1. A Quick Introduction to Khovanov Homology, I
2. A Quick Introduction to Khovanov Homology, II
Abstract. As appropriate for a relic from the past, I will tell
the Kauffman bracket story of the Jones polynomial as Kauffman told it in
1987, then the Khovanov homology story as Khovanov told it in 1999, and
finally the "local Khovanov homology" story as I understood it in 2003
(with perhaps a word about alternating tangles). At the end of my two
talks you will understand what is a "Jones homology", how to generalize
it to tangles and to cobordisms between tangles, and why it is computable
relatively efficiently. But we will say nothing about more modern stuff
- the Rasmussen invariant, Alexander and HOMFLYPT knot homologies, and
the categorification of
sl2 and other Lie algebras.
3. Meta-Groups, Meta-Bicrossed-Products, and the Alexander Polynomial
Abstract. I will define "meta-groups" and explain how one specific
meta-group, which in itself is a "meta-bicrossed-product", gives rise to an
"ultimate Alexander invariant" of tangles, that contains the Alexander
polynomial (multivariable, if you wish), has extremely good composition
properties, is evaluated in a topologically meaningful way, and is
least-wasteful in a computational sense. If you believe in categorification,
that's a wonderful playground.
See also my likewise-titled
paper with Sam Selmani.
