Braid Groups

By: Christian Kassel, Vladimir Turaev, O. Dodane (Designed by)

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Hardcover | 22 August 2008

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360 Pages

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23.39 x 15.6 x 2.06

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Braids and braid groups, the focus of this text, have been at the heart of important mathematical developments over the last two decades. Their association with permutations has led to their presence in a number of mathematical fields and physics. As central objects in knot theory and 3-dimensional topology, braid groups has led to the creation of a new field called quantum topology.

In this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence; this is followed by a treatment of the relationship between braids, knots and links. Important results then treat the linearity and orderability of the subject. Relevant additional material is included in five large appendices.

Braid Groups will serve graduate students and a number of mathematicians coming from diverse disciplines.

Industry Reviews

From the reviews: "Details on ... braid groups are carefully provided by Kassel and Turaev's text Braid Groups. ... Braid Groups is very well written. The proofs are detailed, clear, and complete. ... The text is to be praised for its level of detail. ... For people ... who want to understand current research in braid group related areas, Braid Groups is an excellent, in fact indispensable, text." (Scott Taylor, The Mathematical Association of America, October, 2008) "This is a very useful, carefully written book that will bring the reader up to date with some of the recent important advances in the study of the braid groups and their generalizations. It continues the tradition of these high quality graduate texts in mathematics. The book could easily be used as a text for a year course on braid groups for graduate students, one advantage being that the chapters are largely independent of each other." (Stephen P. Humphries, Mathematical Reviews, Issue 2009 e)

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Preface	p. v
Braids and Braid Groups	p. 1
The Artin braid groups	p. 1
Braids and braid diagrams	p. 4
Pure braid groups	p. 18
Configuration spaces	p. 25
Braid automorphisms of free groups	p. 31
Braids and homeomorphisms	p. 35
Groups of homeomorphisms vs. configuration spaces	p. 40
Notes	p. 45
Braids, Knots, and Links	p. 47
Knots and links in three-dimensional manifolds	p. 47
Closed braids in the solid torus	p. 51
Alexander's theorem	p. 58
Links as closures of braids: an algorithm	p. 61
Markov's theorem	p. 67
Deduction of Markov's theorem from Lemma 2.11	p. 71
Proof of Lemma 2.11	p. 83
Notes	p. 91
Homological Representations of the Braid Groups	p. 93
The Burau representation	p. 93
Nonfaithfulness of the Burau representation	p. 98
The reduced Burau representation	p. 107
The Alexander-Conway polynomial of links	p. 111
The Lawrence-Krammer-Bigelow representation	p. 118
Noodles vs. spanning arcs	p. 125
Proof of Theorem 3.15	p. 137
Notes	p. 149
Symmetric Groups and Iwahori-Hecke Algebras	p. 151
The symmetric groups	p. 151
The Iwahori-Hecke algebras	p. 163
The Ocneanu traces	p. 170
The Jones-Conway polynomial	p. 173
Semisimple algebras and modules	p. 175
Semisimplicity of the Iwahori-Hecke algebras	p. 191
Notes	p. 193
Representations of the Iwahori-Hecke Algebras	p. 195
The combinatorics of partitions and tableaux	p. 195
The Young lattice	p. 200
Seminormal representations	p. 207
Proof of Theorem 5.11	p. 210
Simplicity of the seminormal representations	p. 215
Simplicity of the reduced Burau representation	p. 219
The Temperley-Lieb algebras	p. 222
Notes	p. 236
Garside Monoids and Braid Monoids	p. 239
Monoids	p. 239
Normal forms and the conjugacy problem	p. 243
Groups of fractions and pre-Garside monoids	p. 251
Garside monoids	p. 255
The braid monoid	p. 259
Generalized braid groups	p. 266
Notes	p. 272
An Order on the Braid Groups	p. 273
Orderable groups	p. 273
Pure braid groups are biorderable	p. 278
The Dehornoy order	p. 282
Nontriviality of [sigma]-positive braids	p. 287
Handle reduction	p. 291
The Nielsen-Thurston approach	p. 307
Notes	p. 309
Presentations of SL[subscript 2](Z) and PSL[subscript 2](Z)	p. 311
Notes	p. 314
Fibrations and Homotopy Sequences	p. 315
The Birman-Murakami-Wenzl Algebras	p. 317
Left Self-Distributive Sets	p. 321
LD sets, racks, and quandles	p. 321
An action of the braid monoid	p. 322
Orderable LD sets	p. 323
Notes	p. 325
References	p. 327
Index	p. 337
Table of Contents provided by Ingram. All Rights Reserved.

Braid Groups

At a Glance

Hardcover

Industry Reviews

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