Braid Groups

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.

• Burau
• Garside
• Homotopy
• Iwahori-Hecke
• Markov
• Permutation
• Representation theory
• Theoretical physics
• algebra
• configuration spaces
• fibrations
• homeomorphisms of surfaces
• low-dimensional topology
• topology

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)

“This book is a comprehensive introduction to the theory of braid groups. Assuming only a basic knowledge of topology and algebra, it is intended mainly for graduate and postdoctoral students.” (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1208, 2011)

“The book of Kassel and Turaev is a textbook … for graduate students and researchers. As such, it covers the basic material on braids, knots, and links … at a level which requires minimal background, yet moves rapidly to non-trivial topics. … It is a carefully planned and well-written book; the authors are true experts, and it fills a gap. … it will have many readers.” (Joan S. Birman, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)

• Inst. Rech. Math. Avancée, Univ. Louis Pasteur et CNRS, Strasbourg CX, France

  Christian Kassel

• Dept. Mathematics, Indiana University, Bloomington, U.S.A.

  Vladimir Turaev

Dr. Christian Kassel is the director of CNRS (Centre National de la Recherche Scientifique in France), was the director of l'Institut de Recherche Mathematique Avancee from 2000 to 2004, and is an editor for the Journal of Pure and Applied Algebra. Kassel has numerous publications, including the book Quantum Groups in the Springer Gradate Texts in Mathematics series.

Dr. Vladimir Turaev was also a professor at the CNRS and is currently at Indiana University in the Department of Mathematics.

Policies and ethics