ Ai(C2(X)) -i Ai-i(C2(X)) -+ ... defined by Ai(C2(X), a) = {a E Ai(C2(X)) I a( a) = -a} and Si the restriction of 8i to Ai(C2(X),a). We can see easily that the pair (C2(R3), a) equivariantly retracts to the pair (S2, the antipodal map) where S2 is the unit sphere in R3. Therefore we have H2 (C2(R3), a) = H2(S2, the antipodal map) = Z = (t) where t is a fixed generator of the infinite cyclic group . Therefore the map (f x f )p is determined by the image of t, (f x f)p(t). The Wu invariant of f, denoted by C(f), is just

ON SPATIAL GRAPHS 203

defined by £(f) = (f x f)g(t). See [40] for more details. Summarizing the above construction, we have that the Wu invariant comes from the Gauss map from C2(G) to S2 that maps (x, y) to (f (x) - f (y))/II f (x) - f(y) II. For example let L be the disjoint union of two circles. Then we have that C2(L) is the disjoint union of two tori and two (torus minus ( 1,1)-curve)s. Since a ( torus minus ( 1, 1)-curve) is a homotopy circle and has vanishing second cohomology we can forget it. Moreover owing to the action v , we can forget one of the two tori . (For a rough intuitive understanding, we can forget the action a and may consider the usual cohomology.) Thus in this case the Wu invariant is two times of the map degree of the Gauss map from a torus to S2 and is two times of the linking number , see for example [28]. (The two times come from the action on S2, namely we count the preimages of the Gauss map both at the North Pole and at the South Pole.) Thus the Wu invariant is a generalization of the linking number and calculated from any regular diagram of the spatial graph. See section 2 of [40] for more detail . It is notable that the invariance of the Wu invariant under the (extended ) Reidemeister. move V just corresponds to the coboundary relation in C2(G). See Fig. 1 .9. A prototype of this phenomenon is that the invariance of the linking number under the Reidemeister move II just corresponds to the invariance of the map degree . See Fig. 1.10.

ei

V I

Fig. 1.9

204

KOUKI TANIYAMA

III

Fig. 1.10 Now we would like to give a rough sketch proof of Theorem 1.6. The `only if' part is not difficult and we only show the `if ' part . Suppose that £ (f) = £(g). We choose an arbitrary sequence of crossing changes from f to g . If we leave a circle (we call it a Hopf circle) at each crossing change as illustrated in Fig. 1.11, then we have that the Hopf circles represents 2x for some 2-cochain (= 2-cocycle) x of (C2(G), Q). By the assumption we have that 2x is a 2-coboundary. Then we use the following lemma. Lemma 1 .7. [40] 2-torsion elements.

H2(C2(G), a) is a free abelian group . In particular it has no

Therefore we have that x itself is a 2-coboundary. Then the coboundary is realized by attaching handles as illustrated in Fig. 1.12.

Fig. 1.11

ON SPATIAL GRAPHS 205

I

Fig. 1.12 This is just the point where an algebraic condition is realized by a geometric object. We remark that Lemma 1.7 is not trivial but follows from a careful observation of the structure of the C2(G), see [40]. This completes the sketch proof. Thus the Wu invariant is the most fundamental invariant of spatial graphs. We note here that another homology classification is shown by Yasuhara (the designer of the symbol mark of KNOTS '96) as follows. A disk/band surface Ff of f E SE(G) is a compact oriented surface in R3 that contains f (G) in its interior such that f (G) is a strong deformation retract of Ff. This notion is introduced in [12]. A Seifert linking form SF, : H1(Ff) x H1(Ff) -+ Z of Ff is a bilinear form defined by SFf (a, /3) = ek(a, /3+) where Bk denotes the linking number and 3+ is a slight push-off of ,Q along the positive normal direction of Ff. Let F, be a disk/band surface of g E SE(G). We say that the Seifert linking forms SFf and SF, are equivalent if there is a homeomorphism h : SFf -' SF, such that h o f = g and SFf(a,/3) = SFg(h(a),h(/3)) for any a, 3 E Hl(Ff). Theorem 1 . 8. [51] Let f and g be elements of SE(G). Then f and g are homologous if and only if there are disk/band surfaces Ff and F9 off and g respectively such that the Seifert linking forms SFf and SF, are equivalent.

We note that a disk/band surface of f is not unique up to ambient isotopy in general. However two disk/band surfaces of f are transformed into each other by certain twisting operations. Then Yasuhara showed that the latter condition of Theorem 1. 8 is a calculable one. See [51]. Examples. Let L be a disjoint union of circles. Then we have H2(C2(L), u) = Z as noted above and we have that Fig. 1.13 is a complete list of the homology classes of spatial embeddings of L. • • • • • •

Fig. 1.13

206

KOUKI TANIYAMA

Let K. be the complete graph over n vertices. Then we have that H2(C2(K5), a) Z and Fig. 1.14 is a complete list of the homology classes of spatial embeddings of K5.

000

(9(9(a fl

f I fo

• • •

f2

Fig. 1.14 Let Km,n be the complete bipartite graph over m + n vertices . Then we have that H2 (C2(K3,3), v) = Z and Fig . 1.15 is a complete list of the homology classes of spatial embeddings of K3,3.

• • •

• • •

1@ 1

90 91 Fig. 1.15

We remark that in the last two examples there are no disjoint circles in G. Thus we have that the Wu invariant is not determined by the linking numbers in general. However in [33] it is shown that the Wu invariant (and hence the homology class) of a spatial embedding of a planar graph (a graph that has an embedding into the plane R2 ) is completely determined by the linking numbers of the two component links contained in the spatial graph. We also note that the homology classes are determined in the concrete for certain classes of graphs containing all complete graphs and all complete bipartite graphs in [35].

ON SPATIAL GRAPHS 207

Part 2. Knots in a spatial graph The following is a joint work with T. Motohashi , Y. Ohyama and A. Yasuhara and the detail will appear in [41], [21], [24] and [43]. We will study the dependence and the independence of the knot types of the subgraphs in a spatial graph . This study is motivated by the following two pioneering works. The first one is by Conway and Gordon. Theorem 2 .1. [4] (1) For any element f of SE(K6), f (K6) contains a nonsplittable link. In fact the sum of the linking numbers of all two-component links in f (K6) is odd. (2) For any element f of SE(K7), f (K7) contains a nontrivial knot. In fact the sum of the Arf invariants of all knots in f (K7) containing all vertices of f (K7) is odd. See Fig. 2.1 for examples. The spatial K6 in Fig. 2.1 contains just one nontrivial link and the spatial K7 in Fig. 2.1 contains just one nontrivial knot. See [25] for related topics.

Fig. 2.1 The above theorem implies that the knot types of the subgraphs in a spatial graph are mutually dependent in general.

208 KOUKI TANIYAMA We refer the reader to [29], [15] and [26] for related results and [42] for a higher dimensional analogue.

On the contrary, Kinoshita showed the following. Theorem 2.2. [13]

Any given n(n - 1)/2 knot types are realized by a spatial embedding of On at once where On is the graph on two vertices and n edges joining them.

Fig. 2.2 is an example of a spatial theta curve 0 = 03 that realizes the knot types 31i 41 and 51 in the knot table simultaneously.

Fig. 2.2 The above theorem implies that the knot types of a spatial graph is to some extent mutually independent. We note that this result has been extended to certain graphs in [49] and a higher dimensional analogue is shown in [50]. Thus we are interested in the dependence and the independence of all knot types in a spatial graph. The following is a formulation of this problem, cf. [14].

Let 11(G) be the set of all subgraphs of G. Let F be a subset of 1(G). Suppose that for each ry E IF, 0, E SE(-y) is given. We say that a set of spatial embeddings {07 E SE(ry) I y E F} is realizable up to (i)- equivalence if there is an element f of SE(G) such that the restriction map f 17 is (i)-equivalent to 0, for all ry E I where (i)-equivalence is one of the seven equivalence relations in Part 1. Thus we are interested in whether or not a given set of spatial embeddings of subgraphs of a graph is realizable (mainly up to ambient isotopy) or not. As a rough answer to this problem, we give a complete answer to the realizability up to homology as follows.

ON SPATIAL GRAPHS 209

Let 'G = {G(f) E H2(C2(G), a) I f E SE(G)}. For y E I', we see that the inclusion -y C G induces the inclusion C2(y) C C2(G) and this, in turn , induces the homomorphism h..,.: H2(C2(G), a) -, H2(C2(y), a ). Then we have the following result. Theorem 2 .3. [41] The set {07 E SE(y) I -y E F} is realizable up to homology if and only if there is an element x of ZG such that h.y(x) = G(ary) for all y E r. The proof easily follows from Theorem 1.6. The latter condition of Theorem 2.3 is a calculable one. We show a typical example here. Example. Let G be a disjoint union of a theta curve and a circle. Then under the orientations illustrated in Fig. 2.3 we have that the linking numbers satisfy Pk(f (a), f (d)) + Ek(f (b), f (d)) + 2k(f (c), f (d)) = 0 for any f E SE(G). This follows directly from the definition of the linking number . This is a prototype of the dependence of knot types in a spatial graph.

Q

d

Fig. 2.3 Now we consider the Vassiliev type (or Vassiliev-Gusarov type or finite type) invariants of the subgraphs in a spatial graph . We refer the reader to [44], [2], [1], [34] [9], [17], [23] etc. for Vassiliev type invariants. Let R be a commutative ring with unit 1. Let v : SE(G) -* R be an ambient isotopy invariant . Namely v is a map from SE(G) to R such that v(f) = v(g) for any ambient isotopic embeddings f and g. Let SEi(G) be the set of all i-singular embeddings of G into R3 where an i-singular embedding is a continuous map whose multiple points are exactly i "transversal " double points of edges. Such a double point is called a crossing vertex. Then, under a given edge orientation of G, v is uniquely extended to an ambient isotopy invariant vi : SEi (G) --' R by the recursive formula vi(fo) = vi-1(f+) - vi-1(f-)

where fo, f+ and f- are related as illustrated in Fig. 2.4.

210

KOUKI TANIYAMA

fo

f+

f

Fig. 2.4 Here we only consider ambient isotopies that preserve a small flat plane at each crossing vertex . Let n be a natural number . We say that v is an (order < n) Vassiliev type invariant if v„+1 : SE„}1 (G) -* R is a zero map. (Then we have that vin is a zero map for m > n + 1.) It is easy to check that the definition of (order < n) Vassiliev type invariant is independent of the choice of the edge orientations. Now we fix a ring R and a natural number n . Suppose that for each -y E Sl(G), an (order < n) Vassiliev type invariant v7 : SE(-y) --> R is given. (Possibly v., is a zero map.) Let w : 11(G) R be a map. Then we define an ambient isotopy invariant v = v({v7 },w) : SE(G) --> R by v(f) _

w('Y)v7(f

I7)•

7E0(G)

Then it is elementary to check the following proposition. Proposition 2.4. [24]

v is an (order < n) Vassiliev type invariant.

An i-configuration on G is a pairing of 2i points on the edges of G. A realization of an i-configuration is an element of SE;(G) whose crossing vertices correspond to the pairing. Then we have the following three parallel theorems. Theorem 2.5. [24] The following conditions are mutually equivalent. (1) v(f) = v(g) for any f and g in SE(G),

(2) vi(f) = 0 for any f in SE2(G) with 1 < i < n, (3) for any i-configuration c on G with 1 < i < n, there is a realization ff E SE1(G) of c such that vi(ff) = 0. Theorem 2.6. [24] The following conditions are mutually equivalent. (1) v(f) = v(g) for any edge-homotopic embeddings f and g in SE(G), (2) vi(f) = 0 for any f in SEI(G) with 1 < i < n that has at least one "selfcrossing vertex", (3) for any i-configuration c on G with 1 _< i _< n that has at least one pair of points on the same edge, there is a realization fc E SEI(G) of c such that vi(ff) = 0.

ON SPATIAL GRAPHS 211

Theorem 2.7. [24] The following conditions are mutually equivalent. (1) v(f) = v(g) for any vertex-homotopic embeddings f and g in SE(G), (2) vi(f) = 0 for any f in SE;(G) with 1 < i < n that has at least one "crossing vertex of adjacent edges", (3) for any i-configuration c on G with 1 < i < n that has at least one pair of points on the adjacent edges, there is a realization fc E SEi(G) of c such that vi(fc) = 0.

Remark. (1) Since the mod 2 linking number (resp. the Arf invariant) is an (order < 1) (resp. (order < 2)) Vassiliev type invariant, Theorem 2.1 can be considered as an example of Theorem 2.5 where the ring R = Z/2Z. (2) Since the second coefficient of the Conway polynomial of a knot is an (order < 2) Vassiliev type invariant, the edge-homotopy (resp. vertex-homotopy) invariants defined in [39] is are examples of Theorem 2.6 (resp. Theorem 2.7). Here we give further examples of above theorems . An i-cycle is a graph with i vertices and i edges that is homeomorphic to a circle . An i-cycle of a graph G is a subgraph of G that is an i-cycle. Let G,,, be the graph obtained from an m-cycle by doubling each edge of it . Let dl, el, d2, e2, • • • , d,,,, e,,, be the edges of G,,, such that d; and e; form a pair of multiple edges of Gm for each i. Let w : S2 (Gm) -* R be the map defined by

w(ry) _

1 , d,,,} if -y is an m-cycle that contains even number of {dl, -1 if -y is an m-cycle that contains odd number of {d1i • • • , dm} 0 otherwise.

We fix an (order < n) Vassiliev type R invariant of knots VK. For 7 E 1(Gm), let

1

_ J vK if ry is an m-cycle 0 otherwise.

7J7-

Then we have the following theorem. Theorem 2.8. [24]

If n < m/2 then v = v({v7}, w) is the zero map.

The proof follows by checking the condition (3) of Theorem 2.5. By the assumption n < m/2 we have that at least one pair of multiple edges of G,,, have none of the 2i points of the configuration . Then the i-singular embedding that maps the pair of multiple edges in a parallel-fashion is the required one. As a corollary we have that the knots in a spatial embedding of G5 are dependent. In fact the second coefficients of the Conway polynomial of them satisfy a certain condition. Let C(G) c S1(G) be the set of all i-cycles of G and C(G) = U;C;(G). A graph G is called adaptable if any set of embeddings {07 E SE('y) I ry E C( G)} is realizable

212

KOUKI TANIYAMA

up to ambient isotopy. Then Theorem 2.2 states that B„ is an adaptable graph. Then we have the following corollary.

Corollary 2.9. [241

The graph G5 is not adaptable.

We note that the graph G5 is the first planar graph which is known to be nonadaptable . We will soon discuss it for non -planar graphs. If we restrict our attension to certain class of embeddings we can deduce the conclusion for higher order Vassiliev type invariants . The following is an example. We say that f E SE( G,,,) is edge-homotopically Brannan if the restriction off to G,n - {di, ei} is edge- homotopic to a planar embedding for each i where G,n - {di, ei} denotes the subgraph of G,n obtained by removing the interiors of the edges di and ei. Theorem 2.10. [241 Suppose that n = m - 1. Let f,g E SE(G,n) be edgehomotopically Brannan embeddings . If f is edge-homotopic to g then v (f) = v(g).

The proof follows from a certain recursive calculation . As an example we have that the spatial embedding illustrated in Fig . 2.5 is not edge-homotopic to a planar embedding.

Fig. 2.5 The next two theorems are examples of a complete answer to the realizability up to ambient isotopy of the knot types in a spatial embedding of G where G is K5 and K3,3,. Let a2(J) be the second coefficient of the Conway polynomial of a knot J. Theorem 2 .11. [431

A set of spatial embeddings {0ry E SE(-y) I ry E C(K5)} is

ON SPATIAL GRAPHS 213 realizable up to ambient isotopy if and only if there is an integer m such that m(m - 1) [^ (^717)) - (^7(y)) = a2 a2

2

7EC5(K5) 7EC4(K5)

Theorem 2. 12. [43] A set of spatial embeddings {¢7 E SE(y) I y E C(K3,3)} is realizable up to ambient isotopy if and only if there is an integer m such that a2(0,(-t)) -

E

a2

(07(y))

= m(m - 1).

2

7EC5(K3,3) 7EC4(K3,3)

i-1 I m E Z} is a proper subset of Z, we have that K5 and K3,3 Since the set are not adaptable . Then by the Kuratowski graph planarity criterion [16] we have the following theorem. Theorem 2 . 13. [21]

A nonplanar graph is not adaptable.

The proofs of `only if' parts of Theorem 2.11 (resp. Theorem 2.12) follows from the following three facts 2.14, 2.15 and 2.16 (resp. 2.17, 2.18 and 2.19). Fact 2.14. ([39])

For f E SE(K5), let vK5(f)=

a2(f(y))- a2 > 7EC5( K5) 7EC4(K5)

(f (-Y))

Then vK5 is a vertex-homotopy invariant. Fact 2.15. Vertex- homotopy coincides with homology for K5. Hence the embeddings { fm I m E Z} in Fig. 1.14 form a complete list of the vertex-homotopy classes of SE(K5). Fact 2.16.

vK5(fm) = m 2-1

Fact 2.17.

For f E SE(K3,3), let vK3,3 (f)= E a2(f(y))- a2(f(Y)) 7EC6( K3,3) 7EC4(K3,3)

Then vK3,3 is a vertex-homotopy invariant. Fact 2.18. Vertex- homotopy coincides with homology for K3,3. Hence the embeddings {9m I m E Z} in Fig. 1 . 15 form a complete list of the vertex-homotopy classes of SE (K3,3).

214 KOUKI TANIYAMA

Fact 2.19. v K3.3 1(9 m ) _ m m2-1 We refer the reader to [21] for the proofs of these facts. The proofs of `if' parts of Theorem 2.11 and Theorem 2.12 require certain realization technics of knots in a spatial graph. Following the technics in [13] and [49] Yasuhara established an excellent realization technic in [52]. The following example will suggest the whole technic. We intend to construct a spatial K4 such that the knots in it are all trivial knots except just one trefoil knot. A local knot construction is not sufficient for the purpose. In fact the spatial K4 in Fig. 2.6 contains four trefoil knots. Fig. 2.7 serves a better construction. However it still contains two trefoil knots. An answer is given in Fig. 2.8. Roughly speaking, Fig. 2.6 represents a "one-string interaction" while Fig. 2.7 and Fig. 2.8 represent a "two-string interaction" and a "three-string interaction" respectively. The realization technic is based on the following theorem that every knot is a result of three string interactions. Theorem 2.20. [22] ambient isotopies.

Fig.

Knots are transformed into each other by "delta moves" and

2.6

Fig.

2.7

Fig.

2.8

A delta move is a local move as illustrated in Fig. 2.9. We note that in Fig. 2.9 if we forget one string then the move is identical.

Fig. 2.9

ON SPATIAL GRAPHS 215

The proofs of `if' parts of Theorem 2.11 and Theorem 2.12 require the following remarkable theorem by Habiro. Theorem 2 .21. [10] Two knots Jl and J2 are transformed into each other by "clasp-pass moves" and ambient isotopies if and only if a2(Jl) = a2(J2). A clasp-pass move is a local move as illustrated in Fig. 2.10. We note that this move is a four-string interaction.

I I

Fig. 2.10 The details of the proofs of `if' parts of Theorem 2.11 and Theorem 2.12 will appear in [43]. Using the realization technic we have certain results. As an example we have the following theorem that is a converse of Theorem 2.1 (1).

Theorem 2 .22. [43] Let F C Sl(K6) be the set of disjoint cycles of K6. Then a set of spatial embeddings {q,y E SE(ry) I ry E P} is realizable up to ambient isotopy if and only if Y tk(0ry(ry)) __ 1 (mod 2). -fEr

Proofs and other examples will appear in [43].

REFERENCES 1. D. Bar-Natan : On the Vassiliev invariants, Topology, 34, 423-472, 1995. 2. J. Birman and X .-S. Lin: Knot polynomials and Vassiliev 's invariants , Invent. Math., 111, 225-270, 1993. 3. A. Casson : Link cobordism and Milnor 's invariant, Bull. London Math . Soc., 7, 39-40, 1975. 4. J. H. Conway and C . McA. Gordon: Knots and links in spatial graphs, J. Graph Thory, 7, 445-453, 1983.

5. K. Endo: Isotopy invariant of spatial graphs, Kobe. J. Math., 12, 123-137, 1995. 6. R. Fox and J. Milnor : Singularities of 2-spheres in the 4-sphere , Osaka Math. J., 3, 257-267, 1966.

