New Ideas in Low Dimensional Topology PDF
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New Ideas in Low Dimensional Topology 9348_9789814630610_tp.indd 1 8/1/15 11:37 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk K E Series on Knots and Everything — Vol. 56 New Ideas in Low Dimensional Topology Edited by Louis H Kauffman University of Illinois at Chicago, USA V O Manturov Bauman Moscow State Technical University, Russia & Chelyabinsk State University, Russia World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 9348_9789814630610_tp.indd 2 8/1/15 11:37 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data New ideas in low dimensional topology / edited by L.H. Kauffman (University of Illinois at Chicago, USA), V.O. Manturov (Bauman Moscow State Technical University, Russia & Chelyabinsk State University, Russia). pages cm. -- (Series on knots and everything ; vol. 56) Includes bibliographical references. ISBN 978-981-4630-61-0 (hardcover : alk. paper) 1. Low-dimensional topology. 2. Topological manifolds. I. Kauffman, Louis H., 1945– II. Manturov, V. O. (Vasilii Olegovich) QA612.14.N49 2015 514'.32--dc23 2014035528 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, elec- tronic 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. Printed in Singapore EH - New Ideas in Low Dimensional.indd 1 8/1/2015 9:02:26 AM January8,2015 16:20 NewIdeasinLowDimensionalTopology 9inx6in b1970-fm pagev Introduction This book consists in a selection of articles devoted to new ideas and developments in low dimensional topology. Low dimensions refer to dimensions three and four for the topology of manifolds and their submanifolds. Thus we have papers related to both manifolds and to knotted submanifolds of dimension one in three (classical knot theory) and two in four (surfaces in four dimensional spaces). Some of the work involves virtual knot theory wherethe knots are abstrac- tions of classical knots but can be represented by knots embedded in surfaces. This leads both to new interactions with classical topology and to new interactions with essential combinatorics. The first paper in this volume, by J. Scott Carter, is a pictorial introductiontoknottedfoamsinfourdimensionalspace,ananalogof knotted trivalent graph embeddings in three dimensional space. The second paper, by J. Scott Carter and S. Kamada, is an introduction to the construction of manifolds in many dimensions via branched coverings. The third paper by R. Fenn is a description of some of the variations on knots that occur in virtual knot theory and its generalizations. The fourth paper, by S. Gukov and I. Saberi, is an introduction to the remarkable ideas in physics that are related to constructions of link homology. Linkhomology itself is a newsubject in the study of invariants of knots and links. Inthisapproach,homologytheoriesareassociatedwithknotsand links that categorify classical link invariants so that a graded Euler characteristic of the homology reproduces the classical invariant (e.g. the Alexander polynomial or the Jones polynomial). Such v January8,2015 16:20 NewIdeasinLowDimensionalTopology 9inx6in b1970-fm pagevi vi New Ideas in Low Dimensional Topology categorifications have their roots in certain physical ideas in the sense that they are related to Floer homology and its concept to use the Chern–Simons functional as a Morse function on the moduli space of connections on a three manifold. But the new relations to physics are subtle and involve delicate conjectures in string theory. The fifth paper is by A. Haydys and concerns the structure of Dirac operators in relation to the Seiberg–Witten equations that have been revolutionary in handling invariants of four manifolds. The sixth paper, by D. P. Ilyutko, V. O. Manturov and I. M. Nikonov is a study of graph links. Graph links are a generalization of knot theory that comes from studying knots and virtual knots in terms of their Gauss codes. It is a new and significant development in combinatorialtopology.Theseventhpaper,byA.Juha´sz,isaconcise and detailed survey of Heegaard–Floer homology. The eighth paper, byJ.JuyumayaandS.Lambropoulou,isadescriptionoftheirrecent research on framed braids and Hecke algebras. The ninth paper, by L. H. Kauffman, is an introduction to new ideas in virtual knot cobordism, including a description of a generalization of the Lee homology and Rasmussen invariant to virtual knots and links due to H. Dye, A. Kaestner and L. H. Kauffman and based on work of V. O. Manturov. The tenth paper, by H. R. Morton, is a survey of classical and quantum methods for distinguishing mutant knots and links. Mutants have long been a test case for invariants, as many invariants are unable to distinguish them. The 11th paper, by J. Przytycki, is a study of homology theories generalizing cyclic homology thatarerelated toalgebraic structuresinknottheory. The 12th paper, by D. Rolfsen, is a study of the ordering of knot groups, a consideration that has led to numerous good results in recent years. The 13th paper, by D. Ruberman and N. Saviliev, is a study obtainingCasson-typeinvariantsfromtheSeiberg–Wittenequations. It should be clear to the reader that many if not all of the developments described in this volume are related to physics or motivated by physical considerations. We are looking forward to the further developments that will make these relationships between January8,2015 16:20 NewIdeasinLowDimensionalTopology 9inx6in b1970-fm pagevii Introduction vii the pure mathematics of low dimensional topology and physical phenomena even more intimate. Louis H. Kauffman and Vassily O. Manturov September 2014 May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January8,2015 16:20 NewIdeasinLowDimensionalTopology 9inx6in b1970-fm pageix Contents Introduction v 1. Reidemeister/Roseman-Type Moves to Embedded Foams in 4-Dimensional Space 1 J. Scott Carter 2. How to Fold a Manifold 31 J. Scott Carter and Seiichi Kamada 3. Generalised Biquandles for Generalised Knot Theories 79 Roger Fenn 4. Lectures on Knot Homology and Quantum Curves 105 Sergei Gukov and Ingmar Saberi 5. Dirac Operators in Gauge Theory 161 Andriy Haydys 6. Graph-Links: The State of the Art 189 D. P. Ilyutko, V. O. Manturov and I. M. Nikonov 7. A Survey of Heegaard Floer Homology 237 Andr´as Juh´asz 8. On the Framization of Knot Algebras 297 Jesu´s Juyumaya and Sofia Lambropoulou ix
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