Yet more conference blogging: Rasmussen and Rozansky

This morning’s talks were given by Jake Rasmussen on “The HOMFLY Polynomial of Kazhdan-Lusztig Basis Elements” (an auspicious title if I’ve ever heard one. I have to admit, I was kind of worried that Jake might have done some research overlapping with mine more than I would have liked when I heard that, though it turns out to not be true. The stuff we’re doing is pretty complementary) and Lev Rozansky on “Virtual Crossings and Categorification”.

Jake talked (in a mostly elementary way) about the triply graded Khovanov-Rozansky homology of braid graphs (which are braids, except you have intersection points where you would usually have crossings), which is the main ingredient in constructing the analogous homology for knots.

The basic object under consideration is what is called a Soergel bimodule: basically, you start with a sequence of integers smaller than n, \mathbf{i}=(i_1,\ldots,i_m) and make a bimodule in polynomials in n variables R=\mathbb{Q}[x_1,\ldots, x_n] out of this.

Let s_i: R\to R be the map swapping x_i with x_{i+1} and fixing all other variables. These generate the obvious S_n-action on R. Each s_i has a subring of fixed points R^{s_i}, which is generated by the x_j‘s for j\neq i, i+1 and by the symmetric polynomials x_i+x_{i+1} and x_ix_{i+1}.

Consider the R-R-bimodule

R_{ \mathbf{i}}\cong R\otimes_{R^{s_{i_1}}}R\otimes_{R^{s_{i_2}}}\cdots \otimes_{R^{s_{i_m}}}R.

OK, now calculate its Hochschild homology: that is, take a free resolution of this (as a bimodule). Such a resolution is rather easy to write down, this is really what Khovanov-Rozansky did in their original definition of HOMFLY homology, and Lev makes this pretty explicit in his talks, though I’m not so sure about papers. However, it is rather hard to compute wit. More on that later.

Apply the functor M\mapsto M/[M,R] where [M,R]=R\cdots\{r\cdot m-m\cdot r\}\cdot R to this chain complex. The homology of this is called the Hochschild homology of our module, or alternatively the K-R homology of the singular braid in which the sequence i specifies the order of intersections of pairs of adjacent strands (just as a work in the braid group would specify the crossings). Forget for a moment that you probably have idea what this is and bear with me.

Now, while this definition is rather explicit (its the homology of a rather explicitly described chain complex that you can sic a computer on), this bimodule is a pretty slippery bugger to really get your hands on. Considering that decomposing this Soergel bimodule into indecomposibles is strictly harder than computing the structure coefficients of the KL basis in the Hecke algebra (which is rather hard), it feels a little hopeless to try to get your hands on it, or worse yet its Hochschild homology, but in fact there is more than one good way.

Jake’s perspective is that you should use the MOY relations of the Hilbert series of these (see Section 4 of his paper “Some Differentials on Khovanov-Rozansky Homology” or the video of his talk today) to iteratively calculate this Hochschild homology. This approach has the advantage of being very combinatorial, and proves some interesting results. Perhaps the most interesting for me is that the Hochschild homology is free over as an R-module. This is a really important fact for my geometrical stuff, and I had missed it the first time I ready Jake’s paper. I’m afraid that even in the internet age, talks do have their uses. I won’t go into detail now, but though this sounds like a harmless, cute little theorem, it has powerful geometric consequences.

As Jake has discovered, this approach works great in small examples, but starts to break down as your rank gets bigger and the structure coefficients of the Kazhdan-Lusztig basis get worse (and they get pretty bad, though admitedly in a controlled way). Anyways, this approach is rather computerable, and could lead to some good results in the braid index 4 and 5 cases (certainly an the understanding of the braid index 3 case obtained by hand has allowed me to do computations by hand that others expected to be rather hard).

Now I have a different perspective, which I’ll hopefully roll out in a post soon, but rewriting the slides for that talk is a higher priority at the moment. The short version is that you should do everything geometrically, and as geometric approaches usually are, it’s good for finding surprising points of general structure and not so brilliant for calculating. Oh well, what can I say? I’m a slave to geometry. You can insert your own sexist metaphor (“geometry” does seem to be feminine gender in most European languages with such a notion).

