CONTEMPORARY MATHEMATICS

233

Low Dimensional Topology Proceedings of a Conference on

Low Dimensional Topology January 12-17, 1998

FunchaL Madeira, Portugal

Hanna Nencka Editor

Selected Titles in This Series

233 Hanna Nencka, Editor, Low dimensional topology, 1999

232 Krzysztof Jarosz, Editor, Function spaces, 1999 231 Michael Farber, Wolfgang Liick, and Shmuel Weinberger, Editors, Tel Aviv

topology conference: Rothenberg Festschrift, 1999

230 Ezra Getzler and Mikhail Kapranov, Editors, Higher category theory, 1998

229 Edward L. Green and Birge Huisgen-Zimmermann, Editors, Trends in the representation theory of finite dimensional algebras, 1998

228 Liming Ge, Huaxin Lin, Zhong-Jin Ruan, Dianzhou Zhang, and Shuang Zhang, Editors, Operator algebras and operator theory, 1999

227 John McCleary, Editor, Higher homotopy structures in topology and mathematical physics, 1999

226 Luis A. Caffarelli and Mario Milman, Editors, Monge Ampere equation: Applications to geometry and optimization, 1999

225 Ronald C. Mullin and Gary L. Mullen, Editors, Finite fields: Theory, applications, and algorithms, 1999

224 Sang Geun Hahn, Hyo Chul Myung, and Efim Zelmanov, Editors, Recent progress in algebra, 1999

223 Bernard Chazelle, Jacob E. Goodman, and Richard Pollack, Editors, Advances in discrete and computational geometry, 1999

222 Kang-Tae Kim and Steven G. Krantz, Editors, Complex geometric analysis in Pohang, 1999

221 J. Robert Dorroh, Gisele Ruiz Goldstein, Jerome A. Goldstein, and Michael Mudi Tom, Editors, Applied analysis, 1999

220 Mark Mahowald and Stewart Priddy, Editors, Homotopy theory via algebraic geometry and group representations, 1998

219 Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov, Editors, Secondary calculus and cohomological physics, 1998

218 Jan Mandel, Charbel Farhat, and Xiao-Chuan Cai, Editors, Domain decomposition methods 10, 1998

217 Eric Carlen, Evans M. Harrell, and Michael Loss, Editors, Advances in differential equations and mathematical physics, 1998

216 Akrarn Aldroubi and EnBing Lin, Editors, Wavelets, multiwavelets, and their applications, 1998

215 M. G. Nerurkar, D.P. Dokken, and D. B. Ellis, Editors, Topological dynamics and applications, 1998

214 Lewis A. Coburn and Marc A. Rieffel, Editors, Perspectives on quantization, 1998 213 Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter,

Editors, Studies on composition operators, 1998 212 E. Ramirez de Arellano, N. Salinas, M. V. Shapiro, and N. L. Vasilevski,

Editors, Operator theory for complex and hypercomplex analysis, 1998

211 J6zef Dodziuk and Linda Keen, Editors, Lipa's legacy: Proceedings from the Bers Colloquium, 1997

210 V. Kumar Murty and Michel Waldschmidt, Editors, Number theory, 1998

209 Steven Cox and Irena Lasiecka, Editors, Optimization methods in partial differential equations, 1997

208 MichelL. Lapidus, Lawrence H. Harper, and Adolfo J. Rumbos, Editors, Harmonic analysis and nonlinear differential equations: A volume in honor of Victor L. Shapiro, 1997

(Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/233

Low Dimensional Topology

CoNTEMPORARY MATHEMATICS

233

Low Dimensional Topology Proceedings of a Conference on

Low Dimensional Topology January 12-17, 1998

Funchal, Madeira, Portugal

Hanna Nencka Editor

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeThrck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume is the record of a conference in Low Dimensional Topology, which took place in Funchal, Madeira, Portugal on January 12-17, 1998.

1991 Mathematics Subject Classification. Primary 57Mxx, 57Nxx, 57Pxx, 57Qxx, 57Rxx, 57Sxx, 57Txx.

Library of Congress Cataloging-in-Publication Data Conference on Low Dimensional Topology (1998 : Sao Joao da Madeira, Portugal)

Low dimensional topology : proceedings of a Conference on Low Dimensional January 12--17, 1998, Funchal, Madeira, Portugal / Hanna Nencka, editor.

p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 233) Includes bibliographical references. ISBN 0-8218-0884-2 (alk. paper) 1. Low dimensional topology Congresses. I. Nencka, Hanna, 1956- II. Title.

Contemporary mathematics (American Mathematical Society) ; v. 233. QA612.14.C63 1998 .514'.2 -dc21

Topology,

III. Series:

99-14986 CIP

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams. or g.

