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.
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.
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?