Table Of ContentNew Ideas in Low
Dimensional Topology
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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
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Published by
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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
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required from the publisher.
Printed in Singapore
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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
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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
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Introduction vii
the pure mathematics of low dimensional topology and physical
phenomena even more intimate.
Louis H. Kauffman
and
Vassily O. Manturov
September 2014
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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
