247 Graduate Texts in Mathematics EditorialBoard S. Axler K.A. Ribet Graduate Texts inMathematics 1 TAKEUTI/ZARING.IntroductiontoAxiomatic 38 GRAUERT/FRITZSCHE.SeveralComplex SetTheory.2nded. Variables. 2 OXTOBY.MeasureandCategory.2nded. 39 ARVESON.AnInvitationtoC-Algebras. 3 SCHAEFER.TopologicalVectorSpaces. 40 KEMENY/SNELL/KNAPP.Denumerable 2nded. MarkovChains.2nded. 4 HILTON/STAMMBACH.ACoursein 41 APOSTOL.ModularFunctionsandDirichlet HomologicalAlgebra.2nded. SeriesinNumberTheory.2nded. 5 MACLANE.CategoriesfortheWorking 42 J.-P.SERRE.LinearRepresentationsofFinite Mathematician.2nded. Groups. 6 HUGHES/PIPER.ProjectivePlanes. 43 GILLMAN/JERISON.RingsofContinuous 7 J.-P.SERRE.ACourseinArithmetic. Functions. 8 TAKEUTI/ZARING.AxiomaticSetTheory. 44 KENDIG.ElementaryAlgebraicGeometry. 9 HUMPHREYS.IntroductiontoLieAlgebras 45 LOE`VE.ProbabilityTheoryI.4thed. andRepresentationTheory. 46 LOE`VE.ProbabilityTheoryII.4thed. 10 COHEN.ACourseinSimpleHomotopy 47 MOISE.GeometricTopologyinDimensions2 Theory. and3. 11 CONWAY.FunctionsofOneComplex 48 SACHS/WU.GeneralRelativityfor VariableI.2nded. Mathematicians. 12 BEALS.AdvancedMathematicalAnalysis. 49 GRUENBERG/WEIR.LinearGeometry. 13 ANDERSON/FULLER.RingsandCategories 2nded. ofModules.2nded. 50 EDWARDS.Fermat’sLastTheorem. 14 GOLUBITSKY/GUILLEMIN.StableMappings 51 KLINGENBERG.ACourseinDifferential andTheirSingularities. Geometry. 15 BERBERIAN.LecturesinFunctionalAnalysis 52 HARTSHORNE.AlgebraicGeometry. andOperatorTheory. 53 MANIN.ACourseinMathematicalLogic. 16 WINTER.TheStructureofFields. 54 GRAVER/WATKINS.Combinatoricswith 17 ROSENBLATT.RandomProcesses.2nded. EmphasisontheTheoryofGraphs. 18 HALMOS.MeasureTheory. 55 BROWN/PEARCY.IntroductiontoOperator 19 HALMOS.AHilbertSpaceProblemBook. TheoryI:ElementsofFunctionalAnalysis. 2nded. 56 MASSEY.AlgebraicTopology:An 20 HUSEMOLLER.FibreBundles.3rded. Introduction. 21 HUMPHREYS.LinearAlgebraicGroups. 57 CROWELL/FOX.IntroductiontoKnotTheory. 22 BARNES/MACK.AnAlgebraicIntroduction 58 KOBLITZ.p-adicNumbers,p-adicAnalysis, toMathematicalLogic. andZeta-Functions.2nded. 23 GREUB.LinearAlgebra.4thed. 59 LANG.CyclotomicFields. 24 HOLMES.GeometricFunctionalAnalysisand 60 ARNOLD.MathematicalMethodsinClassical ItsApplications. Mechanics.2nded. 25 HEWITT/STROMBERG.RealandAbstract 61 WHITEHEAD.ElementsofHomotopyTheory. Analysis. 62 KARGAPOLOV/MERIZJAKOV.Fundamentals 26 MANES.AlgebraicTheories. oftheTheoryofGroups. 27 KELLEY.GeneralTopology. 63 BOLLOBAS.GraphTheory. 28 ZARISKI/SAMUEL.CommutativeAlgebra. 64 EDWARDS.FourierSeries.Vol.I.2nded. Vol.I. 65 WELLS.DifferentialAnalysisonComplex 29 ZARISKI/SAMUEL.CommutativeAlgebra. Manifolds.2nded. Vol.II. 66 WATERHOUSE.IntroductiontoAffineGroup 30 JACOBSON.LecturesinAbstractAlgebraI. Schemes. BasicConcepts. 67 SERRE.LocalFields. 31 JACOBSON.LecturesinAbstractAlgebraII. 68 WEIDMANN.LinearOperatorsinHilbert LinearAlgebra. Spaces. 32 JACOBSON.LecturesinAbstractAlgebraIII. 69 LANG.CyclotomicFieldsII. TheoryofFieldsandGaloisTheory. 70 MASSEY.SingularHomologyTheory. 33 HIRSCH.DifferentialTopology. 71 FARKAS/KRA.RiemannSurfaces.2nded. 34 SPITZER.PrinciplesofRandomWalk.2nded. 72 STILLWELL.ClassicalTopologyand 35 ALEXANDER/WERMER.SeveralComplex CombinatorialGroupTheory.2nded. VariablesandBanachAlgebras.3rded. 73 HUNGERFORD.Algebra. 36 KELLEY/NAMIOKAETAL.Linear 74 DAVENPORT.MultiplicativeNumberTheory. TopologicalSpaces. 