Ordering Braids

The braid group is one of those objects that appears throughout mathematics. It had its humble beginnings as a group that, legend has it, Emil Artin designed early in the twentieth century to show high school students an example of an infinite nonabelian group. Since then it has grown into a rich theory which is now finding uses in (among other places) knot theory, mathematical physics, the study of the inverse Galois problem, and cryptography.

In the 1990s, Patrick Dehornoy showed that in addition to the other aspects of their algebraic structure, the elements of the braid group admit a natural ordering. The revelation of this additional structure has led to even more applications of the braid group. The book under review, entitled simply Ordering Braids, is a summary and proof of the fact that the braid group is ordered. The book was written by Dehornoy along with several coauthors, and gives a fascinating introduction to the topic.

The book assumes very little in the way of prerequisites. It opens with a chapter defining the braid group from several different perspectives, including algebraic, topological, and combinatorial. In particular, it shows how the braid group is generated by a natural set of generators σ₁,...,σ_n along with some easily stated relations. The second chapter of the book defines the notion of an ordering and then defines the Dehornoy Ordering of the braid group. To understand this ordering one must first define the concept of a word in the generators being "σ-positive", which means that the σ_i with lowest index i which appears in the word only appears to positive powers. The Dehornoy ordering then says that the element β is less than the element β' if ββ' is "σ-positive". The rest of chapter two shows that if we can prove three relatively straightforward facts then it will quickly follow that this is a well-defined ordering which satisfies many nice properties. These three desired lemmata are:

• Property A - A braid word which is σ-positive is nontrivial.
• Property C - Every nontrivial braid can be written in terms of the generators in a way which is either σ-positive or σ-negative (i.e. the σ_i which appears with lowest index only appears to negative powers).
• Property S - Every braid of the form β^-1σ_iβ is σ-positive.

After a chapter explaining some applications of the braid ordering and consequences of well-orderability, the bulk of the remaining thirteen chapters is spent proving the three properties stated above. Or, to be more specific, using the different facets of the braid group to give a number of different proofs of each of the three. For example, by my count the authors give at least eight distinct proofs of Property C, and a similar number for the other two properties.

The fact that there are so many different approaches comes from the many different ways of thinking about braid groups, and the tour through various proofs involve topics ranging from hyperbolic geometry to automorphisms of free groups, as well as connections with colorability and relaxation algorithms. These latter chapters are meant to be read in any order, and as I jumped between them I felt that the authors did as good job of keeping the chapters self-contained as one could hope when pulling ideas from so many different branches of mathematics.

If one’s goal is simply to prove that the braid group is well-ordered then there are surely more efficient ways than the approaches given by Dehornot and his coauthors in Ordering Braids. But the reader that goes through the entire book will walk away with a good understanding of not only the main theorem of the book but also of many different ways of thinking about the braid group and the connections between then.