216 KOUKI TANIYAMA

7. C. H. Giffen: Link concordance implies link homotopy, Math. Scand., 45, 243-254, 1979. 8. D. L. Goldsmith: Concordance implies homotopy for classical links in M3, Comment. Math. Helvetici, 54, 347-355, 1979. 9. M. Gusarov: A new form of the Conway-Jones polynomial of oriented links, Nauchn. Sem. Len. Otdel. Mat. Inst. Steklov, 193, 4-9, 1991. 10. K. Habiro : Clasp-pass moves on knots , in preparation. 11. H. Inaba and T. Soma: On spatial graphs isotopic to planar embeddings, Proceedings of Knots 96, (S. Suzuki ed.), World Scientific Publ. Co., 1-22, 1997. 12. L. Kauffman, J. Simon, K. Wolcott and P. Zhao: Invariants of theta-curves and other graphs in 3-space, Topology Appl., 49, 193-216, 1993. 13. S. Kinoshita: On 9,; curves in R3 and their constituent knots, in Topology and Computer Science edited by S. Suzuki, Kinokuniya, 211-216, 1987. 14. S. Kinoshita: On spatial bipartite Km,,,'s and their constituent K2,,,'s, Kobe J. Math., 8, 41-46, 1991. 15. T. Kohara and S. Suzuki: Some remarks on knots and links in spatial graphs, Knots 90, ed. A. Kawauchi, Walter de Gruyter, Berlin-New York, 435-445, 1992. 16. C. Kuratowski: Sur le probleme des courbes gauches en topologie, Fund. Math., 15, 271-283, 1930. 17. X. -S. Lin: Finite type link invariants of 3-manifolds, Topology, 33, 45-71, 1994. 18. J. Milnor: Link groups, Ann. Math., 59, 177-195, 1954. 19. J. Milnor: Isotopy of links, Algebraic Geometry and Topology; A Symposium in honor of S. Lefshetz, ed. Fox, Spencer and Tucker, Princeton University Press, 280-306, 1957. 20. K. Miyazaki: The theta-curve cobordism group is not abelian, Tokyo J. Math., 17, 165-169, 1994. 21. T. Motohashi and K. Taniyama: Delta unknotting operation and vetex homotopy of graphs in R3, Proceedings of Knots 96, (S. Suzuki ed.), World Scientific Publ. Co., 185-200, 1997. 22. H. Murakami and Y. Nakanishi : On a certain move generating link-homology, Math. Ann., 284, 75-89, 1989. 23. Y. Ohyama: Vassiliev invariants and similarity of knots, Proc. Amer. Math. Soc., 123, 287-291. 24. Y. Ohyama and K. Taniyama: Vassiliev type invariants of knots in a spatial graph, in preparation. 25. T. Otsuki: Knots and links in certain spatial complete graphs, J. Combin. Theory Ser. B., 68, 23-35, 1996. 26. N. Robertson, P. Seymour and R. Thomas: Linkless embeddings of graphs in 3-space, Bull. Amer. Math. Soc., 28, 84- 89, 1993. 27. D. Rolfsen: Isotopy of links in codimension two, J. Indian Math. Soc., 36, 263-278, 1972. 28. D. Rolfsen: Knots and Links, Math. Lecture Series 7, Publish or Perish Inc., Berkeley, 1976. 29. H. Sachs: On spatial analogue of Kuratowski 's theorem on planar graphs, Lecuture Notes in Math . 1018, Springer-Verlag, Berlin-Heidelberg, 230-241, 1983. 30. J. Simon and K. Wolcott: Minimally knotted graphs in S3, Topology Appl., 37, 163-180, 1990. 31. T. Soma: Spatial-graph isotopy for trivalent graphs and minimally knotted embeddings, Topology Appl., 73, 23-41, 1996. 32. T. Soma: Spatial-graph isotopy and the rearrangement theorem, preprint. 33. T. Soma, H. Sugai and A. Yasuhara: Disk /band surfaces of spatial graphs, to appear in Tokyo J. Math.. 34. T. Stanford : Finite-type invariants of knots, links and graphs, Topology, 35, 1027-1050, 1996. 35. M. Suzuki : Classification of the spatial- graph homology classes of a complete graph, (in Japanese) Master Thesis at Tokyo Denki Univ., 1996. 36. K. Taniyama: Cobordism of theta curves in S3, Math. Proc. Camb. Phil. Soc., 113, 97-106,

ON SPATIAL GRAPHS 217 1993. 37. K. Taniyama: On embeddings of a graph into R3, Comtemporary Math. 164, 239-246, 1994. 38. K. Taniyama: Cobordidim, homotopy and homology of graphs in R3, Topology, 33, 509-523, 1994. 39. K. Taniyama: Link homotopy invariants of graphs in R3, Rev. Mat. Univ. Complut. Madrid, 7, 129-144, 1994. 40. K. Taniyama: Homology classification of spatial embeddings of a graph, Topology Appl., 65, 205-228, 1995. 41. K. Taniyama: Knotted subgraphs in a spatial graph, in preparation. 42. K. Taniyama: Higher dimensional links in a simplicial complex embedded in a Euclidean space, in preparation. 43. K. Taniyama and A. Yasuhara: Realization of knots and links in a spatial graph, in preparation. 44. V. Vassiliev: Cohomology of knot spaces, Theory of singurarities and its applications (V. Arnold ed.), Advances in Soviet Mathematics, 1, AMS Providence, RI, 1990. 45. K. Wolcott: The knotting of theta curves and other graphs in S3, Geometry and Topology (C. McCrory and T. Shifrin ed.), Marcel Dekker, 325-346, 1987. 46. W. T. Wu: On the isotopy of complexes in a eucledian space I, Science Sinica, 9, 21-46, 1960. 47. W. T. Wu: A theory of imbedding, immersion, and isotopy of polytopes in a euclidean space, Science Press, Peking, 1965. 48. Y.Q. Wu: Minimally knotted embeddings of planar graphs, Math. Z., 214, 653-658, 1993. 49. M. Yamamoto: Knots in spatial embeddings of the complete graph on four vertices, Topology Appl., 36, 291-298, 1990. 50. A. Yasuhara: On higher dimensional 9-curves, Kobe J. Math., 8, 191-196, 1991. 51. A. Yasuhara: Disk/band surface and spatial-graph homology, Topology Appl., 69, 173-191, 1996. 52. A. Yasuhaxa: Delta-unknotting operation and adaptability of certain graphs, Proceedings of Knots 96, (S. Suzuki ed.), World Scientific Publ. Co., 115-121, 1997. DEPARTMENT OF MATHEMATICS , COLLEGE OF ARTS AND SCIENCES , TOKYO WOMAN'S CHRISTIAN UNIVERSITY , ZEMPUKUJI 2-6-1, SUGINAMIKU , TOKYO, 167, JAPAN

E-mail address : [email protected]

This page is intentionally left blank

Lectures at Knots 96 edited by Shin'ichi Suzuki ©1997 World Scientific Publishing Co. pp. 219-234

ENERGY AND LENGTH OF KNOTS GREGORY BUCK AND JONATHAN SIMON

1. INTRODUCTION

Knots are idealized 1-dimensional loops that tangle themselves in 3-space. They have been studied, for more than 100 years, primarily as abstract mathematical objects even though the original interest in the subject seems to be based in physics. There is now interest in re-investing the mathematical abstractions with physical-like properties such as thickness [L] [Si] [DEJ] [St] [LSDR] or self- repelling energy [Fu] [01-04] [BS] [B01-2] [Si2] [Lo] [DEJ]. The motivation is partly chemistry/biology and partly the lure of the mathematics itself. By modeling knots with physical properties, new invariants of knots can be defined and there is hope for better understanding of how knotted and tangled filaments (simple loops, links of several loops, or tangled spatial graphs) behave in real systems such as DNA gel electrophoresis [DSKC] [DC] [DC2] [Ketc] [SSC] This paper considers and relates several notions of energy and other measures of geometric complexity for knots. Some of the results described here were announced in [B] and [Si4]. The discussion of Theorem 4 given here is a summary; the detailed proof is presented in [BS2]. We may hope to show that various energies are related to each other, e.g. by inequalities saying that one energy is less than some function of another, and that they also are related to intuitive geometric measures of knot complexity such as compaction (a long knot contained in a small ball) or average crossing number. Another idea, thickness (or, rather, its reciprocal, the rope-length of a knot) may be viewed either as an energy or as a naive geometric measure of complexity; in any case, it also may be related to the others by various inequalities. The general pattern of the theorems is that knots which seem complicated according to one measure, also must be complicated according to others. This leads us to believe that while the various energies etc. are defined differently (and are different), they all are capturing, The authors are supported in part by NSF Grant #DMS9407132 and #DMS9420088. Many thanks are due to the organizers and ot4her hosts of the MSJ International Conference on Knots for their mathematical enthusiasm and warm hospitality.

219

220 GREGORY BUCK AND JONATHAN SIMON

at least approximately, the same intuitive idea of one knot being more complicated than another . One theorem common to the various energy functions is that there are only finitely many knot types that can be realized by knots below a given energy level, and that all knots below some level are unknotted. There also are interesting questions about existence, uniqueness (and/or rigidity) of minimum-energy conformations. The situation is clearer for polygonal knots (minima exist for each knot type, for each number of segments [Si2]) than for smooth knots. But even here the questions of uniqueness or rigidity are open . We know of several examples , found by computer , of apparent distinct local minima for a given knot type under the polygonal energy UMD. Ying-Qing Wu has found [Wu] a tangled unknotted polygon that appears also to be a local minimum. (This was suspected, but still surprizing since the energy has been quite successful at untangling unknots.) For smooth knots, M. Freedman et al [FHW] showed that minima exist for prime knots under O'Hara's energy, and there is a widely believed conjecture that minimum energy conformations do not exist for smooth composite knots; the problem appears to be that the limiting energy is the sum of the energies of the factors , but that as a sequence of knots tries to realize that limit, one factor of the knot gets pulled tight to become singular in the limit. So far we do not know of a (discretized) "smooth" tangled unknot that evolves to a nontrivial local minimum under the gradient of the (discretized) energy Eo [KS]. For the energy EL (see §2; equivalently, for thickness), there are theorems asserting that minima exist , though perhaps with loss of some smoothness. We do not believe that minima will be unique, and they may even be flexible. For non- uniqueness, consider a composition of knots K1# • . #K,,: Topologically, one can permute the factors; however, when the knots are thickened , it appears impossible to exchange factors, so we may have on the order of n! distinct local minima. (But are there distinct global minima?) It seems clear that minimum energy (EL) conformations for links will not be rigid , and this seems to suggest that the same will happen for knots . For example , a standard Hopf link, when thickned , would allow one component to rotate on itself while leaving the other fixed. Thick knots may exhibit the same kind of energy preserving flexibility. For example , a standard torus knot admits a continuous non- rigid "screw symmetry", though we do not believe that the thickest possible position for a particular torus knot would be its standard torus position. If we pass to three -component links (e.g. 2i#2i ) , we see that there appear to be continuous families of minimum energy (EL)conformations that are not geometrically congruent to each other . There are many open questions along these lines. Several computer programs have been developed [Br] [Hg] [Hu] [03] [Wu] to help

ENERGY AND LENGTH OF KNOTS 221

visualize and manipulate knots, and to see how various energy functions behave. These are important tools for discovering and testing conjectures, and such projects invariably require the authors to confront substantial algorithmic and other theoretical problems. The fact that one type of knot seems more tight or more complicated than another manifests itself in the laboratory. When DNA loops (of the same length) are tied into different types of knots, the loops move with different velocities in gelelectrophoresis experiments. Initially, it was observed that the crossing numbers of the knot types largely determine relative velocity; that is, five crossing knots move faster in the gel than four crossing knots and so on [DSKC]. One reason this observation is surprising is that the crossing number of a knot type is a property of a special `Ideal" conformation of the knot, whereas the loops in the gel are moving and bending in many ways, assuming many conformations. Nevertheless, the tendency for velocity and minimum crossing number to be related is well-documented. However, subsequent studies have shown that while the qualitative correlation between crossing number and gel velocity is excellent for knot-types with few crossings, when one gets to seven and eight crossings, there are eight crossing knot types that move more slowly than most seven crossing types. Also there are observed differences between different knot types with the same number of crossings; for example, twist knots move faster than torus knots [DC2]. Knot energies provide an extra level of discrimination that seems to be a better predictor of gel velocity than crossing number ; in particular , the energies (all of them agree!) successfully predict which of the two five-crossing knots is faster in the gel, and which eight-crossing knots should be the slow ones. Here are some observations of relative gel velocities that are consistent with (so could be predicted from) observed minimum energy values (see [Si3] for table).

• All knots are generally ordered by crossing number, especially through six crossings, but less distinctly as number grows [DSKC]. • Twist knots are slightly faster than (2, n) torus knots with the same crossing number [DC].

• The granny knot is slower in the gel than prime 6-crossing knots [Ketc]. • Homologous family of twist knots are ordered by crossing number [Ketc]. • Homologous family of (2, n) torus knots are ordered by crossing number [SSC]. • The (3,4)-torus knot 819 is slower than almost all 7-crossing knots [Co]. The list of knot-types, in order of observed gel velocity, and also in order of estimated minimum energy (using any of the energies Eo, EN, EL, or UMD described below) is: o trefoil knot 31 o figure-8 knot 41

222 GREGORY BUCK AND JONATHAN SIMON

o (2,5) torus knot 51 o twist knot 52 o granny knot 31#31 (square knot appears to have same min. energies) o prime knots 61, 62, 63 (no reports that these were separated by gel) o prime knots 71 - 77 in that order, with 819 and 820 interspersed. The rankings by various energies are identical through 6 crossings (through 7 for all except perhaps EL, work in progress). The parallels become less evident when one gets to 8-9 crossings . So there is evidence that the various energies are related to each other, but also evidence that the relationships are not simple. It was pointed out [Su] that recent experiments [Le] indicate that under certain conditions, the relative gel velocities of different knot types can be affected to the point that some are re-ordered. So we should realize that any attempt to explain real physical behavior in terms of idealized invariants of knot types cannot succeed completely. Nevertheless, the energy approach seems to provide considerable insight.

2. SEVERAL NOTIONS OF "ENERGY" FOR KNOTS

Many people have speculated informally about what would happen if a knotted string were somehow given an electric charge and allowed to repel itself. Seminal papers by S. Fukuhara [Fu] and by J. O'Hara [01] have helped lead to a large body of work involving different energy functions for smooth or polygonal knots. When we try to make the thought-experiment ("charge the string and let go") mathematically precise, the most naive definition of a potential energy function for a charged knot has mathematically unpleasant properties. So the functions defined by various investigators have departed from what seems like "true" physics, in order to obtain functions that are finite and prevent curves from changing knot type. In addition to the functionals discussed below, others have been proposed and studied ([02] [DEJ] [KS]). The energies that have been studied for smooth knots generally are of the following kind: For each pair of points x, y on the curve K, one computes a number that depends on [some power, usually 2, of] the reciprocal distance between the points, and then integrates over K x K. To prevent a near-neighbor effect that would make the integral infinite, one needs to regularize, either by subtracting something equally divergent (e.g. the same quantity computed for points on a standard circle used to parametrize K) or by multiplying by a factor that has a zero of the appropriate order when points approach each other along the curve. The O'Hara energy [01] [FHW] used the first approach, but one also can define this [KS] using the second method. The "normal energy" (sometimes referred to as the "projection energy" [B] [BO1], and also the "symmetric energy" defined below, use the multiplicative

ENERGY AND LENGTH OF KNOTS 223

factor approach. Each of these functionals, as well as the energy for polygons UMD [Si2], satisfies the following properties (among others of course): 1. E(K) is invariant under Euclidean isometry and change of scale; 2. 0 < E(K) < oo if K is not self- intersecting; 3. For knots of fixed total length (or lengths uniformly bounded away from 0), if parts of a knot are brought together to make the knot self-intersect, the energy tends to +oo. 4. There are only finitely many knot types realized below any given energy level, and below some level, only unknots. Property 1 tells us that E is determined by the shape of a knot, not the size. Properties 2 and 3 tell us that E separates knot-types with infinitely high "potential energy walls". Therefore we might hope to find minimum energy conformations for each knot type; we could think of these as canonical conformations for each type. Property 4 says that energy provides a reasonable taxonomy for knots.

Another kind of "energy" comes from visualizing a knot as actually made of some rope, with a positive thickness. Given a smooth curve K in 3-space, we can associate a number, R(K) that bounds the thickness of a uniform tube that can be placed around K without self-intersection (see definition later). To correct for varying lengths, we either normalize K to have its length equal 1 or define R(K) to be a ratio of radius/length. To get an energy satisfying Properties 1-4 above, we use the reciprocal of thickness, which we may call the rope-length of K. Some theorems about rope-length (which we denote EL) are summarized with the definition in the next section. Notation, Definitions, and Background Results. For most of this paper we will let the conformation space be C2 knots, parametrized by arclength. Since the notions we discuss are independent of parametrization, we often supress the arclength parameter t. Let K be given by the parametrized curve x(t). Then we denote by x, y arbitrary points x(t), y(t) in K and we write i(t) for the derivative. We also use: Pyv = Ix - yj (when x,y are understood, we write only p) =-u rxv - IX-VI

dx = i(t)dt, a line element at x and dx = IdxI.

224

GREGORY BUCK AND JONATHAN SIMON

The simplest attempt at a knot energy that depends on a power of the inverse distance would be: E(a) = if

I dxdy Pa = IJ KxK SxS I7(y) 1 Y(x )

where S is the parameter space. However this quantity diverges for a _> 1, and fails to provide an infinte barrier to self-crossing for a < 1. The naive approach therefore does not work - some modification is required . It is common to take a = 2 to obtain scale invariance and the desired barrier to changing knot-type . However the integral then diverges, so it must still be regularized. With these considerations in mind we define four energy functions , each of which have properties 1-4 above, bound the crossing number, and are independent of parametrization.

The normal energy is (equivalent to [BOl]): IdxxrI2 EN(K)=J J

P2 We sometimes write dx x r as sina, where a is the angle between the chord (y - x) and the tangent direction dx or as cosO, where 0 is the angle between (y - x) and the normal plane to K at x.

The symmetric energy is: Es(K) = f f

Idxxrlldyxrl

JJ p2

The O'Hara energy (also called the Mobius energy or the conformal energy) [01] [BHW] [KS] is: E°(K)=^J P2 Ix 1 2 Ox - yl sl denotes the arclength distance between x and y along K; we may think of a unit-speed parametrization with a circle Sl of some radius as parameter space). We next define the thickness R(K) and rope length EL(K). The thickness is an attempt to make precise the idea of the thickest piece of rope that could be used to follow the shape of K. The definition and results summarized below are taken from [LSDR]. We introduce theorem numbers here for later use. Let N(K; R3) be the normal bundle of K in R3 and let e : N(K; 1R3) -* R3 be the standard projection map. Since we are assuming K is a C2 embedding, we know that for sufficiently small e > 0, the map e is a Cl diffeomorphism of the tube around K x 0 in N(K; R3) of radius a to a neighborhood of K in R3. We define R(K) to be

ENERGY AND LENGTH OF KNOTS 225

the supremum of such E. It may be shown that the following more naive definition is equivalent. For each x E K, let Ny denote the normal plane to K at x; and let D(x, R) denote the solid disk of radius R centered at x contained in N.. Then R(K) = sup{ R : D(x, R) n D(y, R) = 0 Vx # y E K} The rope length or length energy of K is defined as: EL(K) - arclength(K) R(K) Results in [LSDR] on thickness (hence on EL if one normalizes the knot to have length = 1) include the following. Theorem T1.

Maximum curvature of K < R(K).

There actually is a complete characterization of R(K). Define a pair of points (x, y) of K to be critical if the chord vector (x - y) is perpendicular to the tangent at x or perpendicular at y, and let R2(K) be half the minimum of all distances Ix - yI for such pairs. Then we have: Theorem T2. R(K) equals the minimum of the minimum radius of curvature of K and R2(K). In particular, R(K) < each of these. Remark. It is clear that R(K) cannot exceed half the doubly critical self-distance, that is between pairs of points where the chord is perpendicular to both tangents; however [LSDR] goes on to establish the result for half the singly critical selfdistance. This may be a bit surprizing, as the minimum singly-critical self-distance of a curve is, in general, smaller than the minimum doubly-critical self-distance (e.g. for any ellipse that isn't a circle). But when self-distance is "in control" of thickness, then the two minima coincide. In particular, the thickness R(K) is at most half the singly-critical self-distance of K.