3rded. 37 MONK.MathematicalLogic. (continuedafterindex) Christian Kassel Vladimir Turaev Braid Groups Withthegraphical assistanceofOlivierDodane 123 ChristianKassel VladimirTuraev InstitutdeRechercheMathe´matiqueAvance´e DepartmentofMathematics CNRSetUniversite´LouisPasteur IndianaUniversity 7rueRene´Descartes Bloomington,IN47405 67084Strasbourg USA France [email protected] [email protected] EditorialBoard S.Axler K.A.Ribet MathematicsDepartment MathematicsDepartment SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley SanFrancisco,CA94132 Berkeley,CA94720-3840 USA USA [email protected] [email protected] ISBN:978-0-387-33841-5 e-ISBN:978-0-387-68548-9 DOI:10.1007/978-0-387-68548-9 LibraryofCongressControlNumber:2008922934 MathematicsSubjectClassification(2000):20F36,57M25,37E30,20C08,06F15,20F60,55R80 (cid:2)c 2008SpringerScience+BusinessMedia,LLC Allrightsreserved.Thisworkmaynotbetranslated orcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com Preface The theory of braid groups is one of the most fascinating chapters of low- dimensional topology. Its beauty stems from the attractive geometric nature ofbraids andfromtheir close relationsto other remarkablegeometricobjects such as knots, links, homeomorphisms of surfaces, and configuration spaces. Ona deeper level,the interestofmathematicians inthis subjectis due tothe important role played by braids in diverse areas of mathematics and theo- retical physics. In particular, the study of braids naturally leads to various interesting algebras and their linear representations. Braid groups first appeared, albeit in a disguised form, in an article by AdolfHurwitzpublishedin1891anddevotedtoramifiedcoveringsofsurfaces. The notion of a braid was explicitly introduced by Emil Artin in the 1920s to formalizetopologicalobjects that modelthe intertwining of severalstrings in Euclidean 3-space.Artin pointed out that braids with a fixed number n of strings form a group, called the nth braid group and denoted by B . Since n then,the braidsandthe braidgroupshavebeenextensivelystudiedbytopol- ogists and algebraists. This has led to a rich theory with numerous ramifica- tions. In 1983, Vaughan Jones, while working on operator algebras, discovered new representationsofthe braidgroups,fromwhichhe derivedhis celebrated polynomial of knots and links. Jones’s discovery resulted in a strong increase of interest in the braid groups. Among more recent important results in this field are the orderability of the braid group B , proved by Patrick Dehornoy n in 1991, and the linearity of B , established by Daan Krammer and Stephen n Bigelow in 2001–2002. The principal objective of this book is to give a comprehensive introduc- tion to the theory of braid groups and to exhibit the diversity of their facets. The book is intended for graduate and postdoctoral students, as well as for allmathematiciansandphysicists interestedinbraids.Assumingonly abasic knowledge of topology and algebra, we provide a detailed exposition of the more advancedtopics. This includes backgroundmaterialin topologyand al- gebrathatoftengoesbeyondtraditionalpresentationsofthetheoryofbraids. vi Preface Inparticular,wepresentthebasicpropertiesofthesymmetricgroups,thethe- oryofsemisimplealgebras,andthelanguageofpartitionsandYoungtableaux. We now detail the contents of the book. Chapter 1 is concerned with the foundationsofthetheoryofbraidsandbraidgroups.Inparticular,wedescribe theconnectionswithconfigurationspaces,withautomorphismsoffreegroups, and with mapping class groups of punctured disks. In Chapter 2 we study the relation between braids and links in Euclidean 3-space. The central result of this chapter is the Alexander–Markov descrip- tion of oriented links in terms of Markov equivalence classes of braids. Chapter 3 is devoted to two remarkable representations of the braid group B : the Burau representation, introduced by Werner Burau in 1936, n and the Lawrence–Krammer–Bigelow representation, introduced by Ruth Lawrence in 1990. We use the technique of Dehn twists to show that the Burau representation is nonfaithful for large n, as was first established by John Moody in 1991. We employ the theory of noodles on punctured disks introduced by Stephen Bigelow to prove the Bigelow–Krammer theorem on the faithfulness of the Lawrence–Krammer–Bigelow representation. In this chapter we also construct the one-variable Alexander–Conway polynomial