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

**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 SL
_{2}(**Z**) and PSL_{2}(**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)