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© 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

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§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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10 9 8 7 6 5 4 3 2 1 04 03 02 01 00 99

Contents

Preface 1x

1r1' and infinite simple homotopy type in dimension 3 VALENTIN POENARU 1

Cubulations mod bubble moves LOUIS FUNAR 29

Numerical presentations of tortile categories lVELINA BOBTCHEVA AND FRANK QUINN 45

On Quinn's invariants of 2-dimensional CW-complexes lVELINA BOBTCHEVA 69

The quadratic form of a link AKIO KAWAUCHI 97

Quantum SU(2)-invariants of three-manifolds associated with the trivial first cohomology class modulo two HITOSHI MURAKAMI 117

An element of infinite order in TQFT-representations of mapping class groups GREGOR MASBAUM 137

The perturbative S0(3) invariant of homology circles TOMOTADA 0HTSUKI 141

Configuration space integrals and invariants for 3-manifolds and knots ALBERTO S. CATTANEO 153

The multivariable Alexander polynomial for a closed braid H. R. MORTON 167

Boundary links and h-split links KAZUAKI KOBAYASHI 173

Finite type invariants of ribbon 2-knots KAZUO HABIRO, TAIZO KANENOBU, AND AKIKO SHIMA 187

Arrangement of Markov moves for 2-dimensional braids SEIICHI KAMADA 197

vii

viii CONTENTS

A criterion for signed unknotting number PAWEL TRACZYK

On some e-,tensions of Artin's braid relations H. NENCKA

Asymptotic behavior of energy of polygonal knots JUN O'HARA

215

221

235

PREFACE

This volume of the Contemporary Mathematics series is an outgrowth of the conference on "Low Dimensional Topology" which took place at the University of Madeira (Portugal) in January 1998.

The aim of the conference was to bring together active scientists from America, Asia, and Europe, working in the field of low dimensional topology.

The present collection of contributions, which reflects the many-sided develop-ment and actual progress in low dimensional topology, is organized as follows.

The first two papers of this volume are devoted to characterisations of 3-manifolds and PL-manifolds. The first one, by V. Poenaru, considers the following conjecture: If M 3 is a closed 3-manifold with infinite 11"1, then the universal cov-ering space M 3 is simply connected at infinity. This work is strongly related to Poenaru's programme for the Poincare conjecture. The second one, by L. Funar, is devoted to the set of cubical decompositions of a compact manifold mod out by some combinatorial moves. It has been shown that there are obstructions for two cubulations of the same PL-manifold. It has been stated that if A(n) C zn+l is a sublattice, depending only on the dimension, then the obstruction in zn+l I A(n) is not trivial.

The next two contributions are linked to categories in low dimensional topology. First, I. Bobtcheva and F. Quinn consider a numerical approach to tortile categories, which appears as an intermediate structure in conformal and quantum field theory and low dimensional topology. The objective of the paper is to give a setting for direct calculations in categories derived from representations of root-of-unity deformations of Lie algebras of arbitrary rank (quantum groups) and to describe algorithms used to derive all structure information from a smaller set of data. In the following paper, I. Bobtcheva considers a semisimple stable autonomous tensor category A over a field K. Associating an element Q(P) E K, invariant under the Andrews-Curtis moves, to any group presentation with finite number of generators, she has shown that this is the same invariant as the one produced by Quinn's algorithm.

The goal of the next article, by A. Kawauchi, is to characterize the infinite cyclic covering quadratic form of a link as a restriction of the Seifert form (extended over rationals) of every connected Seifert surface of the link, giving a generalization of Erle's result. Kawauchi also shows that, for the sphere-slice link, the induced non-singular form of the quadratic form is null-cobordant, the signature of the canonical linking matrix is zero, and the nullity invariants are all even.

ix

X PREFACE

The next four papers are devoted to topological quantum field theories, quan-tum invariants, and mapping class groups. First, H. Murakami considers a polyno-mial invariant for rational homology 3-spheres, derived from the quantum 8U(2)-invariants associated with the trivial first cohomology class modulo 2. In the next paper, G. Masbaum gives a simplified proof of the following proposition: The image of the mapping class a- 1(3 in the TQFT-representation corresponding to 8U(2) at level k has infinite order except if k=1,2,3, or 8. The goal ofT. Ohtsuki's paper is to construct quantum Hilbert space for the perturbative 80(3)-invariant which is obtained by asymptotic expansion of the quantum 80(3)-invariant, and to describe the two structures of the quantum Hilbert space of T 2. The last contribution of this series is by A. Cattaneo, who tells how to associate invariants of 3-dimensional rational homology spheres and knots to certain combinations of trivarient diagrams.

