# Geometry and triply graded knot homology

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 $C_\beta$ on the group $\mathrm{GL}(n)$ for each braid $\beta$ 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 $C_\beta$

And that’s exactly what we do; it’s simply a matter of producing the right functor. We remember that $C_\beta$ is equivariant for the actions of the upper triangular matrices $B$ on $\mathrm{GL}(n)$ are by left and inverse right multiplication.

Whenever we have a complex of equivariant sheaves $\mathcal{F}$on a space $X$ for a group $G$, we can take equivariant hypercohomology of the complex with respect to $G$ to obtain a graded vector space $\mathbb{H}^*_{G}(X;\mathcal{F})$ (in the equivariant derived category discussed in this post, this is just Hom from the constant sheaf to $\mathcal{F}$). In this case, we’ll be interested in taking cohomology for two different groups: $B\times B$ and $B_\Delta$, 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: $C_\beta$ 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.
Theorem:

• This associated spectral sequence for $B_\Delta$ is a knot invariant.
• The $E^2$-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 $E^2$-page of the spectral sequence for $B\times B$ 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 $\mathfrak{sl}_n$). But there is a geometric one! It’s not even that hard (a stripped down proof would run between 5 and 10 pages).

## 24 thoughts on “Geometry and triply graded knot homology”

1. Great stuff! Of course, as soon as the words “geometric” are mentioned around the words “knot homology” my eyes light up—but then I realize that you mean “geometric representation theory.” Which strikes me as fascinating—but is there any chance that you know of that this point of view will lead to a connection with the geometry of the underlying knot (eg by having the algebraic group in question act on some low-dimensional manifold…)

2. sheesh i didn’t do a very good proofreading job… anyway geometric is clearly one word, not more than one…

3. Well, the direction that might bring this stuff back to topology is what one might call “brave new TQFT” a la Lurie, which allows you to create topological invariants in a way that is deeply topological, and yet can start with a purely algebraic input like an n-category. But that’s pure speculation; BZ? Nad-meister? Are you reading? Any thoughts?

4. I’m just tossing words out there, but I think the geometric input involves a (flat?) G-bundle on your 3-manifold of choice (produced by a construction involving knots or something), perhaps with some extra data like (T or B) reductions on certain loci.

5. Get rid of “et al” in Thm 1.3, if you write a paper your deserve to have your name mentioned when people cite it.

In the end aren’t you getting a quadruply graded theory? 3 gradings for each page, and an additional grading which tells you which page you’re on? Are the higher pages interesting? Your Thm 1.2 says that they’re knot invariants, right? Whatever the answer to this is (yes they’re interesting, no they aren’t, they might be interesting) I’d argue that you should put that answer in the introduction.

6. Are the higher pages interesting?

Maybe, maybe not. They could be very interesting, but no one has slightest clue how to compute them. The E^2 page at least has a well known decategorification. The other ones, not so much. If you look at Jake Rasmussen’s “differentials on KR homology” paper, you’ll see, there are tremendous numbers of spectral sequences relating homological knot invariants that people had already studied, and no one has the slightest clue what their overall structure is (I think Jake is the only person who’s even capable of computing examples).

You’re right, though, that perhaps that should be emphasized.

7. John Baldwin has done a similar thing for the spectral sequence from Khovanov homology to Heegaard floer homology of the branched double cover (shown that the higher pages are knot invariants). Again, it’s an interesting open question to understand them, at least to see what they decategorify to. I wonder how all this stuff will play out?

Ben or someone else: is there a good introduction somewhere to this geometric representation stuff, for a topologist?

8. Geometric input to what?

Your brave new TQFT. Shouldn’t it have something to do with Chern-Simons theory?

9. In my understanding, it’s very difficult to ever find naturally occuring spectral sequences that converge much later than E_2. It was an open problem whether the Dolbeault spectral sequence of a compact complex manifold can take arbitrarily long time to converge; only recently did someone find a construction to do that. What is known as to how late any of these spectral sequences can converge, never mind what information is present?

10. I’m trying to sort out if I remember how this confusing Reidemeister IIb move issue works… Is the point that it’s known that closed diagrams (i.e. links) are invariant under Reidemeister IIb, but it’s not known if the the invariant applied to an arbitrary tangle is invariant under RIIb?

11. Is there a homology version (or categorification) for Vassiliev’s finite type invariants?

12. Noah,

It’s much worse than that; no one knows if the original HOMFLYPT homology is invariant under it. Luckily, you can get between 2 closed braids that represent the same knot without ever using that move, so if only think about braids, you don’t need to prove that invariance.

