Large cardinals and knot theory
Saturday 2008 January 26
I just came across the paper by P. Dehornoy: From large cardinals to braids via distributive algebra. J. Knot Theory Ramifications 4 (1995), no. 1, 33-79, which rather amazingly describes an application
of large cardinals in set theory to knot theory. The connection goes via left distributive algebras:
these are sets with a binary operation a[b] satisfying a[b[c]]=a[b][a[c]]. Typical examples are given by
a group G with a[b] given by the conjugation action of the group on itself: a[b] = aba-1. (By coincidence, Grigor Sargsyan mentioned this a few days ago in his comments on my post about set theory.)
Most of the obvious examples of left distributive algebras satisfy a[a]=a. Some examples not satisfying this come from the elementary embeddings of suitable models of set theory into themselves. There are two natural operations on the elementary embeddings of a suitable model into itself:
composition (corresponding to the group product in the example above) and and action a[b], which can be thought of informally as the image of the elementary embedding b under the action of a (corresponding to the action of a group on itself). The operation a[b] makes the set of elementary embeddings into a left distributive algebra that in general does not satisfy a[a]=a. (The existence of suitable elementary embeddings is essentially a rather powerful large cardinal axiom: the smallest ordinal not fixed by an elementary embedding turns out to be a very large cardinal.)
Dehorney’s paper explains how these new left distributive algebras first constructed using large cardinals were used to prove new results about braid groups in knot theory. A typical application is a definition of a linear order on braid groups, extending several previously known partial orders.
My impression is that most (and maybe all) the results about braid groups and left distributive algebras first proved using large cardinals have also later been proved by more elementary methods. This is rather like the applications of von Neumann algebras to knot theory by Vaughan Jones: they provided the initial motivation, but once the new results were known they were also proved by more elementary means.
