Graduate Texts in Mathematics, vol. 247, Springer, New York, 2008 (340 p. + xii). ISBN 978-0-387-33841-5.

Christian Kassel and Vladimir Turaev

Braid Groups

Mathematics Subject Classification (2000): Primary 20F36, 20F10, 20E05, 20F60, 57M25, 57M27, 57R50, 57R52, 20M05, 20C08; Secondary 05E10, 11F06, 16D60, 16K20, 20B30, 20C30

Presentation (excerpts from the introduction)

The theory of braid groups is one of the most fascinating chapters of low-dimensional topology. Its beauty stems from the attractive geometric nature of braids and from their close relations to other remarkable geometric objects such as knots, links, homeomorphisms of surfaces, and configuration spaces. On a deeper level, the interest of mathematicians in this subject is due to the important role played by braids in diverse areas of mathematics and theoretical 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 Adolf Hurwitz published in 1891 and devoted to ramified coverings of surfaces. The notion of a braid was explicitly introduced by Emil Artin in the 1920s to formalize topological objects that model the intertwining of several strings in Euclidean 3-space. Artin pointed out that braids with a fixed number n = 1, 2, 3,... of strings form a group, called the n-th braid group and denoted B_n. Since then, the braids and the braid groups have been extensively studied by topologists and algebraists. This has led to a rich theory with numerous ramifications. In 1983, Vaughan Jones, while working on operator algebras, discovered new representations of the braid groups, from which he derived his 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_n, proved by Patrick Dehornoy in 1991, and the linearity of B_n, established by Daan Krammer and Stephen Bigelow in 2001-2002.

The principal objective of this book is to give a comprehensive introduction 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 all mathematicians and physicists interested in braids. Assuming only a basic knowledge of topology and algebra, we provide a detailed exposition of the more advanced topics. This includes background material in topology and algebra that often goes beyond traditional presentations of the theory of braid groups. In particular, we present the basic properties of the symmetric groups, the theory of semisimple algebras, and the language of partitions and Young tableaux.

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.

Table of Contents

Introduction
Chapter 1. Braids and Braid Groups
Chapter 2. Braids, Knots, and Links
Chapter 3. Homological Representations of the Braid Groups
Chapter 4. Symmetric Groups and Iwahori-Hecke Algebras
Chapter 5. Representations of the Iwahori-Hecke Algebras
Chapter 6. Garside Monoids and Braid Monoids
Chapter 7. An Order on the Braid Groups
Appendix A. Presentations of SL₂(Z) and PSL₂(Z)
Appendix B. Fibrations and Homotopy Sequences
Appendix C. The Birman-Murakami-Wenzl Algebras
Appendix D. Left Self-Distributive Sets
References
Index

List of errata (updated 26 March 2010)

(9 Frebruary 2024)