Braid Groups

342 Pages·2008·7.888 MB·English

by  Christian Kassel, Vladimir G. Turaev, O. Dodane

247
Graduate Texts in Mathematics
EditorialBoard
S. Axler
K.A. Ribet
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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
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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