There also is a relation between thickness and the ratios of chord length to arclength along K. For points x, y on K, let arc(x, y) denote the minimum arclength along K between the two points. Theorem T3. Suppose K is a (C2) knot of total length 1 and thickness R. Then for any x, y on K, arc(x, y) 1 < Ix - yj 4R

226 GREGORY BUCK AND JONATHAN SIMON

Remark.

In terms of the rope-length EL(K), Theorem T3 says arc(x, y) 1 E

Ix _ yl 4 L

Remark . The supremum of this ratio over all pairs of points of K is called the distortion of K [G] [02]. The following is how one shows that the energy EL(K) satisfies Property 4 above.

Theorem T4. We assume K is normalized to have total length = 1. Given a lower bound on R(K) (i.e. an upper bound on EL(K)), one can deduce an upper bound on the bridge number of K (from curvature); the number of sticks needed to represent K as a polygon (from a close analysis of local behavior); and the minimum crossing number of the knot type [K] (from stick number). This relation between EL(K) and crossing number should be contrasted with the theorem offered below , relating EL(K) to the average crossing number. The results in [LSDR] give a bound on the minimum crossing number of the knot type of order EL2. Here we shall give a bound on (average) crossing number of the given curve (so it also yields a bound on minimum for the knot-type ) and it is of lower order, hence better for very complicated knots.

A simple energy for a polygonal knot K is defined as follows: For each pair X, Y of nonconsecutive segments of K, compute the minimum distance between the segments , MD(X, Y); then take the sum over all non-adjacent pairs X,Y of the numbers length(X) x length(Y) [MD(X, Y)]2 . If one wants simultaneously to consider knots with varying numbers of segments, then it is helpful to normalize for the number of segments, e.g. by subtracting the energy associated to a standard regular n-gon [Si3]. When a knot in 3-space is projected into a plane, for almost all choices of direction, the projected curve is immersed and one can count the number of self-crossings. This can be averaged over all directions to produce the average crossing number. M. Freedman et at showed [FHW] that acn(K) can be computed as double iintegral over K x K which facilitates comparison with energies. The integral is a modification of Gauss's formula for the linking number of two space curves. The integrand for the average crossing number measures the probability that the line elements dx, dy appear to cross from an arbitrary perspective. The definition is:

ENERGY AND LENGTH OF KNOTS 227

I [dx, p2 r] I dxdy , acn(K) = 1 ff SXS where the numerator of the integrand is the norm of the triple scalar product of the three vectors. The number acn(K), being an average, bounds the crossing number of the knot type, denoted here c([K]), which is the minimum number of crossings required in a planar diagram of the knot -type [K]. Therefore acn(K) is another reasonable measure of the complexity of the conformation K. However , the acn (K) does not provide any barrier to the changing of knot -types, and so is not much use as an energy. On the other hand, an energy function which does blow up on self-intersection and also bounds the crossing number would both measure complexity and have canonical minima. Note that bounding the crossing number is in itself a worthy goal, since this invariant of knots seems difficult to analyze. In the next section we connect "compaction " of a knot , the energies EN(K), EL(K), and acn(K). The same results hold with ES in place of EN and some are known for E0.

3. COMPARISON THEOREMS FOR ENERGIES, COMPACTION, AND AVERAGE CROSSING NUMBER

Energy and Compaction. Theorem 1 .

Let K be an arclength L knot contained in a ball of radius D. Then EN(K) > 4D

Remark. We cannot reverse this inequality: a curve of large diameter can be given high energy, keeping diameter large, by pinching it together at one place. Proof of Theorem 1. Assume we have a unit-speed parametrization, so dx is a unit vector. For x E K define the cone at x, denoted W. as the points w E R3 such that the angle B between w - x and Nx (the normal plane to K at x) is less thana. Now for each point xeK, some sub arc of K must span Wy, that is, pass from one nappe of the cone to the other at least once, because K is a closed loop. This will allow a lower bound for the contribution to the energy at x. The basic idea is that if the subarc is close to x then the contribution is large, though the subarc may be short, if the subarc is further away then it must be longer to span the wedge, so we still get at least a constant contribution. Denote the subarc described above by 1. Let p E l be such that Ix - pI is a maximum for y E 1. Let Ix - pl = d < 2D. Then by

228 GREGORY BUCK AND JONATHAN SIMON

the triangle inequality l must have arclength at least d, since the shortest distances to the surfaces of WW would be perpendicular lines. This gives us a contribution to the energy integral sin a 2

ff(P)

of at least d( )2 at each point x E K (we are approximating the inside integral). This simplifies to 2d. But d < 2D and the total arclength is L, so N(K) > 4D. ❑ Lemma 2 .1. Let K be an arclength L knot such that the entire (singular) tube of radius R(K) is contained in a ball of radius D. Then EL(K) > ( 4)(D)2 Proof. The singular tube about K intersects itself only at the boundary, otherwise radii slightly smaller than R(K) would produce self-intersection (cf. proof of Theorem 1 of [LSDR]). Thus we have from Pappus' theorem that the volume of this (singular) solid torus is 7rR2L, and this must be less than the volume of the ball, 37rD3. The desired inequality follows. ❑

Theorem 2.

Let K be an arclength L knot contained in a ball of radius D. Then

EL(K) >

43 _

D)a

Proof. If all of K is contained in the ball the, in particular , a pair of points x, y having maximum distance has Ix-yI < 2D . For such points , the chord vector (x-y) is perpendicular to each tangent (we only need one), so by Theorem T2, R(K) < D. Since each point of the singular tube about K of radius R(K) has distance <_ R(K) to some point of K, we conclude that the whole singular tube is contained in a ball of radius 2D. The theorem then follows from Lemma 2.1. ❑ Energy and Average Crossing Number. Theorem 3. EN(K) > Es(K) > 47racn(K) > 47rc([K]) . Remark. A similar inequality between Eo(K) and acn(K) is established in [FHW]. Once we obtain (Theorem 4) a relation between EN and EL then we can deduce a bound for acn(K) in terms of EL as well (Corollary 4.1). One cannot hope for a

ENERGY AND LENGTH OF KNOTS 229

converse saying that some energy is bounded by a function of acn: Draw a planar curve modeled on part of the graph y = sink; such a curve will have acn(K) = 0 since it is planar, but arbitrarily high energy (of all kinds) since it is packed tightly.

Proof of Theorem 3. Assume a unit-speed parametrization. The numerator of the integrand for acn(K) is the magnitude of the triple scalar product of unit vectors: I [dx, dy, r] 1. For some angles a, /3 , we have sin a = Idx x r i , and sin /3 = I dy x r l. Then I [dx, dy, r] < sin a sin 0 (since the angle from the line of dy to the plane of dx and r is < either of the angles from dy to dx or r) so ES(K) > acn(K). Moreover, 2EN(K)

f rldxxrI2+Idyxrl2 = f^sin2a+sin2/3 J J p2 P2 P2

since this counts every element twice. But sin 2 a+ sin2 /3 > 2 sin a sin /3, so EN(K) ❑ ES(K). Energy and Length (or Thickness). We now wish to relate the energy EN(K) to the rope-length EL(K). Theorem 4 will say that if EL is small then EN is small. Remarks. There cannot be a converse. This (family of) example(s) is based on an example given in [FHW], used there to show that finite energy Eo does not imply C2 smooth. For each n, construct a C2 curve K„ as follows: Start with a round unit circle K0. Replace a small arc of K0 with a bump that is an arc (representing < n degrees) of a circle of small radius r < n. Smooth the corners to make a C2 curve K,,. The minimum radius of curvature of K„ is < r. Thus, from Theorem T2, 2rrn. As we let n aproach oo, the energies the thickness R(K„) < r, i.e. EL(K„) EN, ES, andE0 of K„ approach the energies of a circle, while EL is unbounded. Regarding the formula below, it would not be surprizing if it could be simplified or improved. It would be especially interesting if one could reduce the power 3. Based on computer experiments, it appears that there might be linear inequaities relating the energies Eo and EL. Also there may be linear inequalities relating the energies for minimum-energy conformations under several energies [KS] [St] [B]. We state two bounds: the quadratic bound has smaller coefficient than the (3) power bound, so it should be sharper for low energy situations. Theorem 4.

11EL(K)4/3 > EN(K), 4EL(K)2 > EN(K).

230 GREGORY BUCK AND JONATHAN SIMON

Corollary 4.1. 11EL(K)413

> 47r acn(K),

1EL(K)2 > 4ir acn(K).

Proof of 4.1.

This follows directly from Theorems 3 and 4.

❑

This bound on acn(K) is very large relative to our intuitive sense of what a knot having certain EL looks like. With that caveat, Corollary 4.1 becomes an improvement on the bound for c([K]) obtained in [LDSR] when EL is large, e.g. EL > 94. At that point, the knot is [LSDR] equivalent to a polygon with 30 edges (or fewer) so the minimum crossing number of the knot type is at most (30-12(30-4) = 377. On the other hand, according to Corollary 4.1, the average crossing number of K (hence the minimum crossing number of its knot type) is < an (94) 3 : 374. Proof of Theorem 4 (overview). This proof, which includes discussion and several lemmas, is presented in detail in [BS2]. In the remainder of the present paper, we give an overview of the proof. To begin, normalize the curve so it has thickness R = 1 and, therefore, EL(K) _ L, the actual arclength of K. The essence of the proof is that we divide the interior integral of EN(K) into two summands: the "local" contribution and the "global" contribution. The energy EN(K) _

J Idxxr12 11=K yEK

p2

Let Ix denote the inner integral; we obtain a bound on Ix in terms of EL(K) = L and then multiply by the length L to get our bound on EN. For each x E K, define two sets (recall arc(x, y) = minimum arclength along K between x and y): L. = {y E Kjarc(x, y) < 7r} and Gx = {y E Kjarc( x, y) > ir} Define "local" and "global" integrals as follows:

Ix - lip+ ly" - f

2 2 Idxxr1+ /' Idxxr1

JyELz p2

JyEG. P 2

We first bound Ili. (This part of the argument also will yield the overall quadratic bound.) Each point in R3 lies on a circle tangent to K at x, having some radius o > 0. (The tangent line to K may be viewed as or = oo.) The contribution of a point y E K to Ix depends (see Lemma 4b below) on the radius a for that y. We show in Lemma 4a that a > 1 for all y E K.

ENERGY AND LENGTH OF KNOTS 231 At the given point x E K, for each radius 0 > 1, let Q.,,(0) be the union of circles of radius a that are tangent to K at x. In particular, let Qx denote Qx(l). The set Qx(a) is a singular torus (it is pinched at the point x). Define Wx to be the interior of Qx, and let Vx denote R3 - (Qx U W, ).

Lemma 4a. For each y E K (in particular y E Lx), y E V. U Qx. That is, K cannot turn enough to get inside Wx. Let B be a closed 3-ball of radius = 1 whose boundary is Sketch proof of 4a. tangent to K at x. The union of all such 3-balls equals the pinched solid torus W. Show that for each such ball B, K fl int(B) = 0. ❑ Lemma 4b.

Let y E K lie on a circle of radius 0 tangent to K at x. Then Idx x rl2 _ 1 p2

402

Proof of 4b. [BO1] Let p be the center of the circle; express 1 d2 and cosines of the angles of triangle pxy.

f

in terms of sines ❑

Remark. If we multiply the above constant integrand by (21ra)(2ira), we obtain the result [BOl] that if C is a round circle (of any radius) then the energy EN(C) _ 7r 2.

We can now obtain the bound on Iia and the overall quadratic bound for EN. From Lemma 4a, we have that each point y E K lies on a circle of radius 0 > 1 tangent to K at x (and so contributes < 1 to the integrand) or lies on the tangent line to K at x (and so contributes zero to the integrand). Thus 1

7r

J-^ 4 2

Similarly, X Ll L I <-f 4= 4 Thus EN(K) = f Ix < L xEK 4

We now proceed towards the (1) power bound by analyzing I9ib. The first observation is that for points y E Gx, Ix - yI > 2 (that is for y E G, f is < a because of the distance between x and y, regardless of the integrand 1 the angle.

232 GREGORY BUCK AND JONATHAN SIMON

Lemma 4c . If K is a C2 knot normalized to have R(K) = 1, then for each x, y E K, ix - yI < 2 = arc(x, y) < ir. That is y E Gx = Ix - yI > 2. Sketch proof of !c. This depends on Theorems Tl and T2 in Section 2. The other ingredient is Schur's Theorem [Ch], which compares the end-to-end distances of certain curves where it is known that the curvature of one curve is at least as large as the curvature of the other. ❑ In analyzing Igiob, we shall ignore Idx x rI , which is < 1 , and bound fa. . The P^l basic idea in the rest of our argument is that the condition R(K) = 1 preve nts too much arclength of K from being too close to the point x E K (compare Theorem 2). We consider spherical shells ( of thickness =1 and inner radii = 1, 2, 3, etc.) about a point x and bound the amount of K that can lie within a given shell. The maximum energy contribution would occur if the hypothetical maximum packing in each case actually occurred ; assuming that (unattainable) shape were attained, we get a bound for the energy contribution from each shell, along with a bound on the number of shells ( since we have only the given total length L available).

The conclusion is that we need at most N shells, where N is the greatest integer [(2L) 1,3], and

Iglob < 2.92+9.04N < 2.92+9.04[(3L)1/3] < 2.92+9.04(3L)1/3 4 4 We then combine and simplify the bounds on Iia and Igiob to obtain Ix = I,. + Igiob < llL113 . Multiplying by the length L to bound the outer integral, we obtain EN(K)=

J

Ix<11L4/3

xEK

This completes the proof of Theorem 4.

0

REFERENCES [Br] K. Brakke (further devel. by J. Sullivan), EVOLVER, Geometry Center, University of Minnesota , http ://www.geom . umn.edu/software /download/evolver.html, Program for visualizing and energy minimizing knots. [B] G. Buck, On the energy and length of a knot, talk in Special Session on Physical Knot Theory, Amer. Math. Soc., Iowa City, March 1996. [BOl] G. Buck and J. Orloff, A simple energy function for knots, Topology and its Appl. 61 (1995), 205-214. [BO2] G. Buck and J. Orloff, Computing canonical conformations of knots, Topology and its Appl. 51 (1993), 246-253. [BS] G. Buck and J. Simon, Knots as dynamical systems, Topology and its Appl. 51 (1993), 229-246.

ENERGY AND LENGTH OF KNOTS 233 [BS2] G. Buck and J. Simon, Thickness and crossing number of knots, submitted (preprint 10/96). [Chl S. S. Chern, Curves and surfaces in euclidean space, Studies in global Geometry and Analysys, MAA Studies in Mathematics vol. 4, 1967. [Co] N. Cozzarelli, pers. comm. 7/94. [DSKC] F.B.Dean, A. Stasiak, T. Koller, and N.R. Cozzarelli, Duplex DNA knots produced by escherichia coli topoisomerase I, J. Biological Chemistry 260 (1985 ), 4975-4983. [DEJ] Y. Diao, K. Ernst, and E.J.J. VanRensburg, Energies of knots, (preprint, 3/95); Knot energies by ropes, (preprint 4/96). [DC2] P. Droge and N.R. Cozzarelli, Topological structure of DNA knots and catenanes, Methods in Enzymology 212 (1992), 120-130. [DC] P. Droge and N.R. Cozzarelli, Recombination of knotted substrates by Tn3 resolvase, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 6062-6066. [FHW] M. Freedman, Z.-X. He, and Z. Wang, Mobius energy of knots and unknots, Annals of Math. 139 (1994), 1-50. [Fu] S. Fukuhara, Energy of a knot, A Fete of Topology: Papers Dedicated to Itiro Tamura (Y.T. Matsumoto and S. Morita, ed.), Academic Press, New York, 1988, pp.443-451. [Hg] M. Huang, Univ. Illinois-Chicago, http://www.eecs.uic.edu/ mhuang/research.html, Program for visualizing and energy minimizing knots. [Hu] K. Hunt, KED, University of Iowa, http://www.cs.uiowa.edu/ hunt/, Program for visualizing, manipulating, and energy minimizing polygonal knots. [Ketc] R. Kanaar, A. Klippel, E. Shekhtman, J.M. Dungan, R. Kahmann, and N.R. Cozzarelli, Processive recombination by the phage Mu Gin system: Implications for the mechanisms of DNA strand exchange, DNA site alignment, and enhancer action, Cell 62 (1990), 353-366. [KS] R. Kusner and J. Sullivan, Mobius energies for knots and links, surfaces and manifolds, Geometry Center Research Report GCG64 (1993 rev. 1994), University of Minnesota, to appear in Proc. of 1993 Georgia Top. Conf.. [L] R. Litherland, Thickness of knots, talk in Workshop on 3-Manifolds, Univ. Tennessee, 1992. [Le] S.D. Levene and H. Tsen, Analysis of DNA knots and catenanes by agarose-gel electrophoresis, Protocols in DNA Topology and Topoisomerases vol I (M. Bjornsti and N. Osheroff, ed.), Humana Press, 1996. [LSDR] R. Litherland, J. Simon, 0. Durumeric, and E. Rawdon, Thickness of knots, submitted (preprint 6/96, based on [L] and [Sill). [Lo] S. Lomonaco, The modern legacies of Thompson's atomic vortex theory in classical electrodynamics, (preprint, 4/95). [M] J. Milnor, On the total curvature of knots, Annals of Math. 52 (1950), 248-257. [Mo] H. K. Moffatt, The energy spectrum of knots and links, Nature 347 (Sept. 1990), 367-369. [N] Alexander Nabutovsky, Non-recursive functions, knots "with thick ropes", and selfclenching "thick" hyperspheres, Comm. Pure Appl. Math. 48(4) (1995), 381-428. [01] J. O'Hara, Energy of a knot, Topology 30 (1991 ), 241-247. [02] J. O'Hara, Family of energy functionals of knots, Topology Appl. (1992), 147-161. [03] J. O'Hara, Energy functionals of knots, Topology-Hawaii (K. H. Doverman, ed.), (Proc. of 1991 Conference), World Scientific, 1992, pp. 201-214, (Computer program by K. Ahara). [04] J. O'Hara, Energy functionals of knots II, Topology and its Appl. 56 (1994), 45-61. [Sc] R. Scharein, Knot-Plot, Univ. British Columbia, www.cs.ubc.ca/spider/schaxein/,

234 GREGORY BUCK AND JONATHAN SIMON Program for drawing, visualizing, manipulating , and energy minimizing knots. [Sil] J. Simon , Thickness of Knots, talk in Special Session on Knotting Phenomena in the Natural Sciences, American Mathematical Society, Santa Barbara, Nov. 1991; Lecture Notes on Physical Knot Theory, (notes by H. Naka), 1993 , from summer 1991 course, Kwansei Gakuin University. [Si2] J. Simon, Energy functions for polygonal knots, J. Knot Theory and its Ramif., 3 (1994), 299-320. [Si3] J. Simon , Energy functions for knots: beginning to predict physical behavior, preprint 11/94, to appear 1996 in Proc. of 1994 IMA Conference on Geometry and Topology of DNA. [Si4] J. Simon, Thickness and energy of knots, talk at IMA Conf. on Polymers 6/96. [SSC] S. Spengler, A. Stasiak, and N.R. Cozzarelli, The stereostructure of knots and catenanes produced by phage A integrative recombination: implications for mechanism and DNA structure, Cell 42 (1985), 325-334. [St] A. Stasiak, Ideal forms of knots, talk in Special Session on Physical Knot Theory, Amer. Math. Soc., Iowa City, March 1996. (Su] D. Sumners , pers. comm. 6/96. (T] J. A. Thorpe, Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics , Springer-Verlag, New York, 1979. [WC] S. Wasserman and N.R. Cozzarelli, Supercoiled DNA-directed knotting by T4 topoisomerase, J. Biol. Chem. 266 (1991 ), 73-95. [Wu] Y: Q. Wu, MING, University of Iowa, http:// www.math . uiowa.edu/ wu/, Program for visualizing, manipulating , and energy minimizing polygonal knots. DEPARTMENT OF MATHEMATICS, ST. ANSELM COLLEGE, MANCHESTER NH 03102 E-mail address : [email protected] DEPARTMENT OF MATHEMATICS, UNIVERSITY OF IOWA, IOWA CITY, IA 52242 E-mail address : jsimonmath.uiowa.edu