The final part of the volume is devoted to knots, links, braids, and free groups. The first contribution in this series is by H. Morton, who proposes a method for finding the multivariable Alexander polynomial of a link presented as the closure of a braid. He shows that the Alexander polynomial of the closed braid can be derived from the Torres-Fox formula. In the next paper, K. Kobayashi considers four kinds of links, that is link-homotopically trivial links in the strong sense, boundary links, h-split links, and Brunnian links. The relations among them are considered. Some geometrical conditions to become h-split link are shown as well. K. Habiro, T. Kanenobu, and A. Shima construct a Vassiliev-like filtration on the free abelian group generated by the set of ribbon 2-links in 4-space in two equivalent ways. A finite type invariant for a ribbon 2-knot is given, and the notion can be extended to ribbon 2-links. It has been shown that the higher derivatives of the normalized Alexander polynomial of ribbon 2-knots at t = 1 are of finite type. S. Kamada is interested in the arrangement of Markov moves for 2-dimensional braids, and he considers quasi-commutativity of 2-dimensional Markov moves. In particular, if two simple 2-dimensional braids represent the same surface link type, then they are related by a Markov moves equivalence through simple 2-dimensional braids. A relation between unknotting numbers of surface links and those of 2-dimensional braids is given. A criterion for signed unknotting number is studied by P. Traczyk in his contribution. Various versions of signed unknotting number are investigated, and tables for prime knots up to nine crossing are given. The goal of the next paper, by H. Nencka, is to give generalisation of Artin's braid relations. A class of relations among the generators of braids groups is proposed which partially solves Xu's problem. The last paper, by J. O'Hara, deals with the asymptotic behavior of energy of polynomial knots. A regularization of "modified electrostatic energy" of charged knots is studied. New energy functionals for smooth knots are defined, and formulae for the asymptotic behaviour of the energy of polynomial knots are given.

The organizers of the conference were particularly pleased to be able to hold the meeting on the island of Madeira, and in receiving financial help from diverse sources which will be named explicitly below.

I wish to express my gratitude to: Francisco Santos (Regional Secretary of Ed-ucation), Jose Manuel Castanheira da Costa (Rector of the University of Madeira), Ludwig Streit (Director of the Center of Mathematical Science) and Rita Vasconce-los (President of the Mathematics Department), for making the conference possible.

PREFACE xi

I am particularly grateful to: Valentin Poenaru, Frank Quinn, and Vladimir Thraev for their advice, suggestions, observations, and help, during the conference and the editing of the volume.

I especially thank Philippe Combe for his help and advice in the editing of the book.

I would like to thank Antonio Pires and Gabriel Gomes, my colleagues from the University of Madeira, and Ana Paula Fernandes, our secretary, for their help in the organization of the conference.

The conference was made possible thanks to the support of several organi-zations. I wish to acknowledge support from Fundac;ao de Ciencia e Tecnologia, PRAXIS XXI, Pluriannual project, Fundac;ao Luso-Americana, British Council, Camara Municipal do Funcha.l, BANIF, and the Secret aria Regional de Thrismo e Cultura.

Funchal, January 1999, Hanna NENCKA

Selected Titles in This Series (Continued from the front of this publication)

207 Yujiro Kawamata and Vyacheslav V. Shokurov, Editors, Birational algebraic geometry: A conference on algebraic geometry in memory of Wei-Liang Chow (1911-1995), 1997

206 Adam Koranyi, Editor, Harmonic functions on trees and buildings, 1997 205 Paulo D. Cordaro and Howard Jacobowitz, Editors, Multidimensional complex

analysis and partial differential equations: A collection of papers in honor of Franc;ois Treves, 1997

204 Yair Censor and Simeon Reich, Editors, Recent developments in optimization theory and nonlinear analysis, 1997

203 Hanna Nencka and Jean-Pierre Bourguignon, Editors, Geometry and nature: In memory of W. K. Clifford, 1997

202 Jean-Louis Leday, James D. Stasheff, and Alexander A. Voronov, Editors, Operads: Proceedings of Renaissance Conferences, 1997

201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F. Dias, J.-M. Ghidaglia, and J.-C. Saut, Editors, Mathematical problems in the

theory of water waves, 1996 199 G. Banaszak, W. Gajda, and P. Krason, Editors, Algebraic K-theory, 1996 198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial

mechanics, 1996 197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry,

1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective

endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie algebras and their

representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and

related topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Tuong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally,

Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F.

Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A.M. Marcantognini, G. A. Mendoza, M.D. Moran, A. Octavio, and W. 0.

Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors,

Homotopy theory and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, S£(2) representations of finitely presented groups,

1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara,

and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995

185 Raul E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995

184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on Algebra, 1995

183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995

182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995

(See the AMS catalog for earlier titles)

v

Low Dimensional Topology Hanna Nencka, Editor

This volume presents the proceedings from the conference on low dimensional topology held at the University of Madeira (Portugal). The event was attended by leading scientists in the field from the U.S., Asia, and Europe. The book has two main parts. The first is devoted to the Poincare conjecture, characteri-zations of P L -manifolds, covering quadratic forms of links and to categories in low dimensional topology that appear in connection with conformal and quantum field theo-ry. The second part of the volume covers topological quantum field theory and polynomi-al invariants for rational homology 3-spheres, derived from the quantum SU(2)-invariants associated with the first cohomology class modulo two, knot theory, and braid groups. This collection reflects development and progress in the field and presents interesting and new results.

ISBN G-8218-0884-2

9 780821 808849 CONM/233