# The Unapologetic Mathematician

## AMS Sectional, Day 2

Still no wireless, so I’ll again jot a little something about the noteworthy talks.

Louis Kauffman gave a talk about an invariant of “virtual” knots and links, which are described in his paper that I linked to the other day. The invariant is an extension of the Kauffman bracket that I extend to tangles. An obvious question is how to do both extensions, getting functors on the category of virtual tangles.

Heather Dye, Kauffman’s former student, then spoke on virtual homotopy. This may well be related to Allison Henrich’s work on Legendrian virtual knots. It’s all tangled (har) up together.

I gave my talk after that. I’ll make a separate post with the link to my slides.

After lunch, Alexander Shumakovitch gave a very clear (though not yet complete) combinatorial categorification of HOMFLY evaluations. There’s a parameter $n$ in his theory, and setting it to 2 gives back the combinatorial version of Khovanov homology. Setting it to higher values should correspond to what Josh Sussan — currently finishing his Ph.D. here at Yale — has done in the representation-theoretic picture for $U_q({\mathfrak sl}_n)$.

The last talk that really grabbed me was Michael “Cap” Khoury’s explanation of a new definition for the Alexander-Conway polynomial. The really interesting thing here is that it really looks like he’s realizing it as some sort of representable functor on some sort of category. Almost, but not quite. We talked for a bit about what’s missing, and I don’t think it’s impossible to push it a bit and get that last lousy point.