of links. Chapter4is concernedwiththesymmetricgroupsandthe Iwahori–Hecke algebras, both closely related to the braid groups. As an application, we construct the two-variable Jones–Conway polynomial of links, also known as the HOMFLY or HOMFLY-PT polynomial, which extends two fundamental one-variablelinkpolynomials,namelytheaforementionedAlexander–Conway polynomial and the Jones polynomial. Chapter 5 is devoted to a classificationof the finite-dimensional represen- tations of the generic Iwahori–Hecke algebras in terms of Young diagrams. As an application, we show that the (reduced) Burau representation of B n is irreducible. We also discuss the Temperley–Lieb algebrasand classify their finite-dimensional representations. Chapter 6 presents the Garside solution of the conjugacy problem in the braid groups. Following Patrick Dehornoy and Luis Paris, we introduce the concept of a Garside monoid, which is a monoid with appropriate divisibility properties. We show that the braid group B is the group of fractions of a n Garsidemonoidofpositivebraidsonnstrings.Wealsodescribesimilarresults for the generalized braid groups associated with Coxeter matrices. Chapter 7 is devoted to the orderability of the braid groups. Following Dehornoy, we prove that the braid group B is orderable for every n. n The book ends with four short appendices: Appendix A on the modu- lar group PSL (Z), Appendix B on fibrations, Appendix C on the Birman– 2 Murakami–Wenzl algebras, and Appendix D on self-distributive sets. The chapters of the book are to a great degree independent. The reader may start with the first section of Chapter 1 and then freely explore the rest of the book. Preface vii The theory of braids is certainly too vast to be covered in a single vol- ume. One important area entirely skipped in this book concerns the con- nections with mathematical physics, quantum groups, Hopf algebras, and braided monoidal categories. On these subjects we refer the reader to the monographs [Lus93], [CP94], [Tur94], [Kas95], [Maj95], [KRT97], [ES98]. Other areas not presented here include the homology and cohomology of the braid groups [Arn70], [Vai78], [Sal94], [CS96], automatic structures on the braid groups [ECHLPT92], [Mos95], and applications to cryptogra- phy [SCY93], [AAG99], [KLCHKP00]. For further aspects of the theory of braids, we refer the reader to the following monographs and survey articles: [Bir74], [BZ85], [Han89], [Kaw96], [Mur96], [MK99], [Ver99], [Iva02], [BB05]. Thisbookgrewoutofthelectures[Kas02],[Tur02]givenbytheauthorsat the Bourbaki Seminar in 1999 and 2000 and from graduate courses given by the first-namedauthoratUniversit´eLouisPasteur,Strasbourg,in2002–2003 andbythesecond-namedauthoratIndianaUniversity,Bloomington,in2006. Acknowledgments. It is a pleasure to thank Patrick Dehornoy, Nikolai Ivanov, and Hans Wenzl for valuable discussions and useful comments. We owe special thanks to Olivier Dodane, who drew the figures and guided us through the labyrinth of LATEX formats and commands. Strasbourg Christian Kassel March 3, 2008 Vladimir Turaev Contents Preface ........................................................ v 1 Braids and Braid Groups .................................. 1 1.1 The Artin braid groups .................................. 1 1.2 Braids and braid diagrams................................ 4 1.3 Pure braid groups ....................................... 18 1.4 Configuration spaces..................................... 25 1.5 Braid automorphisms of free groups........................ 31 1.6 Braids and homeomorphisms.............................. 35 1.7 Groups of homeomorphisms vs. configuration spaces ......... 40 Notes....................................................... 45 2 Braids, Knots, and Links .................................. 47 2.1 Knots and links in three-dimensional manifolds.............. 