Lectures at Knots 96 edited by Shin'ichi Suzuki ©1997 World Scientific Publishing Co. pp. 235-261

CHERN- SIMONS PERTURBATIVE INVARIANTS TOSHITAKE KOHNO

Introduction The purpose of these lecture notes is to describe recent developments in topological invariants of 3-manifolds arising from the perturbative expansion of the partition function of the Chem-Simons action . Let M be a closed oriented 3-manifold and G a compact Lie group . We denote by A the space of G connections on a principal G bundle over M. The Chern-Simons action CS : A --+ R is defined by

CS(A) = :F71 fMTr (AAdA+ 3AAAAA) where A is a G connection . Witten's invariant is defined formally by the path integral as the partition function Zk(M) = f exp (ik CS(A)) VA for an integer k in the seminal work [W]. Based on Witten 's investigation on the relation between the Chem -Simons gauge theory for 3-manifold with boundary and the Wess-Zumino-Witten conformal field theory on Riemann surfaces, Witten's invariant was constructed rigorously using the monodromy of conformal field theory (see [RT] and [Ko3]). In this lecture we deal with the perturbative aspects of the theory. Namely, we look at the asymtotic expansion of the partition function Zk(M) as k tends to infinity. In a finite dimsnsional case it is known that the main contribution to the asymptotic behaviour of such integral comes from the critical points of f by the stationary phase method . An explicit form of the semi-classical limit of Zk(M) was already speculated in [W]. The main term involves the Ray-Singer analytic torsion, the eta invariant and the Chern- Simons invariant . This speculation has been verified in several explicit examples (see [FG], [J] and [Ro]). As higher terms of the perturbative expansion we obtain amplitudes associated with Feynman diagrams, which are expressed as the integral of the wedge product of Green forms on the configuration space of the manifold M. A topological invariance 235

236 TOSHITAKE KOHNO

of such amplitude was shown by Axelrod-Singer [AS]. We refer the reader to [K1], [T] and [F] for different approaches to the subject. Let C be a closed oriented curve in M. We fix a finite dimensional representation R of G and we consider the function on the space of connections WR(C) = TrRHo1C(A) which is by definition the trace of the holonomy of the connection A in the representation R. In the case the manifold M contains a link L = C1 U • . C,., Witten's partition function is Zk(M) = J exp (ik CS(A)) fl WR;(C=) DA. :-1 As the perturbative expansion we obtain various Vassiliev type topological invariants of links in M with integral representations. These lecture notes are organized in the following way. In section 1, we review basic facts on classical Chern-Simons theory. To explain the nature of the perturbative theory we will explain in section 2 the stationary phase method in a finite dimensional case. In section 3, we compute the regularized determinant of the Hessian of the Chern- Simons action at a flat connection and we explain the reason why it is expected that the main term of the perturbative expansion of the Chern-Simons partition function contains the Ray- Singer torsion and the eta invariant . Section 4 is an introduction to the method developed by Axelrod and Singer. In section 5 we look at the framework to show the toplogical invariance of the integrals associated with Feynman diagrams . This leads us to a new approach in integral geometry based on the graph complex. In order to formulate the Chern- Simons perturbative theory for a 3-manifold with boundary, we discuss as a first step the case of the product of a Riemann surface and the unit interval in section 6 . In particular , we investigate the Poisson structure of chord diagrams on surfaces. Finally, in section 7, we deal with the elliptic KZ system , which gives an analogue of the Kontsevich integral in the case of elliptic curves.

1. Review of Chern- Simons theory Let M be a closed oriented 3-manifold and G a compact Lie group. We denote by g the Lie algebra of G. Let P be a principal G bundle over M. In this lecture we deal with the case G = SU(2). Since all SU(2) bundles over M are isomorphic to the topologically trivial bundle M x SU(2), the group of gauge transformations g is of the form 4 = Map(M, SU(2)). Let A denote the space of G connections on P, which is identified with 111(M, g), the space of g valued 1-forms on M. The classical Chern-Simons action

CS:A -+ R

CHERN-SIMONS PERTURBATNE INVARIANTS 237

is defined by (1.1)

CS(A) = 47r LTr (AAdA+ 3AAAAA)

where A is a G connection on P. The group of gauge transformations G acts on A by A9=9'A9+ 9 ld9, 9 E G and with respect to this action we have

CS(A9) = CS (A) + 27rn for some integer n, which is the mapping degree of g. Hence the Chern- Simons action gives a well-defined function CS:A/G-*R/Z. For A E A we consider a family of connections At = A + to with a parameter t. We have

(1.2) CS(A + ta) = CS(A) + 2- fM Tr (FA A a) + where FA denotes the curvature form of the connection A. Hence we conclude that the critical points of the Chem -Simons action are the flat connections. Let us investigate the infinitesimal gauge transformation . We suppose that g E 9 is written in the form g = etf for small t. We have A9=A+t(df+[A,f])+••• and we see that the infinitesimal gauge transformation is expressed by the covariant derivative as dAf = df + [A,f]• We notice that the Chern-Simons action is invariant under this infinitesimal gauge transformation. Let us suppose that a is a flat connection and we denote by ga the associated flat g bundle over M. We have the de Rham complex 0 + 52a-,

- Q.,->S2n-^0

with the differential da, where 1Z denotes the space of j forms with values in ga. We have d,', = 0 since a is a flat connection. In this lecture we shall assume that the flat connection a satisfies H* (M, 9a) = 0. By Poincare duality the conditoins essentially reduce to H°(M, 9a) = 0,

H1(M, 9a) = 0.

238 TOSHITAKE KOHNO

Here H°(M, gam) = 0 corresponds to the irreduciblity of the representation of the fundamental group associated with the connection a and H' (M, g,) = 0 corresponds to this representation being isolated. Under this assumption we study the determinant of the Hessian of the Chern-Simons action at the flat connection a. 2. Stationary phase method Let us review briefly the classical stationary phase method. We start from the Gaussian integral 00 e vx 'dx=,^ p>0. yl^ By an analytic continuation we have M adx = it f e-tam AI e+ A E R. f VF

Let us suppose that Q is a non-degenerate quadratic form in x1i x2, , x,,. We have n/2

(2.1) ei4(2i,... x,) dx1 ... dx n

I det Q1

R^

e

is

gn Q

where sgn Q is the signature of the quadratic form Q. Let f be a real valued function in x1i x2, • • • , xn and we suppose that f has only finitely many non-degenerated real isolated critical points. Let us now discuss the asymptotic behaviour of the integral of the form ...

✓ R^

,xn) eik f (x1, dxl•••dxn

as k tends to infinity. For simplicity we first deal with the integral in one variable g(k) = J eikf(x) dx. The rapid oscillation of eikf(x) will tend to cancel large contribution to the integral in general , however this cancellation will not occur at critical points of f (x). Let us suppose that xo is the only critical point of f (x) and that f"(xo) > 0. We have g(k) ti

Ix

o +e &kf

(x) dx =

xo-e

fua

eik (

al

nxo) +u2)

where we put f(x) - f(xo) = u2. Since we have -m x

2u

f,(x) =

2u du

j7(x)

2 f"(xo)

CHERN-SIMONS PERTURBATIVE INVARIANTS 239

we obtain that 2 - r00 eiku2 +ikf(xo) 6;-

9(k)

(xo) J- co

dx.

Combining with the Gaussian integral, we have eikf(x) dx N

K K

27f

eikf(xo)+ 4

Ty "(xa)

as k tends to infinity. In general, under the above assumption for f , we have the asymptotics (2.2) eekf(xi,... 7rn/2eikf(a) ,x e 4gn Q ) dx 1 ... dx n ^k o0 R." a kn /2 I det H(f, a)I

where the sum is for all critical points a of f and H(f, a) denotes the Hessian of f at a. The above investigation in a finite dimensional case is the motivation of our study of the determinant of the Hessian of the Chern-Simons action at a flat connection. This subject will be discussed in the next section. 3. Semi-classical approximation

Let us go back to the situation where M is a closed oriented 3-manifold and A is the space of G connections on M. We fix a flat connection a. As in section 1, we denote by (1 , da) the associated de Rham complex. The tangent space T0A is equipped with the inner product defined by =- f Tr (AA*B) M

and with respect to this inner product we have an orthogonal decomposition T«A= Imda®Kerda where d,, is the differential da:Q0--$Z«. The subspace Im da is idetified with the tangent space of the gauge orbit To(ga).

Let us consider a one-parameter family of connections at = a + to and compute the Hessian of the Chern-Simons action at the flat connection a. We have (3.1)

z CS(at) = CS(a) + _ L Tr (Q A d00) +

which means that the Hessian is written as the quadratic form /

240 TOSHITAKE KOHNO

Let us recall that the Chern-Simons action is invariant under the infinitesimal gauge transformation . The quadratic form Q degenerates on the gauge orbit Im d« and it can be shown that Q defines a non-degenerate quadratic form on the quotient space

Q.' /d.Sto.. We are going to compute the determinant of the quadratic form Q defined on the space S2a/da12°. For this purpose we introduce the operator P=e(da*+*da) acting on Q1 ED Q3 Q1 a) QO where c=1on Staand e=-1 on Q.1. On

St° ® Im da ® Ker d^ the operator P acts as

0 dd 0 a 0 0 0 0 Q

P and we have

p2 = AO a (D Ala where is is the Laplace operator d*,da + dad* acting on the space of j-forms Sta. Let us recall the definition of the regularized determinant of the Laplace operator. Let 0 be a Laplace operator with positive eigenvalues A. The zeta function of the operator 0 is defined by

Co(S) which is an analytic function if the real part of s is sufficiently large. We define the reguralized determinant of 0 by det A = exp(-((0)). Let us notice that formally we have

(A ' (0) = ^(- log A). A In our situation we have det P2 = det 0°0 det Da. Hence I det PI is well-defined. On the other hand we have det PI = det(d*.da)l det Q1.

CHERN-SIMONS PERTURBATWE INVARIANTS 241

Hence we obtain (det 0l)1/2 (3.2) IdetQ1 _ (det AO ) 1/2 Now we are in a position to describe the relation between I det QI and the RaySinger analytic torsion. Let us recall that the Ray-Singer torsion for the flat conncection a is by definition (det AO )3/2 l )1/2 T. _ (det A From the above discussion we obtain the following proposition. Proposition 3.3. The determinat of the Hessian of the Chern-Simons action at the flat connection a is related to the Ray-Singer torsion by the formula detd-.da = T112

FJ detQI Let us give a geometric interpretation of the above formula from the viewpoint of the stationary phase method. Let us recall again that the Chern-Simons action CS degenerates on the gauge orbit and we see that det dada is the ratio of the volume form of the gauge orbit and the volume form of the gauge group. Comparing with a finite dimensional case the square root of the Ray-Singer torsion might be interpreted as the term corresponding to I det H(f, a) 1-1/2 in the formula (2.2).

We discuss briefly the phase factor ewe. We have sign P = sign Q where sign P is the one studied by Atiyah, Patodi and Singer and is defined in the following way. We introduce the eta function

,I(S) = E (JAI-esgnA) ago

and by analytic continuation we define signP = q(0). We refer the reader to [FG] and [J] for a precise statement of the Witten conjecture on the semi-classical limit of Zk(M). 4. Higher loop amplitude Before starting the discussion on SU(2) Chern-Simons theory let us deal with the Abelian gauge theory where G = U(1). In this case the Chern-Simons action is quadratic and the partition function is computed as the Gaussian integral.

242 TOSHITAKE KOHNO

Let L = K1 U K2 be a two-component link in R3. Let A denote the space of U(1) connections on R3, which is identified with the space of R valued 1-forms on R3. The partition function we are going to consider is of the form (4.1)

P (^,^k 1 Zk = fA ex 4x JR3 A n dA + v^ fK^ A + fK2 A) DA.

The integrand contains a quadratic form in A and a linear form in A. We consider again a finite /dimensional analogue. Let Q(xl

, ...

xn)

= 2 E As9xsx9 4,)

be a non-degenerate quadratic form and we try to evaluate the integral dxl ... dx,,. fRn

Completing the square, we obtain that the above integral reduces to (4.2)

e-^2 Eij

up to a constant multiple , where (0) denotes the inverse of the matrix (a;j) Let us go back to the situation of our infinite dimensional integral Zk. The quadratic form appearing in the integral is given by the differential . Following our consideration in the finite dimensional case, one would expect that the integral Zk is expressed in terms of the inverse of the above differential . The inverse of such differential operator is known to be an integral operator with kernel L of the form (d-1^o)(x ) = JER.3 L(x, y) A W(y), where the kernel L is called the Green form . In the case of R3, the Green form L is given by

L(x, y) = w(x - y) using the 2-form w defined in the following way. For x E R3 \ {0} we put w(x) = 1 xldx2 A dx3 + x2dx3 A dx1 + x3dx1 A dx2

4v IIX113 which is the volume form for S2 normalized so that fs2 w(x) = 1. We see from the above consideration that formally the main part of our integral corresponding to the integral in the formula (4.1) is expressed as exp

k > LIJQ R4

CHERN-SIMONS PERTURBATNE INVARIANTS 243

where Lr,, is the integral defined by

I

(4.3) LT =

y).

w(x -

XEKp,yEK9

In particular, we recover as the 1-loop amplitude L12 the Gauss formula for the linking number of K1 and K2. To investigate the higher loop amplitudes of SU(2) Chern-Simons theory, let us review briefly techniques of Feynman diagrams commonly used in physics. We consider the integral (4.4)

eTkt(x1, •• ,z+.) dx1 ... dx,

Zk Rn

where f contains a cubic term and is written as ... xn) +

Q(x1,

f (x1, ... , xn) =

F, Aijkxixjxk ijk

with the the quadratic form Q(xi, • • • , xn) = 2 Ei; Ai;xixj. By a change of variables we see that Zk = k-n /2

J

°° e^Q(x1, ...,xn ) E M ==0

Rn

m

m (^) 2

[ ` \ijkxixjxk

m k /

ijk

dx1 ... dxn

In order to describe the asymptotic behaviour of Zk as k tends to infinity we are going to evaluate the integral (4.5)

J

lm e^Q(x1, ... xn) (^ \ijkxix; xk I dx1 ... dxn. ilk )

Rn

The above integral is expressed as f (EJk,kDDjDk +7k \, L

M

IRn

e "V/ -+(Q(x1,

where

...

,xn)+Ek Jkxk) J=0

1 a

D;

7-7 a As we have seen in (4.2), The integral eV +(Q(x1,... ,xn)+Ek Jkxk) Rn J

is equal to e ^' ^^ a^^JiJ' up to a constant multiple. Hence up to constant the integral (4.5) reduces to m Eij A' J'J,

(4.6) ijkAt,kD,D,Dk ) KE

e -s J=0

244 TOSHITAKE KOHNO

This is a polynomial in aij and Aijk, which is up to a constant multiple written as the sum of AijkAiijikiAtt A)J A kk ijki' j'k'

and (4.8)

E ^ijk '\

i,j,kr^ij `kki `iT

ijki'j'k'

We express them as trivalent graphs with 2 vertices. Namely, to each trivalent vertex we associate the factor Aijk, with the index i, j, k corresponding to each edge meeting at the vertex, and to each edge connecting the vertices we associate the factor Aij with the index i, j corresponding to the initial point and the terminal point of the edge. It is not hard to see that in general m the quantity (4.6) is expressed as the sum of terms corresponding to trivalent graphs with m vertices. Let us notice that the first Betti number of such graph is equal to m. We call such graph a Feynman diagram with m loops. We conclude from the above discussion that the m-th order term of the asymptotic expansion of the integral (4.4) is expressed as the sum of terms corresponding to all Feynman diagrams with m loops. An important feature in the case of SU(2) Chern-Simons gauge theory is that the action degenerates on the gauge orbit. To explain the situation, we consider again a finite dimensional analogue. Let us suppose that an l dimensional Lie group G acts on R" as isometries and that f (xj, "' , xn) = Q(xi, "' , x,)+Eijk Aijkxixjxk is invariant by this action. We suppose moreover that there exists a smooth function F : R" -+ R' such that F has a unique zero at each G-orbit. For x with F(x) = 0 we denote by J(x) the Jacobian of the action of G at x composed with F. We see that det J(x) is the ratio of the volume form of the G-orbit at x and the volume form of the Lie group G. Now the integral (4.4) without the redundant part over the orbits of G is written as

f

R^

e,lf(xl,...,:„)b(F(x)) det J(x) dxl ... dx,,.

The first additional term is replaced by its Fourier representation b ( F(x )) _ (2x)l fRt e

i Fj(' ' del ...del

A way of dealing with the second additional term used in the physics literature is to introduce anti-commuting variables {ci} and {cj } such that det J is up to a constant multiple equal to

f

Eij ''J'jcj e dcl ... dcidcl ... dci.

Such variables are called ghosts and we refer the reader to [R] for detailed description of such techniques.

CHERN-SIMONS PERTURBATWE INVARIANTS 245

A program to compute the asymptotics of the Chern-Simons partition function was performed by Axelrod and Singer in [AS]. Our new Lagrangian has two additional terms, one coming from gauge fixing and the other coming from ghosts. Instead of presenting a detailed treatment of this formal computation, we just describe 2-loop amplitudes arising from this computation. Let a be a flat SU(2) connection as in section 1. We fix a Riemannian metric on M. We introduce a 2-form LESt2(MxM; gAg) which is the kernel of the operator da(Da)-1. Let { Ia}, a = 1 , 2,3 be an orthonormal basis of the Lie algebra g and we write L in the form L = E Lab(x, Y)Ia A Ib. Then we have

(4.9) (d.-1c7)a(x) = f EM Lb(x,y) A
A faibici fa2b2c2 Laici (X1, x1) A La2c2 ( x2, x2 ) A Lbib2 (xl , X2)

and Ir2(M,a) _ 12 EfMxM\0

faibici fa2b2c2Laia2 ( x1, x2 ) A Lc1

(x1, x2 ) A Lbib2 ( xl, x2)

respectively where A denotes the diagonal set. The following result was shown in [AS]. Theorem 4 .10 [AS]. Let M be a closed oriented 3-manifold and let a be an acyclic SU(2) connection and s a framing of M. Then 1 Ir 1 ( M , a) + Ira (M, a )

-

1

CSgrav (9,

s)

is independent of the metric and is a topological invariant of M. In the above theorem CSgrav(g, s) denotes the Chern-Simons action for the metric connection associated to the metric g defined by using the framing s. Essential part of the proof consists of the finiteness of the integral and the computation of SIr3 (M, a), j = 1, 2 on the space of metrics on M.

246 TOSHITAKE KOHNO

5. Graph complex In this section we introduce techniques to show topolgical invariance of the integral associated with Feynman graphs. We deal with topological invariants for knots in W. We refer the reader to [BT] and [Ko4] for a more extensive treatment. For a manifold M, we denote by Con f(M) the configuration space of ordered n points in M. Namely, we set Confn(M)=

{(x1i".,xn

)Ix1,•••,xnEM,

Xi #xj if i #j }.

Let us consider the configuration space Confn(R3) The differential form w(xi x3), 1 < i # j < n is considered to be a 2-form on Confn(R3) where w denotes the volume form of S2 as in the previous section. In the following, we consider a finite graph IF whose boundary consists of finitely many points on a circle, satisfying the following conditions: (1) For each vertex which is not on the boundary, there are at least 3 edges meeting at the vertex. (2) There is no edge adjoining the same internal vertex. We suppose that each edge of the graph is directed. We set OF={p1,...,Pk}. The circle is parametrized as e2iit, 0 < t < 1. Here we fix a base point 0 on the circle corresponding tot = 0, and Pk has coordinate tk, so that 0 < t1 < • • • < tk < 1. These vertices are called external vertices. We fix an ordering for the other vertices called internal vertices and we denote them by {pk+1, ' ' pk+t }

We denote by m the number of edges of the graph r. We define the degree of the graph r by degF=-k-31+2m, which may also be expressed as E(i - 1)ki + 3)1i i>2 i>4

where ki and 1; denote the number of the external vertices with valency i and the number of internal vertices with valency j respectively. We define (M \ E)o to be the space of f : S' -+ R3. We define Xk,t by Xk,1 _{(f, t, x) E

(M \ E)o x

Ak

x Con f,R3 ; xj

^

f (S1), 1 < j < 1}.

Here L k is the k simplex Ak = {(t1, ... , tk) 10 < t1 < ... < tk < 1}.