13. Gil,

Not as far as I know. Presumably somebody out there has a graduate student thinking about it.

14. Greg,
Possibly it is interesting to note that the spectral sequence converges to something easy to describe which depends only on the number of link components and their labels.
Hence $E_2$ is nontrivial (and contains at least as much information as colored HOMFLYPT), whereas $E_{\infty}$ can be read easily from the knot or link.
What happens in between is a mystery …

15. 14. So if I start with a general knot diagram in order to turn it into a braid closure I might need to use RIIb, but if I have two braid closures for the same knot then I can get between them without ever using RIIb?

16. So if I start with a general knot diagram in order to turn it into a braid I might need to use RIIb, but if I have two braid closures for the same knot then I can get between them without ever using RIIb?

Yes. This is a consequence of Markov’s theorem. RIIb is always necessary for going from a non-braid to a braid, since it’s the only move that can turn incoherently oriented Seifert circles into coherently oriented ones.

17. One confusing issue in the relation of this story to three-dimensional
topology is that braid groups are coming up in two very different ways: braid groups arise from the topology of Lie groups, which is how they come up in the geometric rep theory story Ben is telling, but the type A braid groups arise from low-dimensional topology. There’s a Hecke category for each G, and its modules give particularly nice braid group actions on categories (Ben, is there a nice way to characterize by some kind of “continuity” or “algebraicity” which braid group actions come from Hecke actions??)
On the other hand representations of braid groups of type A come up naturally when studying any TFT (in dims three or four, depending on what you want to act on). In type A the two stories are related (the relation goes by various names, such as Laplace transform or gl_n-gl_m dual pair etc) but not for other types. So if one is studying 4d gauge theories (which are supposedly behind both geometric Langlands and Khovanov homology) there are really two independent appearances of representations of braid groups.. very confusing!

18. is there a nice way to characterize by some kind of “continuity” or “algebraicity” which braid group actions come from Hecke actions??

I think generally people think about the Hecke algebra condition as requring the twists (Artin generators; lifts of simple reflections) to all have the same pair of possible eigenvalues (which pair depends on the normalization). By the same logic, the symmetric group is when the eigenvalues can only be $\pm 1$. (The important distinction being that $\pm 1$ is a subgroup of the multiplicative group of any field, and no other 2 element subset is).

19. there are really two independent appearances of representations of braid groups.. very confusing!

of course, what’s really confusing is that the Artin groups of other types have a topological interpretation too! The B/C type braid group is the group of braids in a cylinder and thus has to do with invariants of knots in the complement of the torus (and absolutely nothing to do with the Reshetikhin-Turaev invariants for type B/C!)

20. David: I like to think about a toy case in which the Hecke algebra, or even simpler the symmetric group, appears in these two different ways. Recall the main statement of Schur-Weyl duality: V^{tensor n} is both a representation of S_n and GL(V); and if V is k-dimensional, GL(V) contains S_k as well as its Weyl group. The isotypic decomposition of V^{tensor n} with respect to either S_n or GL(V) is the same; the representation factors of S_n are the multiplicity factors of GL(V) and vice-versa. This situation has a quantum analogue in which both of the symmetric groups deformed to the standard Hecke algebra; I think the quantum parameter q has the same value too.

At first glance, Schur-Weyl duality is a duality between two very different things. But in some ways, they are not all that different. For instance, the crystal limit of the decomposition of V^{tensor n} is the RSK algorithm for words on k letters of length n. But if k ≫ n, then most of the letters are distinct, so it then looks more like the RSK algorithm for permutations — which itself arises as the crystal limit of the matrix block decomposition of a single Hecke algebra. The double limit where k and n both go to infinity, and maybe k goes to infinity first, has also been studied. I think that the differences between the two halves of the duality are then even more ephemeral. (But I do not know all that much about it.)

21. Ben – you may have answered my question, I’m not sure.. My question was about actions of the Hecke category, not the Hecke algebra.. namely for an action of the braid group we have a condition on the eigenvalues of generators that says it descends to a Hecke action. On the other hand most of the braid group actions on categories that I’m aware of come from actions of the Hecke category (as is discussed in Rouquier’s paper) – on this level I don’t think there’s a simple “eigenvalue” picture, but maybe I’m being naive, but in any case I’d like to get a sense for when a braid action extends to the action of the whole Hecke category. I’d expect this to be a kind of notion of continuity (ie the braid group generates the Hecke category in some reasonable sense..) Curious for any thoughts!