So, a couple of months ago, I gave a talk at the Max Planck Institute on knot homology, and as motivation I tried to explain why anyone studying the HOMFLY polynomial is inexorably led to the Hecke algebra. Nathan Geer, who was in the audience, asked me afterwards if there was anywhere this construction was written down, and lacking a good answer or the ambition to write a paper about it myself, I thought I would try to explain it in a blog post. It’s just applying an old TQFTologist’s trick, but old tricks often still have some new life in them.

The idea is this. Pick your favorite link invariant , valued in a ring . Then we can extend this to braids by just taking the invariant of the closure, and thus linearly to the group algebra of the n-strand braid group .

Now, interestingly, this map is not just any old linear map; it’s a trace, by the second Markov move. Of course, every algebra A with a trace has a unique largest two-sided ideal on which the trace vanishes, and thus a smallest quotient that factors through.

Thus, starting with a knot invariant, we get a series of algebras for each . One can check that the “adding a strand” map survives here, so we actually get a series of algebras equipped with inclusions and traces . These will have compatibilities that depend on the invariant. For example, if your invariant is multiplicative under disjoint union, the traces will satisfy where k is the value of your invariant on the unknot.

# Skein relations

Of course, in general, this will be very difficult to calculate. If you give a random invariant like bridge number, it will be very hard to do any calculations. But what if you have a skein relation? If you think about a skein relation carefully, you’ll see it tells you something about . If we have the skein relation

where is the knot with a positive crossing at a particular point, is the same knot with a negative crossing there and is the same knot with the crossing unzipped, then this gives a relation

where $\sigma$ is crossing the two strands. Of course, the same relation holds for higher indices for the braid crossing any two adjacent strands. Of course, this might not be all relations (in fact, usually not), but it gives a very interesting set of them.

# Examples

The example I had in mind with this construction was the HOMFLY polynomial. If one gives the HOMFLY polynomial the skein relation

one sees that

The Hecke algebra relations! In fact, you can prove that in this case, these are all the relations, and the associated algebra to the HOMFLY polynomial is the Hecke algebra (with coefficients in ). This explains why a person who wanted to categorify HOMFLY, might start by thinking about the Hecke algebra.

- From the Jones polynomial, one obtains the Temperley-Lieb algebra.
- From the Alexander polynomial, one gets an interesting quotient of the Hecke algebra, roughly the one killing all representations that aren’t hooks.

Does anyone feel like computing any others?

Good places you can send people with questions about this is Jones’s original paper, Ocneanu’s part of the HOMFLY paper, or Jones’s Annals paper. Another typical source for this stuff is Goodman, De La Harpe, and Jones.

Actually an even better source is the Birman-Wenzl paper (which I hadn’t had the chance to look at before). The reason this is better is that in Jones and Ocneanu they’re starting with an algebra they already understand and getting the new knot invariant, whereas in Birman-Wenzl there was already a construction due to Kauffman of the knot invariant and their goal is to get an algebra from it.

Actually, given any link invariant I that takes values in a vector space k, you always get a skein space by a similar construction. Given a choice of boundary X and two tangles A and B that share that boundary, you have an invariant I(A,B) = I(A cup B). You can then think of I(A,.) as a vector-valued invariant, taking values in the space of functions from B to k. For some choices of I, this relative invariant always lies in a subspace Sigma(X) of the set of functions from B to k. So you can rename the target of I(A) = I(A,.) to be Sigma(X). In yet more favorable cases, Sigma(X) is finite-dimensional for all X.

A similar construction automatically gives you not just algebras, but an entire k-linear ribbon category for all link invariants that satisfy a few mild properties. After all, the skein spaces have bilinear operations defined by partial gluings. It’s clear that the category is always “there” no matter how the invariant was defined for you.

If the question is a good reference for these ideas, they are discussed in an arXiv paper by Calegari, Freedman, and Walker, “Positivity of the universal pairing in 3 dimensions”. Another paper is “Universal manifold pairings and positivity” with a bunch of authors including Freedman.

That was basically the trick I was refering to. I believe it was taught to me by Gregor Massbaum. The main point of the post was how nicely this works for HOMFLY, giving you an algebra I’m always trying to convince people is interesting.