Rozansky’s talk was almost pretty good, though a lot of review for me. On the other hand, it contained a rather interesting “rebranding,” if you will, of K-R homology. As many of you know, K-R homology replaces each crossing with a two-term complex in some algebraic category (matrix factorizations, or just plain modules in the HOMFLY case), which seem a little on the unmotivated side. One of the terms in this complex corresponds to simply opening the crossing in an oriented way, and a more complicated one (which appears to have come to Mikhail Khovanov in a dream or something) corresponds to the the two strands intersecting (often people will insert a double line at the point of intersection). Admittedly, this term is more natural from the perspective of Soergel bimodules, but bear with me for a moment. But despite first appearances, this term for the intersection is natural: it’s the unique non-split extension of the open crossing by a term corresponding to a virtual crossing. Since the homology of the two term complex is the module corresponding to the virtual crossing, we can think of the two-term complex for an honest crossing as a “homological deformation” of the virtual crossing (there are two ways of doing this, for the positive and negative crossings).

Of course, this is all woozy philosophy, which doesn’t prove any new results. But often woozy philosophy points the way to new and more interesting results, as happened in this case. It turns out that this perspective is perfect for categorifying the Kaufman polynomial (warning: I don’t mean the bracket polynomial. The Kaufman polynomial is a different quantum invariant) for \mathrm{SO}(2n). For actual details, read their preprint.

Sorry afternoon speakers, I do not have the energy at this late hour to write about your talks. As I may have mention before, y’all can watch the video if you’re curious.


9 thoughts on “Yet more conference blogging: Rasmussen and Rozansky

  1. Thank you for drawing the distinction between the Kauffman polynomial and the Kauffman bracket. Now if we can just get everyone to stop calling the Kauffman bracket the Jones polynomial we’ll be set.

  2. Now if we can just get everyone to stop calling the Kauffman bracket the Jones polynomial we’ll be set.

    Yeah right. And monkeys might fly out of my butt.

  3. While you’re rewriting your slides, three minor typos…

    on the slide “The Rouquier complex”
    You haven’t defined the notation R_e, I think, and you’ve written R_{s_i} where you mean just R_i,.

    on the slide “Reduction to compact form”
    you’re missing “to” from “we get to use”.

    Now in exchange… besides just not being the whole story, what is wrong with the knot invariant we get if we just take co-invariants instead of Hochschild homology? First, I guess — it is actually a knot invariant, right? Just the Koszul level 0 piece of the real thing. I’m guessing the right answer is that there’s no exact triangle, but I don’t remember seeing that said. Is there something more fundamental?

  4. I don’t understand why it would be so important to distinguish between calling something the Jones polynomial and calling it the Kauffman bracket. Isn’t the Kauffman bracket just the original Jones towers of algebras approach to the Jones polynomial plus the key observation that the Temperley-Lieb algebra is given by nifty pictures?

    Sure Jones original approach can be adapted to get a skein theoretic polynomial, similarly the Kauffman bracket can be adapted to get a skein theoretic polynomial. The difference is whether you think of TL as being given by generators and relations or by pictures. I don’t think that’s so big a difference that we should be careful about which name to use.

  5. The difference is that the Kauffman bracket is an invariant of regular isotopy of unoriented links, while the Jones polynomial is an invariant of ambient isotopy of oriented links. There is a well-known relation between the two if you have both structures, but (strictly speaking) they have different domains.

  6. I was going to say “The Kaufman bracket is an invariant of framed links, whereas the Jone polynomial is simply an invariant of links.” Which I think is just another way of saying what John did.

  7. Besides just not being the whole story, what is wrong with the knot invariant we get if we just take co-invariants instead of Hochschild homology? First, I guess — it is actually a knot invariant, right? Just the Koszul level 0 piece of the real thing.

    You’re forgetting the shift. When you do a negative stabilization, you have to identify Hochschild homology with a shift, so that co-invariants on one side become the 1st degree Hochschild homology on the other, so there’s basically no way of making an invariant just with coinvariants.

    Incidentally, you only found typos on pages I’m no longer planning on using in my talk, so not much of a quid pro quo. You’d be better if you appealed to my higher instincts.

  8. What I was saying is that from the original point of view the Jones polynomial is an invariant of braids invariant under Markov 1 and with good properties under Markov 2. That is to say, it is more natural to think of it as giving an invariant of framed links. However, it is easily modified to give an invariant of oriented links, as Jones did. Similarly, Kaufman’s approach first gives an invariant of framed links and can be modified to give an invariant of oriented links. I admit that when people distinguish the two they use the terms the way you’re doing, but given that you have to fix your definitions and normalizations anyway (is it q or -q? do you put in funny shifts because you’re motivating Khovanov homology?) I see no reason to enforce pedantry on this point.

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