Khovanov Homology for Tangles and Cobordisms

Abstract. In my talk I will display one complicated picture and discuss it at length. Applying a certain 2D TQFT, we will get a homology theory whose Euler characteristic is the Jones polynomial. Not applying it, very cheaply we will get an invariant of tangles which is functorial under cobordisms and an invariant of 2-knots.

Why is it interesting?

• It is a knot/link/tangle invariant stronger than the Jones polynomial.
• It may be stronger than the original "Khovanov Homology".
• It has several generalizations, but as a whole, we hardly understand it. It may have significant algebraic and/or physical ramifications.
• It is functorial in the appropriate sense, and Rasmussen (math.GT/0402131 uses it to do some real topology.

Handouts: NewHandout-1.pdf, handout2.pdf.
Transparencies: Reid2Proof.pdf, R3Full.pdf.

The Picture:

An aside: (added May 6, 2004). On day one of my five day visit to Microsoft I refused to sign the Non-Disclosure Agreement and got escorted out of the building. On day two I signed it; had I not, I would have become a burden on my immediate hosts (who were nice and friendly and have nothing to do with Microsoft policies). I kept refusing to sign the Intellectual Property Agreement, though (that agreement would have made all my thoughts during my visit Microsoft property). The sanction was that I was not paid the nice consulting fee I was originally offered. In retrospect, I should have taken the money and donated it to the Free Software Foundation. There was a little more to the story than I have energy to write. In short - my immediate hosts were great; the corporation wasn't.