CHERN-SIMONS PERTURBATIVE INVARIANTS 247 We define differential 2-forms wi , 1 < i # j < k + 1, and wii, 1 < i < k, in the following way. We have a map

0 : Xk,i - Confk+i(R3) defined by

g(f, t, x)

=

(f (t1), ...

, f (tk); x1, ...

x1).

Pulling back by 0, we set wig = O*w(xj - xi). For 1 < i < k, we have a map cpi : Xk,i - S2

defined by

f'(ti)

lei , , Ilf'(ti)II, Pulling back the volume form w by cpi, we set wii = w w. Let r be a graph in the above sense . To each directed edge [pips] we associate denotes a loop adjoining the external vertex pi. wfpipj] = wig. Here [pipi], 1 < i < k, We define wr, by wr

=

A

wif.

)pipj )Eedge(r)

where edge(F) stands for the set of edges of the graph P. We denote by it : Xk,1 --> (M \ E)o the natural projection. The fiber 7r-1(f) is equal to Ak x Conf1M, where we set M = R3 \ f (S1). For a graph F of degree n in the above sense, let us consider the differential form wr. Then, the integration along the fiber f

(5.1)

m(F) = J^ 1(f)

defines a differential form of degree n on (M \ E)o. The finiteness of integral (5.1) can be shown by means of the compactification of Xki developed in [AS] and [BT]. We denote by Dn the vector space over R spanned formally by the graphs of degree n in the above sense. Here the graph obtained from F by making the direction of one edge opposite is identified with -F. Let W be the subspace of D" spanned by graphs with double edges, i.e., graphs containing two distinct vertices, so that there are at least two edges adjoining these two vertices. Let us observe that for a graph F with a double edge, we have wr = 0. We define C" to be the quotient vector space Dn/W.

For F E C" we define SI' as a formal sum of graphs obtained by contracting edges in the following manner. For each directed edge e whose boundary contains an internal vertex, we define a new graph F/e obtained from IF by contracting the edge e. We define the direction of the edges of F/e as the one induced from F. In a similar way, we define F/p,pj+1, 1 < j < k - 1, to be the graph obtained from I' by

248 TOSHITAKE KOHNO

contracting the arc pjpj}1. on the circle, with the direction for edges induced from F. We define 8F by a formal sum k

(5.2)

8F

= y(- 1)'F /pips+1 + e(e) F /e. j =o eEedge(P)

where po = Pk+1 = 0. The symbol e(e) is 1 or -1 determined by the following convention. For 1 < i < j < k + 1, let us consider the edge adjoining the vertices pi and pj. We put E11Xipj]) = e([pjpi]) = (- I)'•

The above convention comes from the orientation for the boundary of the compactified configuration space. After contracting the arc pj_1pj or the edge [pipj], i < j, we delete the index j and relabel the vertices by shifting index n by n -1 for n > j.

example: Let us consider the graph ri with vertices pi, • • • , p4 and the edges [p1p3], [p2p4] and the graph F2 with k = 3 and l = 1. We have 6(I'1

+ I'2) = 0.

It turns out that the integral of wr , + wr, is locally constant on the space of knots and determines a knot invariant. Let us recall that wr, +wr, - 2'-4 defines , as a knot invariant, nothing but the second coefficient of the Conway polynomial (see [GMM] and [BT]).

We denote by A"((M \ E)o) the space of differential n forms on (M \ E)o. By means of the construction of the previous paragraph , we have a map Ir:C"-+A"((M\E)o) by defining v(F) as the integration along the fiber of wr for a graph r of degree n. Let r be a graph. We define its level as m - 1, where in is the number of the edges of r and 1 is the number of the internal vertices of F. Let us observe that the level is preserved by the differential 6. We denote by FjC" the subspace of C" spanned by the graphs with level less than j. The following theorem describes how dµ" (F) can be expressed in terms of the graph (see [BT] and [Ko4]). Theorem 5.3. Let r be a graph of degree n and level j. Then, as a differeintial form on the space of knots, d(p(F)) can be written as a linear combination of p(Fi) with some graphs Fi of degree n + 1. Moreover, modulo the subspace p(F,C"+1) d(p(F)) is expressed as p(6F) up to sign.

CHERN-SIMONS PERTURBATNE INVARIANTS 249

6. Chord diagrams on surfaces First, we describe some basic facts on chord diagrams on surfaces. Let G be a Lie group whose Lie algebra g is equipped with an adjoint invariant symmetric nondegenerate bilinear form B : g x g --> R. Let E be a closed oriented surface of genus g, and consider the moduli space .ME(G) of flat G connections on E. The moduli space ME(G) is identified with the set of conjugacy classes of representations of the fundamental group 1r1(E) into G. The variety ME(G) contains an open set ME(G)° corresponding to the conjugacy classes of irreducible representations of 7r1(E), which has a structure of a symplectic manifold. A chord diagram is a collection of finitely many oriented circles with finitely many chords attached on them, regarded up to orientation preserving diffeomorphisms of the circles. Here we assume that the endpoints of the chords are distinct and lie on the circles. Let D be a chord diagram. We consider a continuous map 7 : D -+ E and we denote by [y] its free homotopy class. We call such [y] a chord diagram on E. Up to homotopy we shrink the chords on E to get loops with transversal intersections. We represent [-y] by loops with specified vertices. Here the specified vertices are considered to be shrunk chords. We denote by DE the complex vector space spanned by all chord diagrams on E and by A(E) its quotient space modulo the 4 term relations (see [BN] and [AMR]). As was shown by in [AMR], A(E) has a structure of a Poisson algebra in the following way. Let yl and 72 be chord diagrams on E where the chords are shrunk and are represented by the specific vertices as explained above. We suppose that yl and rye intersect transversely on E. Let p be one of the intersections of -yi and y2. We denote by 71 UP rye the chord diagram on E which is the union of '11 and rye, with p considered to be the specific vertex corresponding to a shrunk chord. For a chord diagram 7 we denote by [7] its equivalence class in A(E). We have the following proposition. Proposition 6.1 [AMR].

We define the bracket by

{[7'1], [72]} = E 612(1')[71 UP 72] PEryi1Yy2

where e12 (p) is the intersection index of 71 and 72 at p. Then the above bracket is anti-symmetric and satisfies the Jacobi identity. The proof is analogous to the proof of the result of Goldman [G] that the free Z-module Zi`r spanned by the free homotopy classes of loops on E has a structure of a Lie algebra ([G] Theorem 5.3). We observe that the Lie algebra of free homotopy classes of loops on E introduced by Goldman [G] appears as a quotient space of A(E).

250 TOSHITAKE KOHNO

Let D be a chord diagram with n oriented circles C1, C2, • • • , C„ and ry : D -+ E a chord diagram on E, considered up to free homotopy. As in the previous paragraph we shrink the chords on E and represent 1' = [ -y] by n loops on E with transversal intersections and with the specified vertices corresponding to the shrunk chords. We assign finite dimensional representations R1 : G -> Aut(V), 1 < j < n, and the associated representations of the Lie algebra are denoted by rj : g --> End(Vj), 1<j
Let 0 be a flat G connection on E. Associated with 0 and the above representations rj : g -> End(Vi),1 < j < n, we define a function To : An (E) C in the following way. The representation r„ 1 < j < n, is considered to be an element of g` ®V* ®V;. The invariant bilinear form B : g x g -+ R defines Sl E g ®g by identifying g with its dual associated to each chord. By the endpoints of the chords, each oriented circle C„ 1 < j < n, is divided into several arcs C3k, k = 1, 2, • • • . Considering the holonomy along the path -y(C;k) on E we obtain a linear map Hol.y(G;,,) : V; -+ V„ which is considered to be an element of V' ® V j. Our way of defining Top is quite similar to the method to define the weight system in [BN]. Contracting the above three kinds of tensors according to the chord diagram on E we obtain a scalar which is denoted by T,(1'). We call TO(r) the weight system associated with the holonomy 0 and the representations r3 : g -^ End(Vj),1 < j < n. We denote by C(AG(E)) the space of smooth functions on MG(E)°, which has a natural structure of a Possion algebra induced from the symplectic structure on MG(E). We have the following proposition. Proposition 6.2 [AMR]. and defines a linear map

The above T,(F) is compatible with the 4 term relation To:A(E)-+C.

With the Lie group G and the representation r the map

f : A(E) -' C(MG(E)) defined by f ([-y]) (0) = T,(-y) is a homomorphism of Poisson algebras. Let us recall the definition of an invariant of finite order in the sense of Vassiliev [V] for oriented links in an oriented 3-manifold M following [BL] (see also [BN], [Ka] and [L]). Any C valued invariant v of oriented links in M can be extended to be an invariant of immersed circles in M, which are allowed to have transversal intersections , using the rule: v(Lp) = v(L+) - v(L_)

CHERN-SIMONS PERTURBATNE INVARIANTS 251

where Lp denotes a singular link with a transversal double point at p and L+ (resp. L_) stands for the link obtained by replacing p by a positive (resp. negative ) crossing. Here we think of the above graphs as parts of bigger graphs which are identical outside a small sphere. Let k be a non-negative integer . An invariant v of oriented links in M is called an invariant of order k, if v vanishes on singular links with more than k intersections . An invariant v of oriented links in M is called a Vassiliev invariant, or an invariant of finite order , if it is of order k for some non-negative integer k. We denote by Vk the vector space of Vassiliev invariants of order k for oriented links in M. The space of all Vassiliev invariants V = Uk>_oVk is a vector space with the increasing filtration VOCV1C•••VkC•••.

Let us now consider the case when M is the product of a closed oriented surface E and the unit interval I = [0,1]. We have a natural projection map p : M -+ E. Let L be an oriented link with n components in M = E x I. Projecting L onto E by p, we obtain a link diagram drawn on E . The notion of the framing is well-defined for links inExI. Let Vk( E) denote the space of all C valued invariants of order k for oriented framed links in E x I. Let v be a Vassiliev invariant of order k and D a chord diagram with k chords. Let 1' be a chord diagram on E and let -y : D -+ E be a representative of r with with k vertices , which are transverse double points. We define w (v)(I') E C by w(v)(I') = E:kleiv('Y`)

where ryi ranges over all the possible links in E x I obtained from ry by resolving the double points by positive or negative crossings , and the signes ei = 1 or -1 according as the crossing is positive or negative. It follows from the fact that v is of order k that w(v) is well-defined (see for example [BN] or [L]). It can be also checked that the above w(v) is compatible with the 4 term relation and is called the weight system associated with the invariant v. This induces a map w : Vk (E)/Vk 1(E) -+ Homc(Ak(E), C). In the previous section we have shown that associated with representations of a Lie group R; : G -+ Aut (V3),1 < j < n, and a flat G connection 0 on E, we can define To E Homc (AA (E), C). From the viewpoint of the Chern-Simons perturbative theory it would be natural to ask if one can integrate To to construct a Vassiliev invariant v such that w(v) = To. Theorem 6.3. For the above weight system To associated with the flat connection 0, we have a Vassiliev invariant v such that w(v) = To.

252 TOSHITAKE KOHNO

A detailed proof of the above theorem will appear elsewhere. We use representations of the braid group of a Riemann surface constructed based on a flat connection and the Drinfel'd associator. In the next section we discuss an invariant with an explicit integral representation in the case of genus 1. 7. Elliptic KZ system In this section we discuss invariants for links in the product of a torus and the unit interval based on elliptic KZ system (see [Ko5] for details). Let g be a finite dimensional complex simple Lie algebra. First, we recall the definition of the classical Yang-Baxter equation following Belavin and Drinfel'd [BD]. We fix an associative algebra A with unit containing g. Let r(u) be a meromorphic function with values in the tensor product g ® g. The functional equation for r(u) in A ® A ® A of the form [r12(ul -u2), r13 (ul - u3)] + [r12 (ul - u2), r23 (U2 - u3)] + [r13 (ul - u3) , r23 (u2 - u3)] = 0

is called the classical Yang-Baxter equation. Here the meaning of the suffix is as follows. We define the embedding cp12 : g ®g - A ®A ®A by W12 (a (& b) = a ®b ®1 and we put cp12(r(u)) = r12(u). Analogously we define r13(u) and r23(U)-

Non-degenerate solutions of the classical Yang-Baxter equation with the unitarity condition rl2(-u) = -r21(u) have been classified by Belavin-Drinfel'd [BD] into three classes - rational solutions, trigonometric solutions and elliptic solutions. We denote by {Iµ} an orthonormal basis of g with respect to the Cartan-Killing form. We put as in the previous section S2 = Eµ lµ ® I. Then the function r(u) = Il/u is a typical rational solution of the classical Yang-Baxter equation.

Let 7rj : g -+ End(Vj),1 < j < n, be finite dimensional representations of the Lie algebra g. Let r(u) be an arbitrary solution of the classical Yang-Baxter equation. We denote by rij(u) E End(Vi 0 . • • (9 V„), 1 < i, j < n, the operation of r(u) on the i-th and j-th components through the above representations. Let us consider the system of partial differential equation for a function (FJ(zl, • • • , zn) with values in Vl®...0Vnof the form (7.1)

^^

= E r,j(zg - zj) p. j,j#i

A solution of the above differential equation is considered to be a horizontal section of the meromorphic connection w=Erij(zi-zj ) (dz,-dzj) £<j

CHERN-SIMONS PERTURBATIVE INVARIANTS 253

for a trivial vector bundle over C' with fiber V1 ®• • • ® V,,. The following lemma was observed by Cherednik [Ch]. Lemma 7.2. If r(u) is a solution of the classical Yang-Baxter equation, then the equation (7.1) is consistent. Namely, we a havea a = a for any i, j.

Let us notice that the differential equation corresponding to the simplest rational solution r(u) = St/u is nothing but the original Knizhnik-Zamolodchikov equation discovered in [KZ]. We are going to discuss the integrable system defined by using an elliptic solution. In the following of this section, we restrict ourselves to consider the Lie algebra g = sl(N, Q. Let ej, 1 < j < N, be the standard basis of the complex vector space -11N. Let Al and A2 be the matrices defined by CN. We put e = e2""'_

A1e3 = e'-le„ (7.2)

1 < j < N

A2ei = ei +l, 1 < j < N - 1,

A2eN = el.

Then, Al and A2 satisfy A1A2 = eA2A1. Let a1 and a2 be the inner automorphisms of sl(N, C) defined by aj(X)=Aj1XA;, j= 1,2 for X E sl(N, C). We see that a1 and a2 are commuting automorphisms of order N and that they do not have a common non zero fixed vector. For 1, m E Z, we define Il(1,'m) by SZ(t,m) = (ala® ®1)(Sl). It can be checked that we have the relation

(al a2 (9 1)(1l) = (1(9 ai za2m)(SZ)• Putting a = (1, m), and considering a as an element of the direct sum ZN ® ZN, we write 1l' for 11(l, m). Here ZN denotes the cyclic group of order N. We can easily check the following lemma. Lemma 7.4. The above S2", a E ZN ® ZN, satisfies the following properties. (1) We have PSZ" = Q-", where P is the permutation operator defined by P(x®y) _ y®x. (2) The relation [f212 + Q13' Q23] = 0

holds if a - 0 + 'y = 0. Here the meaning of the suffix for S2 is the same as r;3 in the equation (1.1). (3) We have aEZN®ZN

254 TOSHITAKE KOHNO The elliptic solution which we are going to discuss appeared in the work of Belavin [B]. To describe the solution we first recall some basic properties of the Weierstrass C function. Let w1 and w2 be complex numbers with Im w2/w1 > 0 and L the lattice defined by L = {lw1 + mw2 11, m E Z}. The Weierstrass C function is defined by the series

1 1 1 1 zl z- w W w z wer,w70

which is a meromorphic function with simple poles at w E L. We put w3 = w1 + w2. The function ((z) is an odd function of z, with the properties ((z+wj) =((z)+2((?_

' j = 1,2,3. ) In particular , for w1 = 1 and w2 = r with Im r > 0, the function ((z) is also denoted

by C(zir). With the above notation, we put

p(z) = 1C(zI NT) + Q(,-) [((z - 1 - mTI NT) + ((l + mTI NT)] . 0
The following proposition was shown in [BD ](see also [E]). Proposition 7.5. The function p(z) satisfies the following properties. (1) p(z) is a meromorphic function which has only poles of order 1 at 1 + mr with 1, m E Z. The residue of p(z) at z =1 + mr is

(2) p(z + 1) = (al ® 1)p(z),

p(z + r) = (a2 ® 1)p(z)

where a1 and a2 are inner automorphisms of sl(N, C) defined above. (3) p(z) is a solution of the classical Yang -Baxter equation.

Moreover, p(z) is characterized by the above properties (1) - (3). It is known by Belavin and Drinfel'd [BD] that such elliptic solution exists only for sl(N, C). Since a1 and a2 are automorphisms of order N, it follows from the above property (2) that we have

(7.6) p(z + N) = p(z), p(z + NT) = p(z). This implies that p(z) defines a meromorphic function on the elliptic curve EN = C/LN, with the lattice LN = {lN + mNr 11, m E Z}. On the elliptic curve E = C/Z + Zr, p (z) defines a multivalued meromorphic function with only one pole.

CHERN-SIMONS PERTURBATIVE INVARIANTS 255 Let HN denote the Heisenberg group with generators x, y with relations xN = yN = 1, [[x, y], x] = [[x, y], y] = 1.

The central element [x, yJ is denoted by c. We have the following exact sequence 0 -* ZN -+ HN4-+Hl(E, ZN) -+ 0

where Hl (E, ZN) = ZN®ZN has as a basis the homology cycles corresponding to the deck transformations A and p defined by A(z) = z + 1 and p(z) = z + r respectively. The above map p : HN -+ H1(E, ZN) is given by p(x) = A and p(y) = p. We have the embedding t : HN -4 GL(N, C) defined by t(x) = Al, t(y) = A2 and j(c) = EI, where the matrices Al and A2 are given as in (1.7). Let s : H1(T;ZN) -+ HN be the map defined by s(lA + mp) = AiA2 . In the following, we fix the above section s for the above exact sequence. As is the previous section, we denote by EN the elliptic curve C/LN. We fix finite dimensional representations 7rj : sl(N, C) -' End(Vj), 1 < j < n. Let us consider the meromorphic 1-form on Cn with values in End(V1 ® • (D Vn) defined by W = E pij(zi - zz) (dz; - dzj), 2Ir lac 1<{<,j
The Heisenberg group HN acts on EN by x(z) = z+1, y(z) = z+r and c(z) = z, which induces a natural action of the direct sum HNn = HN (D - • • ® HN on EN. On the other hand, HNn acts naturally on V1 ® • • • ® Vn through t : HN -+ GL(N, C). It follows from part (2) of Proposition 7.5 that the connection w is compatible with the action of HNn. Considering the quotient by this action, we obtain a projective local system Z over En. The induced connection does not have poles on Confn(E). We call the above local system Z the elliptic KZ system. The 1-form w defines a projectively flat connection on Z. The holonomy of this connection gives a projectively linear representation of the pure braid group of the torus with n strings 0: ir1(Con fn(E), *) GL(V1(9 ... ® Vn).

256 TOSHITAKE KOHNO

Let us notice that the meromorphic 1-form w defined on C" is written as (! m)

1 27r K 1
-

1

(dzi - dzj) + cp - mT

with a holomorphic 1-form cp. We describe the relations satisfied by the matrices SZ^Z''n). Since S m) = Il(' `') if l = 1, m = m' modulo N, we consider a = (1, m) as an element of ZN ® ZN . It follows immediately from Lemma 7.4 that the matrices Stt^, 1
(2) [Slij + cik, S1 k] for distinct i, j, k with a - /3 + ry = 0, (3) [S1 , S2ki] = 0 for distinct i, j, k,1, (4) E-EZN ® ZN QU = 0. Let us notice that in the case N = 1 the above relations (2) and (3) were called infinitesimal pure braid relations in [Kol] and [Ko2]. Let us describe the monodromy representation 0 in terms of the iterated integral of the 1-form w . We take an element of irl(Confn (E),*), which is lifted to a path ry(t), 0 < t < 1 , in C' with a basepoint 7(0) = (x°, x2) • • • , xn ). We suppose that < xn < 1. For each j, the basepoint satisfies x° • • , xn E R and 0 < x° < 1 < j < n, we denote by l j E L the deck transformation sending -yj(0) to yj(1). Identifying the lattice L with H1 ( E, ZN), we obtain an element of the Heisenberg group s(lj) E HN by means of the section s defined in the previous section. Let us recall that the Heisenberg group HN acts naturally on the representation space Vj. We denote by Xj the linear transformation on V1 ®• • • ® V„ obtained as the action of s(l j ) on the j-th component of V1 ®• • • ® Vn.

