I will be giving two talks at UNC:
After quickly recalling what is the Euler characteristic and what is homology and after briefly describing the Jones polynomial, I will move on to describe in simple terms Khovanov's novel "Categorification of the Jones Polynomial", which is to the Jones polynomial as homology is to the Euler characteristic. Namely, it is much more interesting.
With some added and some removed, my talk will be based on my paper On Khovanov's Categorification of the Jones Polynomial and will mostly follow this handout. The only new material will be on the back side of the handout, which will be printed with a list of reasons for why is categorification interestings and with the sketch of the proof that Khovanov's homology lifts to an invariant of knot cobordisms.
The minor advantage of homology theory over the Euler characteristic is that it is a finer invariant. The major advantage is that it is a functor: Given a map between spaces there is a map between their homologies. Think of almost any major theorem in algebraic topology and you'll find that the functoriality of homology is deeply involved. In my talk, I will explain in elementary terms what seems to be the corresponding property of Khovanov's homology: that it is a functor from the category of links and cobordisms to the category of vector spaces (see Jacobsson's arXiv:math.GT/0206303 and Khovanov's arXiv:math.QA/0207264). My proof of this property is in the spirit of Khovanov's, but it is both simpler and more general. It involves the extension of the theory to the canopoly of tangle cobordisms, with values in several related canopolies.
What's a canopoly? No, that would go in the talk; not here. It's an object with a rather messy formal definition but a very simple visual image.