An introduction to knot theory, w b raymond lickorish

Graduate Texts in Mathematics 175 Editorial Board S Axler F.W Gehring K.A Ribet Springer-Science+Business Media, LLC Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTI!ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAc LANE Categories for the Working Mathematician HUGHESIPIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYs Introduction to Lie Algebras and Representation Theory CoHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALs Advanced Mathematical Analysis ANDERSONIFuLLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GuILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HuMPHREYs Linear Aigebraic Groups BARNES!MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Aigebraic Theories KELLEy General Topology ZARlSKIlSAMUEL Commutative Algebra Vol.I ZARlSKIlSAMUEL Commutative Algebra Vol.lI JACOBSON Lectures in Abstract Algebra Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Aigebras and Several Complex Variables 2nd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRnzscHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENY/SNEuJKNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory 4th ed 46 LoEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/Wu General Relativity for Mathematicians 49 GRUENBERGlWEIR Linear Geometry 2nd ed 50 EOWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERlWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN!PEARCY Introduction to Operator Theory 1: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CRoWEuiFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed continued after index W.B Raymond Lickorish An Introduction to Knot Theory With 114 Illustrations t Springer W.B Raymond Lickorish Professor of Geometric Topology, University of Cambridge, and Fellow of Pembroke College, Cambridge Department of Pure Mathematics and Mathematical Statistics Cambridge CB2 ISB England Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East HalI University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720 USA Mathematics Subject Classification (1991): 57-01, 57M25, 16S34, 57M05 Library of Congress Cataloging-in-Publieation Data Liekorish, W.B Raymond An introduetion to knot theory / W.B Raymond Liekorish p em - (Graduate texts in mathematics ; 175) Including bibliographical references (p - ) and index ISBN 978-1-4612-6869-7 ISBN 978-1-4612-0691-0 (eBook) DOI 10.1007/978-1-4612-0691-0 Knot theory Title II Series QA612.2.LS3 1997 97-16660 S14'.224 dc21 Printed on acid-free paper © 1997 Springer Science+Business Media New York Origina11y published by Springer-Verlag New York Berlin Heidelberg in 1997 Softcover reprint of the hardcover 1st edition 1997 Ali rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly ana1ysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used fteely by anyone Production managed by Steven Pisano; manufacturing supervised by Johanna Tschebull Photocomposed pages prepared from the author's TeX files 765 ISBN 978-14611-6869-7 SPIN 10628672 Preface This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in three-dimensional space Knots can be studied at many levels and from many points of view They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained The study of knots can be given some motivation in terms of applications in molecular biology or by reference to parallels in equilibrium statistical mechanics or quantum field theory Here, however, knot theory is considered as part of geometric topology Motivation for such a topological study of knots is meant to come from a curiosity to know how the geometry of three-dimensional space can be explored by knotting phenomena using precise mathematics The aim will be to find invariants that distinguish knots, to investigate geometric properties of knots and to see something of the way they interact with more adventurous three-dimensional topology The book is based on an expanded version of notes for a course for recent graduates in mathematics given at the University of Cambridge; it is intended for others with a similar level of mathematical understanding In particular, a knowledge of the very basic ideas of the fundamental group and of a simple homology theory is assumed; it is, after all, more important to know about those topics than about the intricacies of knot theory There are other works on knot theory written at this level; indeed most of them are listed in the bibliography However, the quantity of what may reasonably be termed mathematical knot theory has expanded enormously in recent years Much of the newly discovered material is not particularly difficult and has a right to be included in an introduction This makes some of the excellent established treatises seem a little dated However, concentrating entirely on developments of the past decade gives a most misleading view of the subject An attempt is made here to outline some of the highlights from throughout the twentieth century, with a little bias towards recent discoveries The present size of the subject means that a choice of topics must be made for inclusion in any first course or book of reasonable length Such selection must be subjective An attempt has been made here to give the flavour and the results from three or four main techniques and not to become unduly enmeshed in any of them v vi Preface Firstly, there is the three-manifold method of manipulating surfaces, using the pattern of simple closed curves in which two surfaces intersect This leads to the theorem concerning the unique factorisation of knots into primes and to the theory concerning the primeness of alternating diagrams Combinatorics applied to knot and link diagrams lead (by way of the Kauffman bracket) to the Jones polynomial, an invariant that is good, but not infallible, at distinguishing different knots and links This invariant also has applications to the way diagrams of certain knots might be drawn Next, techniques of elementary homology theory are used on the infinite cyclic cover of the complement of a link to lead to the "abelian" invariants, in particular to the well-known Alexander polynomial That is