[0, 1] -* Cn, we set y'w = a(t) dt. We consider the iterated

Pulling back w by integral

1

LOW W m

^o
a(tl)a(t2)

... a(tm) dt1dt2 ... dt m•

We have the following Proposition. Proposition 7.6. The holonomy of the local system ,C over Con f(E) for the horizontal section is expressed as the sum of the iterated integrals 9('y)=X1X2...XnI+J Iw+ f ww+•••+J ww w+ •• with the linear transformations X1i • • , Xn defined above. This determines a projectively linear representation

0: 7rl (Con fn(E), *) -* GL(Vi ® ... (9 Vn).

CHERN-SIMONS PERTURBATIVE INVARIANTS

257

Here the associated 2-cocycle c determined by e(xlx2 ) = C(x1,x2 )0( xl)B(x2 ), x1,x2 E 7r,(Cionfn(E), satisfies c(x1, x2)N = 1.

For a fixed elliptic curve E = C/Z + Z-r, the above construction gives projective representations of the pure braid group of the torus with parameter ic. The term with the iterated integral of length m contains n-m. Now the symmetric group S„ acts diagnally on the local system G, where the action of the fiber Vl ®• • • ® V. is defined by the permutation of the components of the tensor product and the action on the base space Confn(E) is defined by the permutation of the coordinates. For a loop in Con fn(E), which is lifted to a path 'y connecting x and a(x) with x E Confn(E) and a E S,,, the holonomy of the above local system gives a map

0(7):Vi ®...®Vn- V0(1) ®...®Vo(n). In particular, if Vl = . . . = Vn, then we get projective representations of the braid group of the torus. The explicit form of such representations was computed by Etingof [E]. We are going to discuss a slightly modified version of the general framework developed in the previous section in the case of the torus. Let E = C/Z + Zr be an elliptic curve with the basis A, a of Hl (E, Z) as in the previous section. Let g be the Lie algebra sl(N, C) and we fix representations 7rj : g -> End(V ),1 < j < n. The Heisenberg group HN acts naturally on Vj by means of the embedding t : HN -p GL(N, C) defined by the matrices Al and A2 as in (7.2). We consider the projectively linear representation a : H1(E,Z) -+ Aut(Vi), 1 < j < n defined by A ^--+ A, and p '--* A2.

Let F be a chord diagram on the torus with n circles. Then, by means of the process of the contraction of the tensors using the above representations 7rj, 1 < j < n, and the projectively linear representation a of H1(E, ZN), we obtain a scalar, which is denoted by T(I'). Let us notice that since our representation of the fundamental group of the torus is projectively linear, T(F) is only well-defined up to a multiplication of a N-th root of unity. In this section, we deal with the case of the torus with the projective local system defined in the previous section. For a chord diagram P on the elliptic curve E = C/Z + Zr we have defined the weight T(I') satisfying the 4 term relation. We are going to construct a Vassiliev invariant v of an oriented framed link in E x I satisfying w(v) = T. Our method is based on the holonomy of the elliptic KZ system. If the link L is contained in a 3-ball, then the invariants we are going to

258 TOSHITAKE KOHNO construct coincide with the usual Vassiliev invariants of oriented framed links in R3 for sl(N, C). Before explaining our construction of Vassiliev invariants for links, let us first describe tangles in E x I. We set Et = E x {t} C E x I, 0 < t < 1. A tangle T in E x I is a one-dimensional submanifold with boundary of E x I such that the boundary 8T is contained in Eo U El. Let J denote the segment in E defined as the image of the open interval (0, 1) by the covering map 7r : C -4 E. We suppose that 8T fl Eo and 8T fl El consist of distinct points in the segment J. In the following we consider a tangle in E x I such that each connected component is oriented and framed. Let T be an oriented framed tangle in E x I. To each connected component T, of T, we assign a finite dimensional representation of sl(N, C). We consider the parameter t for the unit interval [0, 1] as a height function. Deforming the tangle up to regular isotopy, we may suppose that there exists a partition of the unit interval 0 = to < tl < • • . < ti < ti+l < ... < tp = 1 satisfying the following conditions: (1) For each i, 1 < i < p, Et; intersects transversely with the tangle T, and T fl Et; consists of distinct points in the segment J.

(2) The restricted tangle T fl E x [ti, ti+1] is one of the following three types. (i) a tangle with only one minimal point, (ii) a tangle with only one maximal point, (iii) a braid of E.

We denote by n(i) the number of points in T fl Et; and we put T fl Et; _ {z1, z2, ... ,

zn(i)}

with z1, • .. , zn(i) E J and z1 <

z2 < ... <

zn(i ).

Let Vii,

1 < j < n(i), be the representation of sl(N, C) assigned to the component of T passing through zj E T fl Eti. We correspond to zj E T fl Et, the representation 23 where e(i) is 1 or -1 according as T passes through Et; downward or upward. The notation U'j stands for U; if e = 1 and for the dual representation of V if e=-1. For each ti, 1 < i < p, we consider the tensor product V(ti) =

ll(l) ® 122) ®... ® i,n(i;))

Let us denote by Ti,i+l the tangle T restricted to the interval [ti, ti+1] . We are going to construct a map Zi+1:V(ti)-*V(ti+l ) , 0
Our construction is quite similar to the well-known one due to Reshetikhin and Turaev [RT] and others , except that we are considering braids of the torus. In the case when the tangle Ti,i+1 is a braid of the torus, we assign the linear map Z +1 : V (ti) -* V ( ti+1) obtained as the holonomy of the elliptic KZ system.

CHERN-SIMONS PERTURBATIVE INVARIANTS 259

Let us now consider the case when the tangle Ti,i}1 contains only one minimal point at t = si, ti < si < ti+1. We set V(si) = V(ti) and to the tangle restricted to [ti, si] we assign the identity map. We denote by U the tangle T restricted to [si, ti+1]. Let e be a sufficiently small positive number and we decompose the tangle U into 2 parts, [si , si+e] and [si+e, ti + 1]. We denote by UE the tangle restricted to [si+e, ti+i] To the tangle Ue we assign the linear map fE : V(si + e) -, V(ti+1) obtained as the holonomy of the elliptic KZ system . We have a natural injection e : V (si) --, V (si+e) determined by the canonical embedding C -, VC(') ®V 1). Here V^+i1) is the dual +i representation of V^('). We define Z(Ue) : V (si) --, V (tSt1) to be the composition fE o e. For t satisfying si < t < tit1, we denote by (x1(t), • • • , x, (t)) the coordinates of the points in T fl Et. We may suppose that 0 < x1(t) < • • • < x„(t) < 1 and we put xj + 1 - xj = E. It can be shown as in the case of the KZ system , the solution of the elliptic KZ system is written in the form J+1 exp (2f2j 7rloge h(z1(t) ... ,zn(t)) r. for e > 0, where h is convergent when e tends to 0 (see also [E]). Hence we show as in the proof of Proposition 5.1 in [LM] that the limit

Z(U) =1 o Z(UE) exp (- 2^' K loge) is convergent. This construction defines a linear map Z(U) : V(si) -+ V(ti+1).

By composing the identity map V (ti) -, V(si), we obtain the map Z +1 : V(ti) V (ti}1). In the case when the tangle Ti, i+1 contains only one maximal point, we define Zi+1 in a similar way using Z(fl) = limexp K loge) Z(n,).

E-o 2^r

Now we define Z(T) by the composition 4-1 ... Z1 Zo. Using the integrability of the elliptic KZ system, we can show in a similar way as in [BN] the following proposition. Proposition 7.7. For an oriented framed tangle T in E x I, the map Z(T) : V(0) -, V(1) is invariant by a horizontal move preserving the framing, up to a multiplication of a N-th root of unity. Let L be an oriented framed link in E x I with n components . To each component L; we assign V„ a finite dimensional representation of sl(N, C) and we regard L

260 TOSHITAKE KOHNO

as a colored oriented framed tangle. The above construction gives a linear map Z(L) : C -* C. We denote by the same symbol Z(L), or Z(L; V1, • • , V„) the complex number Z(L) (1). Let C be a trivial knot with 0-framing possessing 2 minimal points and 2 maximal points. We put -y, = Z(C; V,). As in [K1] ( see also [BN] and [LM]), we normalize Z(L) as Z(L) = yl ml ... 7nm. Z(L), where mm, 1 < j < is, is the number of maximal points on the j-th component of L. The above Z(L) has an expansion with respect to h = i- 1 of the form Z(L) = 20(L) + Zk(L)hk. k>O

Theorem 7.8. Let L be an oriented framed link in the product of an elliptic curve E and the unit interval I. Then, up to a multiplication of N-th root of unity, Z(L) satisfies the following properties. ( 1) 2(L) is a regular isotopy invariant of L. (2) 2k(L) is a Vassiliev invariant of order k.

(3) For r a chord diagram on E with k chords , we have w (Zk)(r) = T(r).

REFERENCES [AMR] J. E. Andersen, J. Mattes and N. Reshetikhin, The Poisson structure on the moduli space of flat connections and chord diagrams, Topology 35-4 (1996), 1069-1083. [AS] S. Axelrod and I. M. Singer, Chern-Simons perturbation theory, Proc. XXth DGM Conference, (S. Catto and A. Rocha eds) (1992), 3-45. [BN] D. Bar-Natan, On Vassiliev knot invariants, Topology 34-2 (1995), 423-472. [B] A. A. Belavin, Discrete groups and the integrability of quantum systems, Funkts. Anal. 14-4 (1980), 18-26. [BD] A. A. Belavin and V. G. Drinfel'd, Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funkts. Anal. 16-3 (1982), 1-29. [BL] J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270. [BT] R. Bott and C. Taubes, On the self-linking of knots, preprint, Harvard University. [Ch] I. Cherednik, Generalized braid groups and local r-matrix systems, Sov. Math. Dokl. 307 (1990), 43-47. [E] P. Etingof, Representations of affine Lie algebras, elliptic r-matrix systems, and special functions, Commun. Math. Phys. 159 (1994), 471-502. [F] K. F ikaya, Morse homotopy and Chern-Simons perturbation theory, preprint. [FG] D. Freed and R. Gompf, Computer calculation of Witten's 3-manifold invariant, Commun. Math. Phys. 141 (1991), 79-117. [GMM] E. Guadagnini, M. Martellini and M. Mintchev, Wilson lines in Chern-Simons theory and link invariants, Nucl. Phys. B330 (1990), 575-607. [G] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), 263-302.

CHERN-SIMONS PERTURBATWE INVARIANTS 261 [J] L. Jeffrey, Chern-Simons invariants of lens spaces and torus bundles, and the semi-classical approximation, Commun. Math. Phys. 147 (1992), 563-604. [KZ] V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys. B247, 83-103. [Kol] T. Kohno, Serie de Poincar6-Koszul associee aux groupes de tresses pures, Invent. Math. 82 (1985), 57-75. [Ko2] T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139-160. [Ko3] T. Kohno, Topological invariants for 3-manifolds using representations of mapping class groups I, Topology 31 (1992 ), 203-230. [Ko4] T. Kohno, Vassiliev invariants and de Rham complex on the space of knots, Contemp. Math . 179 (1994), 123-138. [Ko5] T. Kohno, Elliptic KZ system , braid group of the torus and Vassiliev invariants, Topology and its Appl. 20 (1997), 1-16. [Kl] M. Kontsevich , Vassiliev 's knot invariants, Advances in Soviet Mathematics 16 (1993), 137-150. [K2] M. Kontsevich , Feynman diagrams and low dimensional topology, Proceedings of the first European Congress of Mathematicians. [LM] Le 71r Quoc Thang and J . Murakami , Representation of the category of tangles by Kontsevich 's iterated integral, Commun. Math. Phys. 168 (1995), 535 - 562. [L] X.-S. Lin, Finite type link invariants of 3-manifolds , Topology 33 (1994), 45-71. [R] J. M. Rabin , Introduction to quantum field theory for mathematicians, Geometry and quantum field theory (D. Freed and K. Uhlenbeck eds) (1995), 183-269. [RT] N. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups , Invent. Math . 103 (1991 ), 547-597. [Flo] L.Rozansky, Large k asymptotics of Witten 's invariant of Seifert manifolds, Commun. Math . Phys. 171 (1995), 279-322. [T] C. Taubes, Homology cobordism and the simplest perturbarive Chem-Simons 3-manifold invariant, preprint. [V] V. A. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, Amer. Math. Soc. (1992). [W] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math . Phys. 121 (1989), 351-399. DEPARTMENT OF MATHEMATICAL SCIENCES , UNIVERSITY OF TOKYO, KOMABA , MEGUROKU, TOKYO 153, JAPAN

E-mail address : [email protected]

This page is intentionally left blank

Lectures at Knots 96 edited by Shin 'ichi Suzuki ©1997 World Scientific Publishing Co. pp. 263-290

COMBINATORIAL METHODS IN DEHN SURGERY C. MCA. GORDON

1. INTRODUCTION

This expository paper is an expanded version of part of the author's lectures at the Workshop. The remaining part of the lectures covered material that is discussed in Sections 2 and 3 of the survey article [Go].

Here, we give an introductory account of the combinatorial methods developed by John Luecke and the author (mainly in the proof of the Knot Complement Conjecture [GL2]) for studying various questions about Dehn surgery. The setting is that we have two Dehn fillings M(a), M(,.3) on a 3-manifold M, which contain surfaces P, Q respectively, and the main goal is to obtain sharp upper bounds on the intersection number of a and ,Q, under certain hypotheses on P and Q. In Section 2 we describe how P and Q give rise to labelled intersection graphs GP C P, GQ C Q. The idea of using such graphs in the context of Dehn surgery is due to Litherland [L]. In Section 3 we introduce the algebraic notion of a type, which plays a key role in the theory. In Section 4 we show how certain faces or collections of faces of Gp give topological information about M(,Q) (and/or Q), and of course similarly for GQ. Section 5 contains the main combinatorial assertion , Theorem 5.1, which says that, if Gp and GQ are graphs as in Section 2, then either Gp represents all types or GQ contains a configuration we call a (p - X(P))-web. Finally, Section 6 contains some applications, mainly to Dehn surgery on knots in S3, although we also say something about the case where M(a) and M(f3) are reducible. We have tried to keep the discussion as general as possible, allowing P and Q to have arbitrary genus until the last part of Section 5. It must be admitted that this generality is somewhat bogus, inasmuch as in all the applications in Section 6, Q 263

264 C. MCA. GORDON

is a 2-sphere and P is either a 2-sphere or a torus. (For statements which apply to surfaces of arbitrary genus see [T] and [R].) The main point is that if Q is a 2-sphere then, if GQ contains a (p - X(P))-web, it contains one that is innermost in the obvious sense, and in such a great web one can find useful faces, namely Scharlemann cycles. Nevertheless, we hope that presenting the material in this way clarifies the logical structure of the argument.

2. GRAPHS OF SURFACE INTERSECTIONS

Let M be a compact, orientable, irreducible 3-manifold with torus boundary. Let a be a slope, that is, the isotopy class of an essential embedded circle, on OM. Then the result of a-Dehn filling on M is the manifold M(a) = M U V, where V. is a solid torus, glued to M via a homeomorphism aM --+ W. taking a to the boundary of a meridian disk of V.. If a and,3 are two slopes on aM, then 1(a, )l3) will denote the minimal geometric intersection number of a and ,Q. One finds in practice that statements of the following form tend to hold, and this is the main type of result that we shall be seeking: if M(a) and M(,3) have certain special properties, then A(a, i3) is bounded above by some L. The bound L will depend on the properties in question. The situation in which we are particularly interested, and to which our methods apply, is when M(a) and M(,3) contain certain kinds of surfaces. So suppose that P, Q are (closed, orientable) surfaces in M(a), M(/3) respectively. We may assume that P meets V« in a finite collection of meridian disks, so that P = P fl M is a surface in M each of whose boundary components has slope a. Similarly, Q gives rise to a surface Q in M whose boundary components have slope 0. For our machinery to be applicable, it is important that aP and aQ be non-empty. So, we may state our basic assumption as follows: (Al) there are properly embedded surfaces (P, OP), (Q, aQ) C (M, OM) such that aP and aQ are non-empty, and each component of aP (resp. OQ) has slope a (resp. N. By an isotopy of (say) P we may assume in addition that (A2) P and Q intersect transversely, and each component of aP meets each component of aQ in exactly A(a, 0) points.

It is clear, however, that for any a and 0 we can always find surfaces P and Q, of arbitrary genus, satisfying (Al) and (A2), and therefore we must impose

COMBINATORIAL METHODS IN DEHN SURGERY 265

some additional conditions if we are to extract any non-trivial information from this situation . The technical assumption which enables us to do this is: (A3) each arc component of P n Q is essential in P and in Q. By (A2), P n Q consists of a finite disjoint union of circles and properly embedded arcs, the endpoints of the arcs being the points of intersection of aP with aQ. Condition (A3) says that for no arc component ry of P n Q is there an arc S C aP, with ary = aS, and a disk D C P such that aD = ry u S (and similarly for Q). We now describe some conditions under which (A3) holds. Recall that a closed orientable surface S in a 3-manifold N is essential if either S has positive genus, is incompressible in N, and is not parallel to a component of ON, or S is a 2-sphere which does not bound a 3-ball in N. First suppose (a) P is essential in M(a) and M contains no essential surface homeomorphic to P.

Let K0 C M(a) denote the core of the solid torus V0. We choose P (among all essential surfaces in M(a) homeomorphic to P) so that p = IaPI = IP n K«1 is minimal. Then p > 0 by hypothesis. Also, standard arguments show that P is incompressible and boundary incompressible in M. In particular, we may assume that no circle component of P n Q bounds a disk in Q. If Q also satisfies (a) (with M(a) replaced by M(3)), and is chosen to minimize q = I OQI = IQ n Kp I , then, again by standard arguments , we may assume that condition (A3) holds.

Another case of interest is (b) P is a Heegaard surface for M( a) and K0 cannot be isotoped to lie on P. If P satisfies (a) and Q satisfies (b) (for M(,Q)), then Gabai [Ga2] shows that (A3) can be achieved if Kp is put in thin position with respect to Q. (Gabai explicitly treats the case when M(,Q) = S3 and Q = S2, but his argument carries over verbatim to any Heegaard surface Q.)

Finally, if P and Q both satisfy (b) (for M(a) and M(,3) respectively), then Rieck has shown [R], using thin position, that again (A3) can be assumed to hold. (The case M(a) = M(/3) = S3, P, Q = S2 is done in [GL2], and was also proved independently by Gabai (unpublished).) Given surfaces P, Q in M satisfying (Al), (A2) and (A3), we focus on the arc components of P n Q, as they lie in P and in Q. We regard these arcs as defining graphs Gp and GQ in P and Q respectively, in the obvious way. Thus the (fat) vertices of Gp are the disks P - int P, the edges of Gp are the arc components of P n Q as they lie in P, and similarly for GQ. We now encode some additional structure, as follows. Number the components of aP 1, 2, ... , p in the order in which they appear on OM, and similarly number the

266

C. MCA. GORDON

components of aQ 1 , 2, ... , q. This gives a corresponding numbering of the vertices of Gp and GQ. Consider an arc component of P fl Q. Its endpoints are points of aPnaQ, the intersections of (say) the ith component of aP with the jth component of aQ, and the i'th component of aP with the j'th component of OQ. We then label the endpoints of the corresponding edge of Gp with j and j' at vertices i and i' respectively, and similarly for GQ. See Figure 1.

GP

GQ

Figure 1

We next give a sign, + or -, to each vertex of Gp, according to the direction on OM of the orientation of the corresponding component of aP, induced by some chosen orientation of P. Equivalently, this is the sign of the corresponding intersection point of KQ with P (with respect to some chosen orientations of M, P and Ka). In particular, if P separates M(a) then the signs of the vertices 1, 2, ... , p of Gp alternate . Similarly, we give a sign to each vertex of GQ. Note that around each vertex of Gp the edge-endpoint labels occur in order 1, 2,... , q, 1 , 2,... , q__, repeated 0 = A(a, Q) times, the ordering being (say) anticlockwise at a positive vertex and clockwise at a negative vertex . See Figure 2. Similarly, around each vertex of GQ w e see the labels 1 , 2, ... , p, 1, 2, ... , p, ... , repeated 0 times.