47 2.2 Closed braids in the solid torus............................ 51 2.3 Alexander’s theorem ..................................... 58 2.4 Links as closures of braids: an algorithm.................... 61 2.5 Markov’s theorem ....................................... 67 2.6 Deduction of Markov’s theorem from Lemma 2.11 ........... 71 2.7 Proof of Lemma 2.11..................................... 83 Notes....................................................... 91 3 Homological Representations of the Braid Groups ......... 93 3.1 The Burau representation ................................ 93 3.2 Nonfaithfulness of the Burau representation................. 98 3.3 The reduced Burau representation.........................107 3.4 The Alexander–Conwaypolynomial of links.................111 3.5 The Lawrence–Krammer–Bigelowrepresentation ............118 3.6 Noodles vs. spanning arcs ................................125 3.7 Proof of Theorem 3.15 ...................................137 Notes.......................................................149 x Contents 4 Symmetric Groups and Iwahori–Hecke Algebras ...........151 4.1 The symmetric groups ...................................151 4.2 The Iwahori–Heckealgebras ..............................163 4.3 The Ocneanu traces .....................................170 4.4 The Jones–Conwaypolynomial............................173 4.5 Semisimple algebras and modules..........................175 4.6 Semisimplicity of the Iwahori–Heckealgebras ...............191 Notes.......................................................193 5 Representations of the Iwahori–Hecke Algebras............195 5.1 The combinatorics of partitions and tableaux ...............195 5.2 The Young lattice .......................................200 5.3 Seminormal representations...............................207 5.4 Proof of Theorem 5.11 ...................................210 5.5 Simplicity of the seminormal representations ................215 5.6 Simplicity of the reduced Burau representation..............219 5.7 The Temperley–Lieb algebras .............................222 Notes.......................................................236 6 Garside Monoids and Braid Monoids ......................239 6.1 Monoids................................................239 6.2 Normal forms and the conjugacy problem...................243 6.3 Groups of fractions and pre-Garside monoids................251 6.4 Garside monoids ........................................255 6.5 The braid monoid .......................................259 6.6 Generalized braid groups .................................266 Notes.......................................................272 7 An Order on the Braid Groups ............................273 7.1 Orderable groups........................................273 7.2 Pure braid groups are biorderable .........................278 7.3 The Dehornoy order .....................................282 7.4 Nontriviality of σ-positive braids ..........................287 7.5 Handle reduction ........................................291 7.6 The Nielsen–Thurston approach...........................307 Notes.......................................................309 A Presentations of SL (Z) and PSL (Z) ......................311 2 2 Notes.......................................................314 B Fibrations and Homotopy Sequences.......................315 C The Birman–Murakami–Wenzl Algebras ...................317 Contents xi D Left Self-Distributive Sets .................................321 D.1 LD sets, racks,and quandles ..............................321 D.2 An action of the braid monoid ............................322 D.3 Orderable LD sets .......................................323 Notes.......................................................325 References.....................................................327 Index..........................................................337