Quantum Topology conference

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This Week in Quantum Topology, Week 1 (just kidding)

I'm on my flight back from Salt Lake City, after a great week at the AMS summer conference on Quantum Topology. I gave a talk on 'Spiders for sl_n', or 'Generators and Relations for Rep U_q sl_n as a pivotal category' depending on your taste. I think it went well.

The highlight of the conference for me was definitely Wednesday, when we heard a sequence of talks about Khovanov homology. Magnus Jacobsson gave a mostly introductory talk, Dror Bar-Natan gave a talk on his 'local' version for sl_2, Jake Rasmussen told us about some new conjectures relating Khovanov-Rozansky homology to knot Floer homology, and Marco (?) talked about progress categorifying the coloured Jones polynomial.

Dror began by essentially following the first few sections of his recent paper, then giving further details on computational methods, and a 'high altitude theorem' about mutation invariance. (I'm not quite sure on the final status of this 'theorem'; I think the consensus was that 'it's probably true, at least over a field', and we'll hear more shortly!) Most interesting about the computation methods is the 'complex simplification' lemma. Apparently one of the obstacles to practical calculation of Khovanov-Rozansky homology is the lack of an appropriate analogue of this.

Jake began by giving an introduction to knot Floer homology. The most interesting part, however, was a conjecture (due to Jake, Gukov, and ??), (approximately) saying that the latest and greatest triply graded Khovanov-Rozansky theory has a family (indexed by integers) of differentials, satisfying various properties, and that the homologies with respect to these differentials, at certain specialisations, recover not only the original doubly graded Khovanov-Rozansky invariants, but also the knot Floer homology! Wow!

Is a combinatorial version of knot Floer homology just around the corner? They don't know how to construct differentials satisfying their proposed properties, they can make various predictions based on the existence of such differentials, which seem to be borne out.

I'll have to admit I was a bit sleepy by Marco's talk, so I can't say much about it.

The other reason Wednesday was such a great day was that we had the afternoon off; many of us hiked up the hill, then had great fun running, glissading and snowball-fighting our way back down.