Figure 2

COMBINATORIAL METHODS IN DEHN SURGERY 267 Since M, P and Q are orientable, an arc component of P fl Q joins points of intersection of OP with aQ of opposite sign. Hence we have the parity rule: if an edge of Gp joins vertices i and i' and the corresponding edge of GQ joins vertices j and j', then i and i' have the same sign if and only if j and j' have opposite signs. (See Figure 1.)

Condition (A3) becomes, in graph-theoretic language, the condition that Gp and GQ contain no trivial loops. We will denote by q the set of (edge-endpoint) labels 11, 2,... , q} of Gp. We also have the associated q-intervals ( 1, 2), (2, 3), ... , ( q - 1, q), (q, 1).

Note that if D is a disk face of Gp, then aD consists of an alternating sequence of edges and corners, where the edges are edges of Gp (i.e., arc components of P fl Q), and the comers are q-intervals (i, i + 1) (i.e., arcs in aP between consecutively labelled components i, i + 1 of aQ. See Figure 3.

Figure 3 Similar remarks apply to GQ. We may describe the philosophy behind our approach as consisting of two parts: Combinatorial: use the one-one correspondence between the edges of the labeled graphs Gp and GQ to show that either Gp contains certain configurations or GQ contains certain (perhaps other) configurations. Topological: use the existence of certain collections of faces of Gp (resp. GQ) to get topological information about the pair (M(,0), Q) (resp. (M(a), P)). These will be elaborated in Sections 5 and 4 respectively. Remark. For simplicity we have assumed that aM is a torus, but everything carries over without much change if we allow M to have additional boundary components

268

C. MCA. GORDON

(with a and ,3 being slopes on some fixed torus component of 8M). In this more general setting, we may also allow P and/or Q to have non-empty boundary. 3. TYPES We now introduce a purely algebraic concept, that of a type, which plays an important role in the theory under discussion. The reader may choose to skip to Section 4 and return to the present section when necessary. Let q be a positive integer. A q-type is a q-tuple (el, E2, ... , Eq), where E; = f, 1IajI?2. :-1 Condition (2) merely excludes the zero vector, together with vectors all of whose co-ordinates are zero except one, which is ±1. Note that a represents r if and only if it represents -r. Also, there are 2q-1 q-types up to sign , including the trivial type, which by definition has e = E2 = • • • = Eq. Example . There are 22 = 4 3-types (up to sign): +++, -++, +-+, and ++-. (1, 0, -2) represents - + + and + + -. A set A C V represents all q-types if and only if, for each q-type r, there exists a E A such that a represents r.

Examples. (1) If Jkl > 2, then {(O,... , 0, k, 0,... , 0)} represents all q-types. (2) {(1, 0, 1), (-1, 1, 0), (1, 1, -1)} represents all 3-types. (3) If 1 < r < q, then A C V represents all q-types if and only if {(al, ... , a,., 0, a,.+1, ... , aq) : (al, ... , aq) E Al represents all (q + 1)-types.

For A C Vq, let c(A) be the number of co-ordinates 1 < i < q such that a; # 0 for some a E A. Then we may regard A C 7Z,(A) C V in the obvious way. By (3) above, A represents all c(A)-types if and only if it represents all q-types. Thus we may use the phrase A represents all types without ambiguity. The key property of types is the following "all types implies torsion " theorem due to Parry [P]. Theorem 3.1 (Parry). If A C V represents all types then there exists AO C A such that Z.q modulo the subgroup generated by Ao has non-trivial torsion.

COMBINATORIAL METHODS IN DEHN SURGERY 269

We remark that it is not true that Ao can be chosen to represent all types; equivalently, it is not true that if A C V is a minimal set of representatives of all types then V/(A) has non-trivial torsion. (Here, minimal means that no proper subset of A represents all types.) For example, A {(1, 2,1), (-1,1,1), (1, -1, 2), (1,1, -1)} is clearly a minimal set of representatives of all types, but Z3/(A) = 0. A useful fact, proved in [GL4, Lemma 4.4], is that if A C V represents all types then either A contains a basis for Rq or there exists Al C A such that c(Al) < q and Al represents all types. As an immediate consequence we have Lemma 3 .2. If A is a minimal set of representatives of all types then A contains a basis for W(A).

4. THE TOPOLOGY

Consider the solid torus Vp C M(,8). The surface meets U,3 in q meridian disks, cutting Vg into q 3-balls. For each q-interval A = (i, i + 1 ), let H,, denote the 3-ball (1-handle) consisting of that part of Vp between the meridian disks whose boundaries are components i and i + 1 of Q. Let D be a disk face of G. Then D C M C M(,3), where the edges of 8D are arcs in Q, and the corners (i, i+ 1) of OD are arcs in 8P running along the corresponding 1-handle H(t,t+i) • See Figure 4, which shows 8D for the face D illustrated in Figure 3.

Figure 4 Let D be a set of disk faces of Gp. Let c(D) be the set of q-intervals A = (i, i + 1)

270

C . MCA. GORDON

that occur as corners of faces in D. Let e(D) be the set of edges of GQ corresponding to edges belonging to faces in D. Suppose that the edges e(D) are contained in a disk E C Q. (Note that this will necessarily be the case if Q is a 2-sphere.) We take E to also contain the fat vertices of GQ at the endpoints of the edges in e(D). Recall that if Q is an essential surface, then we may assume that no circle component of P fl Q bounds a disk in Q. Hence (int D) fl Q = 0. If Q is a Heegaard surface, this need not hold. Assume for the moment, however, that (int D) fl E = 0; we shall see later (at the end of the present section) that in the cases of interest this can be achieved after some disk exchanges on int D.

Define a 3-manifold N(E, D) C M(,Q) by N(E,D) =nhdlEU U HAUDI . \\ AEc(D)

Note that N(E, D) has a natural handle decomposition, with a single 0-handle (corresponding to E), the 1-handles Ha, and 2-handles whose cores are the elements of D. Also, if W denotes the handlebody nhd(E U UAEC(D) HA), then we can read off the element of 7r, (W) (and hence of Hl (W)) represented by the boundary of a disk D E V by reading the sequence of corners of D (with sign, given by the sign of the corresponding vertex of GP) as we go around 8D in some direction. The idea is to find D such that N(E, D), or perhaps the surface ON(E, D), gives topological information about M(,3), or the surface Q, possibly a contradiction. The simplest example of this philosophy is when D consists of a single face of Gp which is a Scharlemann cycle. This is a disk face D of Gp such that all the vertices of D have the same sign, and all the corners of D are the same q-interval A = (i, i + 1). See Figure 5. Scharlemann cycles first appeared (in a slightly different context) in [Si]. The following theorem, which is contained in [S2, Proof of Proposition 5.6], is immediate; see Figure 6. (Note that vertices i and i + 1 of GQ are of opposite sign, by the parity rule.)

Figure 5

COMBINATORIAL METHODS IN DEHN SURGERY 271

Figure 6 Theorem 4.1. Let D be a Scharlemann cycle in GP, such that the edges e(D) lie in a disk E C Q. Then N(E, D) is a punctured lens space. Note that 7r,(N(E, D)) = Zk, where k is the length of D, i.e., the number of edges in 8D. Let us examine the implications for (M(,Q), Q) of the existence of a Scharlemann cycle in Gp as in Theorem 4.1. (Sa) If Q is a Heegaard surface, we conclude tht M(,3) has a lens space summand. Now suppose that Q is an essential surface. Note that (after a small isotopy of N(E, D)) we may assume that 8N(E, D)nQ = E. Let Ebe the disk 8N(E, D) - E, and let Q' = (Q - E) U E'. Then Q' = Q. Also, IQ' n Kph = q - 2, since the two points of intersection of Kp with Q corresponding to the labels i and i + 1 that appear in the corners of the Scharlemann cycle D, have been eliminated. We therefore have the following. (Sb) If Q is an incompressible surface of positive genus, then Q' is also incompressible. But this is a contradiction, because of the minimality of q. (Sc) If Q is an essential 2-sphere, then M(/3) has a lens space summand. In fact, by the minimality of q, Q decomposes M(,0) as M'# (lens space). (Sd) For a variant of (Sc), let us say that a 2-sphere S in a 3-manifold N is Qessential if S does not bound a Q-homology ball in N. (In particular, this is the case if S is non-separating.) It is clear that if Q is Q-essential in M(/3), then so is Q'. Therefore, if we choose Q so that q is minimal over all Qessential 2-spheres in M(Q), then the existence of a Scharlemann cycle in Gp is a contradiction.

272 C. MCA. GORDON

Although Scharlemann cycles are very useful, their existence is not guaranteed in all situations of interest. A more global concept, which gives similar topological conclusions, is that a set of faces representing all types. A q-type is a q-type r = (el, e2, ... , s9) (see Section 3), where the ith co-ordinate of T is formally associated with the ith q-interval (i, i + 1). Let D be a disk face of Gp. By taking the algebraic sum of the corners of D we get an element [D] E 7G9, the free abelian group on the set q-intervals. We say that D represents the q-type r if and only if (1) [D] represents r (in the sense of Section 3); and (2) (coherence condition) no q -interval (i, i + 1) appears as a corner of D at two vertices of opposite sign.

As an example, the disk D shown in Figure 3 fails to satisfy the coherence condition. A set of (disk) faces D of Gp represents all types if and only if for each q-type r there exists D E D such that D represents r. If such a set of faces exists we say that Gp represents all types.

Remark. The role of the coherence condition (2) is twofold. First, it ensures that condition (2) in the algebraic definition of representing a type (see Section 3) is automatically satisfied (since Gp has no trivial loops). Second, it makes it possible to search for representatives of types in a more organized fashion, as we shall see in Section 5 (cf. Proposition 5.2). If D is a face of Gp such that small neighborhoods in D of the edges e(D) all lie on the same side of Q, we say that D lies locally on that side of Q. To get the topological conclusions that we want from a set of representatives of all types, we first need to observe the following (cf. [GL2, Proof of Proposition 3.2, part (A)]). Lemma 4.2. Let D be a set of faces of Gp representing all types. Then there exists D' C D such that D' represents all types and the elements of D' all lie locally on the same side of Q. Proof. Label the two local sides of Q in M(,3), B and W (black and white). Let A = (i, i + 1) be a q-interval, and recall the definition of the 1-handle Ha at the beginning of this section. Orient (the core of) Ha from the end corresponding to component i of 8Q to the end corresponding to component i + 1 of OQ. Then we have the partition

{q-intervals } =13 LI W LI 13W LI WB , where

COMBINATORIAL METHODS IN DEHN SURGERY 273

B E

SW

B to B if Ha runs from

WB

BtoW W to B

Suppose the conclusion of the lemma is false. Then, if B # 0, there exists a B-type TB such that no member of D represents TB. Similarly, if W # 0, there exists a W-type TW such that no member of D represents TW. Define a q-type r by TIB=TB, if B#0

TlW=TW, if VV #0;

Tea=+d\EBW TI,\ =-,VAEWB.

By hypothesis, there exists D E D such that D represents T. Note that, for some orientation of 8D, every corner of D in BW U WB runs from B to W, and hence D can contain no such comer. Therefore either all corners of D belong to B, or all corners belong to W. It follows that either B 34 0 and D represents TB, or W 34 0 ❑ and D represents TW, contradicting the definitions of TB and TW. Theorem 4.3. Let D be a set of faces of Gp representing all types, such that the edges e (D) lie in a disk E C Q. Then there exists D' C D such that the elements of D' all lie locally on the same side of Q, Hl (N(E, V')) is finite and non-zero, and 8N(E,V) = S2. Proof. (cf. [GL4, proof of Proposition 4.5]). We may assume that D is minimal. Hence, by Lemma 4.2, the elements of D all lie locally on the same side of Q. Also, by Lemma 3.2, there exists D' C D such that {[D] : D E V} is a basis for R`( Let N' = N(E,1 '). Then it follows that H1(N') is finite and ON' = S2. It remains to show that H1(N') 54 0. Let N = N(E, D) = nhd(N' U (D - D')). Since ON' = S2, attaching to N' the 2-handles whose cores are the elements of D - V is equivalent to removing from N' an equal number of open 3-balls. By Theorem 3.1 there exists Do C V such that H1(NO) has non-trivial torsion, where No = N(E,Do). Note that No C N C N'. Suppose H1(N') = 0. Then N' is a homology 3-ball, and the Mayer-Vietoris exact sequence gives Hi(8No) = H1(No)®H1(N'-No), contradicting the fact that H1(No) ❑ has non-trivial torsion. Hence Hl (N') 54 0, as desired. Remark. Since H1(N) = H1(N'), the above proof shows that if D is a minimal set of representatives of all types then H1(N(E, D)) is finite and non-zero. As was noted in Section 3, the analogous statement is not true in the purely algebraic setting. It follows from Theorem 4.3 that if Gp represents all types, and the edges of the relevant faces are contained in a disk in Q, then we get conclusions almost identical to (Sa), (Sb), (Sc) and (Sd) above for Scharlemann cycles.

274 C. MCA. GORDON

(Ta) If (Q is a Heegaard surface, then M(,Q) has a summand 1V' with H1(1V') finite and non-zero.

(Tb) If Q is an incompressible surface of positive genus, this is a contradiction. (We get a new surface Q', obtained from Q by a disk exchange, with IQ' fl Kpl _ q - I S(D)I.) (Tc) If Q is an essential 2-sphere, then Q decomposes M((3) as M' # N', where H1(N') is finite and non-zero. (Td) If Q is a Q-essential 2-sphere, this is a contradiction.

Finally, we return to the assumption, when Q, is a Heegaard surface, that (int D) fl E=0. Let D be a collection of faces of Gp representing all types (this includes the case where D consists of a single Scharlemann cycle). We may assume that (int D) fl E consists of a finite number of disjoint circles and arcs, properly embedded in E. Let -y be such a circle or arc, and let El and E2 be the components of E - 'y. There is a corresponding partition

c(D)=C1UC2UC12i where A E C; if the 1-handle Ha has both ends in E;, i = 1, 2, and A E C12 if H,\ has one end in El and one in E2. Then, exactly as in the proof of Lemma 4.2 (the c(D)-type rr here will be defined by orienting all A E C12 so as to run from (say) E1 to E2), there exists D' C D such that D' represents all types and c(Y) C C1 (say). Since we may assume that D is a minimal set of representatives of all types, we conclude that the vertices of GQ corresponding to the labels that appear in the corners c(D) all lie in El. If ry is an arc, or if -y is a circle and E1 is a disk, we replace E by E1 (moved slightly so as to eliminate the intersection ry with int D). If ry is a circle and E2 is a disk, we use a standard innermost circle argument to cut-and-paste D, using subdisks of E2, so as to reduce (int D) fl E. Thus we eventually get a collection of disks b, whose boundaries are the same as those in D, such that (int b) fl E = 0. Remark. There are situations when it is useful to consider submanifolds of M(,3) of the form nhd(E U U HA U D) where E is a subsurface of Q other than a disk. For example, the case where D consists of a Scharlemann cycle and E is an annulus arises in [GL3, Section 3].

5. THE COMBINATORICS

To state the main result of this section, Theorem 5.1 below, we need the following definition.

COMBINATORIAL METHODS IN DEHN SURGERY 275

A subgraph A of GQ is a k-web (k a non-negative integer) if and only if (1) all the vertices of A have the same sign, and (2) at most k edge-endpoints of GQ at vertices of A are not endpoints of edges of A. See Figure 7 for an example.

A k-web

(p=6, A=1. k=4)

Figure 7

In the following theorem, Gp and GQ are graphs as in Section 2. Theorem 5.1. Assume that 0 > 1- X(P)/p. Then either Gp represents all types or GQ contains a (p - X(P))-web. Most of the remainder of this section will be devoted to a sketch of the proof of Theorem 5.1. First we deal with the case of the trivial type, which is different from the general case. Define A to be the subgraph of GQ consisting of all edges of GQ joining vertices of the same sign. If some component of A is not a (p- X(P))-web, then GQ contains

276 C. MCA. GORDON

more than (p - X(P)) edges joining vertices of opposite sign . By the parity rule, the corresponding edges of Gp join vertices of the same sign . Let E be the subgraph of Gp consisting of all vertices of Gp , and all edges of Gp joining vertices of the same sign. Then X(P) = V - E + >X(f) (summed over all faces f of E )
Hence > X (f) > 0, implying that E has a disk face. This face is then a face of Gp whose vertices all have the same sign, and which therefore represents the trivial q-type. For non-trivial q -types we proceed as follows. Define a graph r C P by: • the vertices of r consist of the fat vertices (i.e., the vertices of Gp), together with dual vertices v(D), one in the interior of each face D of Gp; • the edges of r join each dual vertex v(D) to the fat vertices in the boundary of the corresponding face D. See Figure 8.

Figure 8 Now let r be a q-type . We define the directed graph F (r) C P to be the graph I', with edges oriented according to the following rule. Let e be an edge of F, with one endpoint at the fat vertex v, lying in a q-interval (i, i + 1) at that vertex. Then we orient e inwards at v if (rI(i, i + 1)) • sign v = +, and outwards at v if (rI(i, i + 1)) • sign v = -. As an example , if q = 6 and T = + - + - --, then around the vertices of Gp the edges of F(r) are oriented as shown in Figure 9.

COMBINATORIAL METHODS IN DEHN SURGERY 277

Figure 9 The reason for defining the orientation on the edges of I'(r) in this way is the following, which is immediate from the definition. Proposition 5.2. A disk face D of Gp represents r if and only if the corresponding dual vertex v(D) is a sink or source of I'(r). Our search for representatives of types is thus translated into the study of the directed graphs I'(-r) C P. Here we will use the following very elegant combinatorial formulation of the Poincare-Hopf Index Theorem, due to Glass [Gl]. Consider any directed graph S2 in a closed surface S. For each vertex v of S2, let s(v) be the number of switches (i.e., changes in orientation of successive edges) around v, and for each face f of SZ, let s (f) be the number of switches around of. See Figure 10.

s(v) = 4

Figure 10

278 C. MCA. GORDON

Define the index of a vertex or face by I(v)=1-s (2) , I(f)=X(f)-s(2) In particular , a vertex of positive index is a sink or source (1(v) = 1), and a face of positive index is a disk whose boundary is a cycle (I(f) = 1). Lemma 5.3 (Glass). Let St be a directed graph in the closed surface S. Then E I(v) + E I(f) = X(S). Proof. (See [Gl].) Each corner between adjacent edges at a vertex contributes exactly 1 to E s(v) + E s(f ). Hence

X(S) = V - E + EX(f) = V - # corners + EX(f) 2 = V - (E S(v) + E s (f )) .+ EX(f ) 2 =EI(v)+

E I(f)

A label i which at a fat vertex lies immediately between oppositely oriented edges of r(r) is a switch label of r. (Equivalently, i is a switch label if and only if TI (i -1, i) and rI (i, i + 1) have opposite signs .) If the edges of I' (r) adjacent to i are oriented in a clockwise (resp. anticlockwise) direction around i, then i is a clockwise (resp. anticlockwise ) switch label. Note that this is well-defined, i.e., independent of the sign of the fat vertex under consideration; see Figure 9. Let C(T), A(r) C q be the set of clockwise and anticlockwise switch labels of r, respectively. As an example, for the type 7 illustrated in Figure 9, C(T) = {2, 4}, A(T) = {1, 3}. An edge e of Gp is a clockwise (resp. anticlockwise ) switch edge if the labels at the endpoints of e both belong to C(r) (resp. A(T)). A switch edge is an edge of Gp that is either a clockwise or an anticlockwise switch edge. Note that the faces of I'(T) are in one-one correspondence with the edges of Gp, and that under this correspondence, faces of r(r) of index 1 correspond to switch edges of Gp. See Figure 11.

COMBINATORIAL METHODS IN DEHN SURGERY 279

Figure 11 Now let r be a non-trival q-type, and write c(r) = IC(r)I, a(r)

Thus

c(r) = a(r) > 0.

