So, I thought I would actually get back to blogging my research (by which I mean, shameless self-promotion) a bit. Probably the problem that I’ve focused the most on in the past few years is how to understand knot homology geometrically. While this still has a lot of mysteries, Geordie Williamson and I are finishing up a pair of papers that I think are a big step forward in this area.
The one I’d like to talk about in this post is “A geometric construction of colored HOMFLYPT homology.” (It’s not on the arXiv yet, but we’re close. Comments would be helpful, hint, hint). I’m mostly just going to talk about the consequences of this paper for the triply graded homology of Khovanov-Rozansky, though I think one of its most exciting features is how easily it generalizes to the colored situation.
In my last post, I described a complex of sheaves on the group for each braid of index n. I mentioned at the end of that post that there’s a certain functor which sends this sheaf to the complex of Soergel bimodules Khovanov used to define HOMFLY knot homology. This suggests that somehow, this knot homology can be extracted in a direct geometric way from
And that’s exactly what we do; it’s simply a matter of producing the right functor. We remember that is equivariant for the actions of the upper triangular matrices on are by left and inverse right multiplication.
Whenever we have a complex of equivariant sheaves on a space for a group , we can take equivariant hypercohomology of the complex with respect to to obtain a graded vector space (in the equivariant derived category discussed in this post, this is just Hom from the constant sheaf to ). In this case, we’ll be interested in taking cohomology for two different groups: and , the diagonal (which is acting by conjugation).
Unfortunately, it turns out that both of these are boring; they don’t depend on the braid in any interesting way.
But we have an extra ingredient in the picture: carries a canonical filtration (“the weight filtration”) which there is definitely not room in this margin to define (you’ll just have to read SGA 4 1/2).
Thus there is a spectral sequence that starts at the cohomology of the successive quotients and ends with the cohomology of the whole thing. It’s when you look at this spectral sequence that you strike gold.
- This associated spectral sequence for is a knot invariant.
- The -page of this spectral sequence is Khovanov-Rozansky homology (with the gradings given by the two spectral sequence gradings, and an extra secret grading, the “weight grading.”)
- The -page of the spectral sequence for gives the complex of Soergel bimodules used by Khovanov.
Now, you might be tempted to say “Bah, humbug…what’s this geometric stuff for? I was happier thinking about bimodules. At least I can define those.”
Of course, everyone’s entitled to their opinion, so let me explain why this isn’t my opinion.
- First, this hooks Khovanov homology into a rich and already understood vein of categorification. This categorification of the Hecke algebra goes back decades, and is connected to all kinds of fascinating mathematics, from the Kazhdan-Lusztig conjecture to the recent work of Ben-Zvi and Nadler on character field theories.
- Second, alternative perspectives make it clear how to generalize things. If you phrase things in the right way, then you can construct colored HOMFLY homology (originally defined by Mackaay, Stosic and Vaz) by just replacing upper triangular matrices with block upper triangular matrices everywhere (the block sizes giving the colorings). In this case, while there is an algebraic definition of this invariant, it’s very hard to motivate, whereas the geometry is doing something very canonical. Even worse, there is no algebraic proof that one gets a knot invariant, and that the Euler characteristic of the homology is really the colored HOMFLY polynomial (the HOMFLY polynomial for wedge powers of the standard representation of ). But there is a geometric one! It’s not even that hard (a stripped down proof would run between 5 and 10 pages).