A hunka hunka burnin’ knot homology September 24, 2009
Posted by Ben Webster in category O, Category Theory, combinatorics, homological algebra, link homology, low-dimensional topology, quantum groups, representation theory.
One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?
This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.
At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong).
Let me give a little background: As many readers will know, there is an invariant of knots attached to every representation of a simple complex Lie group, of which the Jones polynomial is a special case for the defining representation of . So, as knot homologies have appeared over recent years categorifying the Jones polynomial, and other special cases of these invariants, like the defining representations of and (both work of Khovanov and Rozansky), some of us (I know I’m not the only one) have started to wonder if this isn’t a bunch of special cases of a construction for all representations. I believe this is one of things Khovanov and Lauda had in mind when they wrote down a categorification of quantum groups (and is an idea that goes back at least as far as Crane and Frenkel).
For a while, the idea has been floating around that one might be able to do this using the geometry of quiver varieties. I’ve been thinking about this idea for about 3 years, and I’m sure I was not the first (I know Hao Zheng has also had it in mind for while). But geometry is hard, so for a long time various problems and confusions stood in the way of me (or anyone else, it seems) actually carrying it through.
Luckily, other people were around to remove the geometry from the picture and somehow cook things down into some combinatorics that you can actually get a handle on. Specifically, there’s the recent work of Khovanov and Lauda, which is essentially a combinatorial rewrite of Lusztig’s geometric theory of canonical bases without a quiver variety or perverse sheaf in sight. Once I actually understood this fact (it took a while to sink in), I knew exactly what I needed to do: take all the intuitions I had about the geometry of quiver varieties, and convert them into the same kind of combinatorics.
The result is this paper, preliminarily titled “Knot invariants and higher representation theory.” It constructs a bigraded vector space for each oriented link with components labeled by representations of a complex simple Lie group G, for any group and any representations, whose Euler characteristic is the corresponding quantum invariant of that link. Better yet, it proposes a functoriality map for cobordisms between knots (but does not prove that this doesn’t depend on the handle decomposition of the cobordism. That will not be a fun thing to prove). The method is all combinatorics and combinatorial representation theory, with geometry only peeking through at moments; computations will not be easy, but they are finite and only involve a lot of taking projective resolutions and tensoring modules.
I know there are a reasonable number of blog readers who will be interested, so I thought I would share with you early in the process and see what happens; there may still be mistakes in the paper (though I am pretty confident at this point that all mistakes will be minor and fixable). I’d love to hear any comments people have on the paper (even if you don’t get very far, it could help the exposition to know where people get stuck). It may not be as ambitious as polymath, but I like to think of it as a minor step toward a more open, internet-based mathematics.