If v is a fat vertex of r(r), then s(v) = A(c(r) + a(-r)) = 20c(r)

Therefore I(v)=1-s(2) =1-Ac(r). Assume that Gp does not represent r. Then by Proposition 5.2, Ev dual I(v) < 0. Therefore, by Lemma 5.3, EI(f) ? X(P) - E. fajj(v) = X(P) - Al - Ac(r)) . Note that since c(r) > 1 and A > 1 - X(P)/p by hypothesis , the last quantity above is > 0. By the remark above following the definition of a switch edge, the number of switch edges of Gp > E I( f ). Hence we may assume without loss of generality that there are at least Z( t) clockwise switch edges , and so the number of endpoints of clockwise switch edges > E 1(f) > p&c(r) - (p - X(P)). Now there are &c(r) clockwise switch labels at each vertex of Cp, giving a total of p&c( r) such labels in Gp, and so there are at most (p - X(P)) of these labels that are not endpoints of clockwise switch edges of Gp . Let A be the subgraph of GQ consisting of those edges that correspond to clockwise switch edges of Gp. (In particular, the vertices of A correspond to a subset of C(r).) As noted above, A is non-empty. Also, there are at most (p - X(P)) occurrences of labels at vertices of A that are not endpoints of edges of A. Hence if all elements of C(r) have the same sign, (*c) and all elements of A(r) have the same sign, (*a) then all the vertices of A have the same sign , and so (a connected component of) A is a (p - X(P))-web, as desired. ( Note that we need to make both assumptions (*c)

280

C. MCA. GORDON

and (*a), as we have no control over whether there are at least 2(f^ clockwise or anticlockwise switch edges.) If at least one of (*c) and (*a) does not hold, then we take derived types until both do hold, and then work backwards. We now briefly discuss this inductive procedure. For this, we have to work in the following more general setting. Consider a nonempty set of labels L C q. Associated with L is the set of L-intervals, which are the intervals (el,12) between successive elements of L. An L-type is defined in the obvious way, to be an ILI-type where the ith co-ordinate is formally associated with the ith L-interval. Define G(L) to be the subgraph of Gp consisting of all edges of Gp with at least one endpoint label belonging to L. Thus G(L) and Gp have the same vertices, and G(q) = G. Note that although the corners of the faces of G(L) are not necessarily L-intervals, each is contained in a unique L-interval. Hence, for each disk face D of G(L), by taking the algebraic sum of the L-intervals containing the corners of D we get an element [D] E ZI LI. Then, very much as before, we say that D represents the L-type r if and only if (1) [D] represents T (in the sense of Section 3); and (2) (coherence condition) no L-interval occurs as the L-interval containing a corner of D at two vertices of D of opposite sign.

We define the directed graph r(r) as before (starting with G(L)), orienting each edge according to the restriction of r to the L-interval containing the corresponding comer (and the sign of the fat vertex in question). Proposition 5.2 continues to hold with Gp replaced by G(L). Let r be a non-trivial L-type. We have C(r), A(r) C L, the set of clockwise (resp. anticlockwise) switches of T, as before. The argument we have just given above carries over verbatim to our present more general setting, so we have the following lemma. Lemma 5.4. Let r be a non-trivial L-type, satisfying (*c) and (*a), such that G(L) does not represent r. Then GQ contains a (p - X(P))-web. The derivative of a non-trivial L-type r is the C(T)-type dr defined as follows. Let (c, c') be a C(r)-interval. Then (c, c') contains a unique element a E A(r). We define dT by drI (c,c')= sign a. Example. Taking q = 16 and starting with the q -type r = + - + + - + - - + + + + - -+, Figure 12 below illustrates how one obtains dr and der.

COMBINATORIAL METHODS IN DEHN SURGERY 281

C (t) = (2.5.7.10.14) A (t)=(3.6 . 9.11.161

C (d 'C) = (5.14 ) ,

A (d T) = (2,71

C(d2t)= ( 5) , A (d2t)= (14)

Figure 12

282

C. MCA. GORDON

The key property of the definition of the derivative of a type is the following proposition, which says that to find a representative of -r it is enough to find a representative of dr. Proposition 5.5. Let r be a non-trivial L-type. Then any face of G(C(T)) representing dr contains a face of G(L) representing r. Sketch Proof. We shall give enough of the proof to at least explain the reason for the definition of dr. Let D be a face of G(C(T)) representing dr. Let FD(r) = 1'(r) n D. Claim.

ID(r) has either a face of index 1 or a dual vertex of index 1.

Proof of Claim. This is proved by doubling (D, l'D(T)) along the boundary, and applying Lemma 5.3 to the double 2rD(-r) C 2D = S2. Note that the faces of 2FD(r) are the faces of FD(T) (each appearing twice), together with a face ff for each edge e C 8D (with e C fe). Hence the conclusion will follow if we show that {fat vertices of 21'D (T) of index 111 < {faces fe of index -1}1 . But this in turn follows from the fact that, by definition of G(C(r)), each e C 8D has at least one endpoint label E C(r) (see [GL2, p.398] for details). A dual vertex of ['D(T) of index 1 will correspond to a face of G(L), contained in D, representing r. So, by the claim above, we are done once we show that no face of rD(T) has index 1. Such a face corresponds to a switch edge e of G(L). Now e cannot be a clockwise switch edge, for such edges are edges of G(C(r)), and hence cannot he in the interior of the face D of G(C(r)). So assume that e is an anticlockwise switch edge, with endpoint labels al, a2 E A(r) at vertices v1, v2 respectively; see Figure 13. Then, by the parity rule, (sign vi)(sign al) # ( sign v2)(sign a2). But, ❑ recalling the definition of dr, this contradicts the fact that D represents dr.

Figure 13 Returning to the proof of Theorem 5.1, suppose that one of the conditions (*c), (*a) fails for the q-type r. Note that C(r) = A(-r). Hence, if we define Tn =

T , if r does not satisfy (*a) St -T if T satisfies (*a) but no t

COMBINATORIAL METHODS IN DEHN SURGERY 283

then (*a) fails for TO. Hence d-ro is a non-trivial type. If dro satisfies (*c) and (*a), stop. If not, define ri

dTO , -dro ,

if dro does not satisfy (*a) if dro satisfies (*a) but not (*c)

Continuing in this way, we obtain a sequence ro, r1, ... , r., n > 1, where ro = fr, ri = ±dri_1 is a non-trivial C(r;_1)-type, 1 < i < n, and Tn satisfies (*c) and (*a). If G(C(ri_1)) does not represent r,,, then, by Lemma 5.4, GQ contains a (p-X(P))web. If G(C(ri_1)) does represent rn, then successive applications of Proposition 5.5 show that Gp represents r. This completes our sketch of the proof of Theorem 5.1. We have seen, in Section 4, that if Gp represents all types then we get useful topological information about (M(/3), Q). What if GQ contains a (p - X(P))-web? One idea is that a web might give rise to Scharlemann cycles in GQ (which then in turn give useful topological information about (M(a), P)). In order to make this work, however, we must impose a further condition. Namely, we define a great k-web in GQ to be a k-web A which is contained in the interior of a disk DA C Q with the property that any vertex of GQ lying in DA is a vertex of A. See Figure 14.

Figure 14

284

C. MCA. GORDON

To guarantee that such objects exist, we need to specialize to the case where Q is a 2-sphere. It turns out that one can then prove the following theorem. Theorem 5.6. Assume that A > 1 - X(P)/p, and that Q = S2. Then either Gp represents all types or GQ contains a great (p - X(P))-web. The proof proceeds as follows. By Theorem 5.1, if Gp does not represent all types then GQ contains a (p - X (P))-web A. Let U be a component of Q - nhd(A), and let D be the disk Q - U, containing A. If A is not a great web, then there are vertices of GQ in D that are not vertices of A. These vertices correspond to a non-empty subset L C q. One then proves by induction on ILI that either GQ contains a great (p - X(P))-web or G(L) represents all L-types. (The last step of the induction, which is the desired statement, is the degenerate case where A = 0, L = q, and D is a disk in Q containing GQ.) This inductive argument is quite subtle, and in particular involves relativizing the notion of the derivative of a type. We refer to [GL3, proof of Theorem 2.5] (which does the case where P is a torus), and thence to [GL2, Section 2], for more details.

6. APPLICATIONS

In this section we sketch the proofs of some results which use the methods we have discussed. The first three are about Dehn surgery on knots in S3, and the proofs of these will all be based on the following statement , which is an immediate consequence of Theorem 5.6 and (Ta) of Section 4. Theorem 6.1. Suppose that M(/3) = S3 and that Q is a Heegaard 2-sphere. If A > 1 - X(P)/p then GQ contains a great (p - X(P))-web. When M(,3) = S3, we shall make a slight change of notation, writing K for K,3, a non-trivial knot in S3, MK for M, the exterior of K, and µK for /3, the meridian of K. The following theorem is proved in [GL2]. It implies that knots in S3 are determined by their complements. Theorem 6.2.

Let K be a non-trivial knot in S3. If MK (a) = S3 then a = AK.

Proof. Suppose that MK(a) = S3, with A(a, AK) > 1. Let P be a Heegaard 2-sphere in MK(a). By Theorem 6.1, GQ contains a great (p - 2)-web, A. Hence there exists a label (in fact at least two labels) i E p such that every occurrence of i at a vertex of A is the endpoint of an edge of A. Moreover, since all the vertices of A have the same

COMBINATORIAL METHODS IN DEHN SURGERY 285

sign , no edge of A can have the same label at both endpoints, by the parity rule. Hence, by always leaving a vertex of A along an edge of A with label i at that vertex, we see that A contains an i-cycle, that is, a cycle of edges which can be consistently oriented so that the tail of each edge has label i (and all the vertices in the cycle have the same sign). See Figure 15.

Figure 15 Now it is easy to see that a cycle of this kind which is innermost in the disk DA (allowing the label i to be an arbitrary element of p) is a Scharlemann cycle in GQ. Since P is a 2-sphere the edges of this Scharlemann cycle necessarily lie in a disk in P. But since P is a Heegaard 2-sphere, this is a contradiction, by (Sa) in Section 4 ❑ (here, the roles of P and Q are interchanged). Remark. We observed in the above proof that a great (p - 2)-web in GQ always contains a Scharlemann cycle. So we may state Theorem 6.3. If P and Q are 2-spheres, and 0 > 1, then either Gp represents all types or GQ contains a Scharlemann cycle. If 0 > 2 then the following stronger conclusion holds. Theorem 6.4. If P and Q are 2-spheres, and 0 > 2, then either Gp or GQ contains a Scharlemann cycle. This is an immediate consequence of [CGLS, Proposition 2.5.61 (which deals with the case where P and Q are disks). Theorem 6.4 already shows that if M(a) = M(,3) = S3 then 0(c , [3) < 1, implying that there are at most two inequivalent knots with homeomorphic complements

286

C . MCA. GORDON

(a fact first proved , by a different argument , in [CGLS]). However , there are examples of graphs Gp, GQ with . = 1 for which the conclusion of Theorem 6.4 fails, illustrating the necessity of introducing the more general concept of representing all types. The next application addresses the question of when a reducible manifold can be obtained by Dehn surgery on a knot in S3. Theorem 6.5.

Let K be a non- trivial knot in S3. If MK (a) is reducible then

(1) MK(a) has a lens space summand; and (2) 0(a, µK) = 1. Part (1) was proved in [GL2]; part ( 2) was proved in [GL1] (by an argument different from that outlined below). Part ( 1) implies the Property R Conjecture, proved by Gabai [Ga2]. Corollary 6.6(Gabai). If K is a non-trivial knot in S3 then MK(O) is irreducible. Part (1) also implies

Corollary 6.7. is prime.

Any homology 3-sphere obtained by Dehn surgery on a knot in S3

We remark that Auckly, using results from gauge theory, has shown that there are prime homology 3-spheres which cannot be obtained by Dehn surgery on any knot in S3 [A]. The situation described in Theorem 6.5 can occur: if K is a torus knot or cable knot then there is a slope a such that MK(a) is reducible. However, the Cabling Conjecture [GS] asserts that these are the only knots with this property. This is still open. Sketch Proof of Theorem 6.5. Here we take P to be an essential 2-sphere in MK(a). (Note that MK is irreducible.) By Theorem 6.1, GQ contains a great (p-2)-web A. As in the proof of Theorem 6.2 above, this implies that GQ contains a Scharlemann cycle, and hence that MK(a) has a lens space summand (see (Sc) in Section 4). This proves (1). To prove (2), assume that 0(a, PK) > 1. Then, for homological reasons, P separates MK(a), and hence p is even. If p = 2 then P is an annulus, and hence K is a cable of a knot K' (which may be trivial). Now it is not hard to show (see the first four lines of the proof of Theorem 3 in [GL1]) that if MK(a) is reducible for some a with A(a, uK) > 1 then MK'(a') is

COMBINATORIAL METHODS IN DEHN SURGERY 287

reducible for some a' with A(a', ILK,) > 1. Hence, by induction it suffices to prove the result when K is not cabled. So assume p > 4. As in the proof of Theorem 6.2 above, there is a label in p such that each of the 0 occurrences of that label at each vertex of A is the endpoint of an edge of A. One can then show, using the fact that A > 2, that A contains Scharlemann cycles of GQ on distinct p-intervals (i, i+l), (j, j+l) (see [GL4, Proof of Theorem 2.3]). This allows one to construct a new essential 2-sphere P' C MK(a) such that IP' fl KaI < p (see [GL4, Theorem 2.4]) contradicting the minimality of p. ❑ For our last application to Dehn surgery on knots in S3 we consider the situation where MK(a) contains an incompressible torus. Here, we must assume that MK contains no essential torus, or equivalently, that K is not a satellite knot. Theorem 6.8 [GL3]. Let K be a knot in S3 that is not a satellite knot. If MK(a) contains an incompressible torus then A(a, ILK) < 2. Examples with 0(a, ILK) = 2 have been constructed by Eudave-Munoz [E2]. There are many examples with A(a, ILK) = 1. Sketch Proof of Theorem 6.8. Let P be an incompressible torus in MK(a), and assume (for a contradiction) that 0(a, ILK) > 3. In particular, this implies that P separates MK(a), so that p is even.

If p = 2 then one can show that the knot K is strongly invertible (see [GL3, Section 8]) in which case the theorem is proved by Eudave-Muiioz [El]. So assume that p > 4. By Theorem 6.1, GQ contains a great p-web A. An easy euler characteristic argument (see [GL3, Section 4]) shows that there are at least four labels i E p such that A contains an i-bigon; see Figure 16. Now such an ibigon is either a Scharlemann cycle, or contains within it an extended Scharlemann cycle, that is, a Scharlemann cycle flanked by bigons; see Figure 17. However, a topological argument shows that an extended Scharlemann cycle in GQ would give rise to a new incompressible torus P' in MK(a) with (P fl Kai < p, contradicting the minimality of p (see [GL3, Theorem 3.2]). We conclude that A contains two Scharlemann cycles D, D' of length.2 on disjoint p-intervals (i, i + 1) and (i', i' + 1). By (Sb) in Section 4, the edges of neither D nor D' he in a disk in P, hence each pair of edges forms an essential loop on P; see Figure 18. Shrinking the 1-handle H(i,i+i) to its core has the effect of gluing together the two (i, i+1) corners of D (see Figure 19), giving a Mobius band B. Similarly, D' gives rise to a Mobius band B'. Since 8B and 8B' are parallel on P we may join B and B' by an annulus in P to get a Klein bottle F in MK(a). We now replace the torus P by the boundary Po of a regular neighborhood of F. (Note that Po is incompressible , since MK(a) contains

288

C. MCA. GORDON

an incompressible torus by hypothesis , and is irreducible by Theorem 6.5(2).) But similar combinatorial arguments , applied to Po fl Q, in this case lead to a contradiction (see [GL3, Section 61). 0

0

Figure 16

Figure 17

Figure 18

Figure 19 Finally, we show how the methods we have discussed give some information about the question of when M(a) and M(,Q) are reducible . Let us say that a 3-manifold is Q-reducible if it contains a Q-essential 2-sphere.

COMBINATORIAL METHODS IN DEHN SURGERY 289

Theorem 6.9.

(i) If M(a) and M(/3) are Q-reducible then a = /3.

(ii) If M(a) is reducible and M(/3) is Q-reducible, with a # /3, then M(a) has a lens space summand. (iii) If M(a) and M(/3) are reducible, with a # 3, then either both M(a) and M(,0) have summands Mo,Mp with Hi(MQ) and H1(MO) finite and nonzero, or one of M(a), M(0) has a lens space summand. Remarks. (1) Part (i) is a special case of a result of Scharlemann [S3] proved using the theory of sutured manifolds. The case where M(a) and M(/3) are connected sums of copies of Sl x S2 was proved by Gabai [Gal]. (2) Scharlemann [S3] proves that under the hypotheses of (ii), M(/3) = Sl x S2 # W, and M(a) = W' # W", where W, W' and W" are Q-homology spheres, and that either W' is a lens space or Hl (W) # 0. (3) It is shown in [GL4], by an elaboration of the methods discussed here, that if M(a) and M(/3) are reducible then 0(a„3) < 1. A different proof has been given by Boyer and Zhang [BZ]. Proof of Theorem 6.9. (i) Let P,Q be Q-essential 2-spheres in M(a), M(/3) respectively with p and q minimal. By Theorem 6.3, either Gp represents all types or GQ contains a Scharlemann cycle. But the first contradicts the minimality of q (see (Td) in Section 4), while the second contradicts the minimality of p (see (Sd) in Section 4). (ii) Let Q be as in (i), and let P be an essential 2-sphere in M(a). By Theorem 6.3 (and (Td)) GQ contains a Scharlemann cycle. Hence M(a) has a lens space summand. (iii) Let P, Q be essential 2-spheres in M(a), M(/3) respectively. The result then follows from Theorem 6.3, applied as stated and also with P and Q interchanged, ❑ (together with (Sc) and (Tc) in Section 4).

REFERENCES [A] D. Auckly, Surgery numbers of 3-manifolds: a hyperbolic example, Geometric Topology (W.H. Kazez ed.), Proceedings of the 1993 Georgia International Topology Conference, AMS/IP Studies in Advanced Mathematics Vol.2, 1997 (Part 1), pp. 21-34. [BZ] S. Boyer and X. Zhang, The semi- norm and Dehn filling, preprint. [CGLS] M. Culler, C.McA. Gordon, J. Luecke and P.B. Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987), 237-300. [El] M. Eudave-Muiioz, Essential tori obtained by surgery on a knot, Pacific J. Math. 167 (1995), 81-117. [E2] , Non-hyperbolic manifolds obtained by Dehn surgery on hyperbolic knots, Geometric Topology (W.H. Kazez ed.), Proceedings of the 1993 Georgia International Topology Conference, AMS/IP Studies in Advanced Mathematics Vol.2, 1997 (Part 1), pp. 35-61.

290 C. MCA. GORDON [Gal] D. Gabai, Foliations and the topology of 3-manifolds. II, J. Diff. Geom. 26 (1987), 461478. [Ga2] , Foliations and the topology of 3-manifolds. III, J. Diff. Geom. 26 (1987), 479536. [G1] L. Glass, A combinatorial analog of the Poincare Index Theorem , J. Combin. Theory Ser. B 15 (1973), 264-268. [GS] F. Gonzalez-Acuiia and H. Short, Knot surgery and primeness, Math. Proc. Camb. Phil. Soc. 99 ( 1986), 89-102. [Go] C.McA. Gordon, Dehn filling: a suroey, Proceedings of the Mini Semester in Knot Theory, Stefan Banach International Mathematical Center, Warsaw, Poland, 1995 (to appear). [GL1] C.McA. Gordon and J. Luecke, Only integral Dehn surgeries can, yield reducible manifolds, Math. Proc. Camb. Phil. Soc. 102 (1987), 97-101. [GL2] , Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415. [GL3] , Dehn surgeries on knots creating essential tori, I, Communications in Analysis and Geometry 3 (1995), 597-644.

[GL4] , Reducible manifolds and Dehn surgery , Topology 35 (1996), 385-409. [L] R.A. Litherland, Surgery on knots in solid tori, II, J. London Math. Soc. (2) 22 (1980), 559-569. [P] W. Parry, All types implies torsion, Proc. Amer. Math. Soc. 110 (1990), 871-875. (R] Y. Rieck, Heegaard structures of manifolds in the Dehn filling space, preprint. [S1] M. Scharlemann , Smooth spheres in 1k4 with four critical points are standard, Invent. Math. 79 (1985), 125-141. , Unknotting number one knots are prime, Invent. Math . 82 (1985 ), 37-55. [S2]

, Producing reducible 3-manifolds by surgery on a knot, Topology 29 (1990), 481-

[S3] 500.

[T] I. Torisu, Boundary slopes for knots, Osaka J. Math. 33 (1996), 47-55. DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF TEXAS AT AUSTIN, AUSTIN, TX 78712-1082 E-mail address : [email protected]