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, and make a bimodule in polynomials in n variables out of this.
Let be the map swapping with and fixing all other variables. These generate the obvious -action on R. Each has a subring of fixed points , which is generated by the ‘s for and by the symmetric polynomials and .
Consider the R-R-bimodule
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 where 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 . 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.