Questions about the MOY relations

I have some rather specific questions about the MOY knot invariant which have come up as Sabin Cautis and I have been thinking about Khovanov-Rozansky homology. I’ll start by explaining the following “theorem” and then I’ll ask some questions about it. Hopefully someone (for eg. Scott) will be able to answer them.

Consider closed crossingless Reshetikhin-Turaev diagrams for sl(n) where all strands are labelled by 1 or 2. This means that we look at oriented planar graphs with all edges either labelled 1 or 2, and each vertex trivalent with either two single edges and one double edge and matching orientations.

We can think of such a graph as being related to the representation theory of (quantum) sl(n). The single strands correspond to the standard representation \mathbb{C}^n and the double strands to \Lambda^2 \mathbb{C}^n The vertices correspond to morphisms of representations (for example \mathbb{C}^n \otimes \mathbb{C}^n \rightarrow \Lambda^2 \mathbb{C}^n ). Within this representation theory context, each graph G can be evaluated to a Laurent polynomial c(G) (an element of the trivial representation of quantum sl(n)).

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Soergel bimodules

The next ingredient in my paper with Geordie is understanding a bit about Soergel bimodules.

Soergel bimodules (or “Soergelsche Bimoduln” auf Deutsch) are a remarkable category of bimodules over a polynomial ring. The main thing that’s remarkable about them is that they categorify the Hecke algebra of S_n (for those of you who don’t know any other Hecke algebras, pretend I just said “Hecke algebra”).

Theorem (Soergel). The split Grothendieck group of the category of Soergel bimodules for S_n is the Hecke algebra \mathcal{H}_n of S_n.

We’ll unpack that a bit later. For the moment, bear with me. Of course, one of the most important things about Grothendieck groups is that they have a natural basis. If one uses a split Grothendieck group (like a normal Grothendieck group, except we only have a relation that says [M]+[N]=[M\oplus N], not for non-trivial extensions), then this basis will be the classes of the indecomposable elements.

Theorem (Soergel). The basis of indecomposable objects is the famous basis of Kazhdan and Lusztig.

Now that I’ve told you why people like Soergel bimodules, I guess I had better define them.

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Hochschild homology

I finally got around to writing a bit about my paper with Geordie, and realized that this was considerably more than a one post story. I figured I would start by saying a bit about Hochschild homology. This is a pretty standard bit of homological algebra, but has suddenly bursted out of the deformation theory ghetto into link homology.

So, what is Hochschild homology? Unfortunately, in most books, you seem to get a horrifying and useless definition (a bit like what happens with group cohomology). So let me attempt to explain what it really is. Continue reading

Freiburg! im Breisgau!

For those of you still wondering whether I’ve made it home yet, the answer is: nope. I’m in Freiburg, Germany, to try and get a little work done with my friend and collaborator Geordie Williamson (keep an eye peeled for our paper on the arXiv, though it’s actually been posted on my webpage for a month now). We seem to have made some good progress, both on this mostly done paper and some more speculative projects (I can’t reveal any details, but there were affine Grassmannians involved), and I’m giving a talk tomorrow to edumacate the algebra-types around here about knot theory a little (it’s one of those weird perspective twists that comes from being at Berkeley that I tend to expect algebraists to know some rough version of the Jones polynomial story and some knot theory basics, which is clearly insane).

The rough sketch of my talk is as follows:

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Getting canopolises right.

Leaving aside the issue of thorny question of the correct plural form of canopolis, lets try and get the definition right!

The motivating idea here is that a planar algebra is a nice (the right?) formalism for tensor categories with duals, and that there’s a reasonable hope that the definition will extend nicely to something like an n-planar algebra (a noodle algebra, a foam algebra?), which we should think of, very roughly, as an n-category with lots of duals. A canopolis is a first step along the way – it adds an extra categorical dimension to a planar algebra, but without demanding any more duals.

The term canopolis was introduced by Bar-Natan, although there called a canopoly, then used by Ben and by me and Ari Nieh. It still doesn’t have a good definition written down anywhere. The idea is simple; a canopolis is meant to be a planar algebra of categories. I’m still not too sure how much devil there is in the details though, and would love some help. Continue reading

Deliberately vague talk-blogging

I hope our readers haven’t been too expectant about awaiting talk blogging which has not been forth-coming. I’ve been in Europe for 3 and a half weeks now, and I think the general exhaustion of traveling is catching up to me.

This conference is set up to have 4 “major speakers” who give a series of 3 hour long lectures. These were originally supposed to be Khovanov, Seidel, Gukov and Ozsvath, but in an amazing stroke of bad luck, both of the first two canceled for health reasons (Paul Seidel did so two days before the start of the conference). Jake Rasmussen is pitch hitting for Khovanov, but they couldn’t find any replacement for Seidel on such short notice.

So, about those talks…

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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”.

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