reinforced by the association of that polynomial invariant with the Conway polynomial, as well as by a study of the fundamental group ofa link's complement The use of (framed) links to describe, by means of "surgery", any closed orientable three-manifold is explored Together with the skein theory of the Kauffman bracket, this idea leads to some "quantum" invariants for three-manifolds A technique, belonging to a more general theory of three-manifolds, that will not be described is that of the W Haken's classification of knots That technique gives a theoretical algorithm which always decides if two knots are or are not the same It is almost impossible to use it, but it is good to know it exists [42] One can take the view that the object of mathematics is to prove that certain things are true That object will here be pursued A declaration that something is true, followed by copious calculations that produce no contradiction, should not completely satisfy the intellect However, even neglecting all logical or philosophical objections to this quest, there are genuine practical difficulties in attempting to give a totally self-contained introduction to knot theory To avoid pathological possibilities, in which diagrams oflinks might have infinitely many crossings, it is necessary to impose a piecewise linear or differential restriction on links Then all manoeuvres must preserve such structures, and the technicalities of a piecewise linear or differential theory are needed One needs, for example, to know that any two-dimensional sphere, smoothly or piecewise linearly embedded in Euclidean three-space, bounds a smooth or piecewise linear ball This is the SchOnflies theorem; the existence of wild homed spheres shows it is not true without the technical restrictions What is needed, then, is a full development ofthe theory of piecewise linear or differential manifolds at least up to dimension three Laudable though such an account might be, experience suggests that it is initially counter-productive in the study of knot theory Conversely, experience of knot theory can produce the incentive to understand these geometric foundations at a later time Thus some basic (intuitively likely) results of piecewise linear theory will sometimes be quoted, sometimes with a sketch of how they are proved Perhaps here piecewise linear theory has an advantage over differential theory, because up to dimension three, simplexes are readily visualisable; but differential theory, if known, will answer just as well That apologia underpins the start of the theory Significant direct quotations of results have however also been made in the discussion of the fundamental group of a link complement That topic has been treated extensively elsewhere, so the remarks here are intended to be but something of a little survey Preface vii Also quoted is R C Kirby's theorem concerning moves between surgery links for a three-manifold Furthermore, at the end of a section extensions of a theory just considered are sometimes outlined without detailed proof Otherwise it is intended that everything should be proved! W B Raymond Lickorish Contents Preface v Chapter A Beginning for Knot Theory Exercises 13 Chapter Seifert Surfaces and Knot Factorisation Exercises 15 21 Chapter The Jones Polynomial Exercises 23 30 Chapter Geometry of Alternating Links Exercises 32 40 Chapter The Jones Polynomial of an Alternating Link Exercises 48 Chapter The Alexander Polynomial Exercises 49 64 Chapter Covering Spaces Exercises 66 78 Chapter The Conway Polynomial, Signatures and Slice Knots Exercises 79 91 41 Chapter Cyclic Branched Covers and the Goeritz Matrix Exercises 93 102 Chapter 10 The Arf Invariant and the Jones Polynomial Exercises 103 108 IX x Contents Chapter 11 The Fundamental Group Exercises 110 121 Chapter 12 Obtaining 3-Manifolds by Surgery on S3 Exercises 123 13 Chapter 13 3-Manifold Invariants From The Jones Polynomial Exercises 133 144 Chapter 14 Methods for Calculating Quantum Invariants Exercises 146 164 Chapter 15 Generalisations of the Jones Polynomial Exercises 166 177 Chapter 16 Exploring the HOMFLY and Kauffman Polynomials Exercises 179 191 References 193 Index 199 Exploring the HOMFLY and Kauffman Polynomials 189 It is arduous but straightforward to check directly that this is a solution It is, however, not hard to see that R- = _q-I "~ E-1,1 'tY 10- E-1,1 + " 10- E· 10- E· ~ E-I,j 'tY j,1 + (q - q-I) " ~ E-1,1 'tY j,j' I#j i>j so that R - R -I = (q -I - q) 1v® v This is then the minimal polynomial equation for R Let IJ- = diag(IJ-I, 1J-2, , IJ-m) where IJ-i = q2i-m-l, and let ex = _qlll and f3 = A routine check shows that this provides an enhancement for R Thus by Theorem 16.11, these data provide an oriented link invariant T(L) which, by Proposition 16.12, satisfies where L+, L_ and Lo are related in the usual way Re-normalising to get the polynomial of the unknot to be one, gives for g E BIl , P(~) " (iq"',i(q-I_q » _ ( - -q )-mw(~) Trace(ifJ(g)lJ-®f1) Trace IJ- As m varies, the evaluations of the HOMFLY polynomial at these special values of the variables do, of course, determine the whole two-variable polynomial It is interesting to note that the Alexander polynomial Il d t) = P (L ) (i ,I (11/2 _1-1/2» does not feature as one of the special values (as m ~ 1); from this standpoint Ildt) occurs only by way of creating the entire two-variable polynomial from the whole sequence of special values A full version of this Yang-Baxter equation approach to the HOMFLY and Kauffinan polynomials is given in [127] More complicated R -matrices lead to descriptions of those invariants for "coloured" links that are linear combinations of invariants for satellites and parallels From the above example, Jones [52] produced a "states model" for each of the above values of the HOMFLY polynomial His result is as follows: Fix n ~ 0, let D be a diagram of an oriented link L, and let D* be D less the crossings of D A maps: {segments of D*} -+ {-n, -n +2, -n +4, , n -2, n} is a labelling of D if near each crossing the values of s conform to one ofthe three types shown in Figure 16.2 a b a b a a a a a a a a a a b b a )