Rarely does a mathematical object have as many different descriptions as does the braid group on n strands. The simplest description is simply that the group Bn is generated by elements
s1, s2, ..., sn–1
subject to the relations
|sisj = sjsi
||if |i – j| ≥ 2
|sisi+1si = si+1sisi+1
||if 1 ≤ i ≤ n–2
I am partial to two other descriptions of the braid group Bn. First, the group Bn is isomorphic to the group whose elements are “braids” on n strands modulo isotopy which fixes the endpoints of the graph and certain “Reidemeister moves”. The figure on the left is a diagram of an element of B4. Two such elements are multiplied by stacking their diagrams and connecting the two braids.
Second, the n-strand braid group is also the mapping class group of an n-punctured disc, that is, the group of homeomorphisms of the punctured disc which fix the boundary modulo the subgroup of homeomorphisms isotopic to the identity.
Details on these and several other descriptions of the braid groups are carefully provided by Kassel and Turaev’s text Braid Groups. As is to be expected from any text on braid groups, it highlights the relationship between braid groups and links and gives detailed proofs of many classical results. (For example, the braid groups are torsion-free.) Most of these classical results are also contained in Birman’s famous Braids, Links, and Mapping Class Groups (Princeton, 1975). Braid Groups will, therefore, be most valuable for its treatment of more recent work. A few of these recent results appealed to my interests; these are the topics that I will mention in this review.
For a long time few representations of the braid group were known. The most famous of these was the Burau representation. For n ≥ 6, the Burau representation of Bn was proven in 1993 to be unfaithful (i.e. non-injective) by Long and Paton. In 1999, Bigelow improved this by showing that the Burau representation of B5 is unfaithful. For n = 3, the Burau representation is faithful and for n = 4 it is still not known whether or not the Burau representation is faithful. Braid Groups reproduces elements in the kernel of the Burau representation for n = 5 and n = 6. More helpfully, the authors also prove the results (using the methods of Long-Paton and Bigelow) using a description of the Burau representation in terms of an action of Bn on a homology group.
The homological view of representations of Bn also led to a resolution of the long-standing conjecture that the braid groups are linear, that is, that there exists a faithful representation of Bn into a group of matrices. When n ≤ 3 this is nearly trivial. In 2000, Krammer showed that a certain representation (found by Lawrence) of B4 was faithful. In 2001, Bigelow showed that Lawrence’s representation was faithful for all n. Krammer, at nearly the same time, extended his methods to obtain the same result. This representation is now known as the Lawrence-Krammer-Bigelow representation.
Using Bigelow’s methods, the authors of Braid Groups describe this representation in terms of an action on a homology group and prove it to be faithful. This is one of the few occurences of the term “noodle” in mathematics. Bigelow’s original proof used both forks and noodles, but (sadly) the forks turn out to be unnecessary for this tasty mathematical meal.
My other two favourite sections of Braid Groups are the section on Garside groups (which solves the word problem for braid groups) and the section on the orderability of the braid groups. This last section presents a construction of Dehornoy of a left-invariant total order on Bn. (A left invariant total order on a group is a total order ≤ such that if g and h are elements of the group such that g ≤ h then for all group elements k, kg ≤ kh.)
Braid Groups is very well written. The proofs are detailed, clear, and complete. The topics covered are now fundamental to the study of braid groups and other groups (and algebras) related to the braid groups. (A partial list includes mapping class groups, fundamental groups of configuration spaces, Iwahori-Hecke algebras, Temperley-Lieb algebras, Garside groups, Coxeter groups, and Artin groups.)
A particular strength of the text is its pattern of introducing a general concept (such as the Garside group), proving one or two theorems, and then immediately showing the applicability of the concept to the braid groups. This text is an excellent introduction to many of these concepts.
My only complaint is that there are not enough pictures. More pictures would certainly make the text easier to read, particularly for arguments concerning diagrams.
The text is to be praised for its level of detail. This is a field where it is easy to lapse into giving frustratingly imprecise proofs. The authors in all cases have avoided doing so. There are a few occasions where they refer the reader to a paper or another text for the proof of a theorem. This is to be expected in a reasonably sized textbook about a subject as broad and deep as the study of braid groups.
The authors review certain graduate-level topics (such as fibrations and group representations), but unless one has seen them before the sections of the text that rely on those concepts are not likely to be accessible. Indeed, this book would be very challenging for anyone without a substantial algebra or topology background. For people with that background who want to understand current research in braid group related areas, Braid Groups is an excellent, in fact indispensable, text.
Scott Taylor is Visiting Assistant Professor of Mathematics at Colby College in Waterville, ME.