Dror Bar-Natan: Talks: HUJI-021230:

Why is Categorification Interesting?

because it is orthogonal to everything we know or expect
because there is so much we still don't know about it
because it lifts to an invariant of knot cobordisms

Like in the case of the Jones polynomial, we don't have a topological interpretation of categorification.

Unlike the case of the Jones polynomial, we don't have a characterization of categorification, only a construction.

We don't have a "physical" explanation of categorification (like Witten's Chern-Simons explanation of many other knot invariants).

We don't know how to repeat the story in the case of other knot polynomials (though we have high expectations in some cases).

We don't know to generalize categorification to the case of knots inside other 3-manifolds.

We don't know if the story generalizes to the case of invariants of 3-manifolds.

Categorification doesn't seem to generalize to virtual knots.

We don't understand why the rational homology for all the knots for which it was computed always decomposes as a sum of many "knight moves" and a single "pawn move" at height 0. At the right are the dimensions of the rational homology of the knot 10₁₀₀ at height r and degree m.

Khovanov's homology is a functor from the category of knots with cobordisms to the category of vector spaces! Here's how 4-dimensional invariance is proven:

Theorem. (in progress) There is a canopoly map Kh between the canopoly of cobordisms between tangle diagrams and the canopoly of homotopy classes of morphisms between complexes in the additive canopoly freely generated by the canopoly of canned surfaces modulo local relations. Furthermore, Kh maps isomorphisms between tangles to homotopic complexes.

	Like in the case of the Jones polynomial, we don't have a topological interpretation of categorification.
	Unlike the case of the Jones polynomial, we don't have a characterization of categorification, only a construction.
	We don't have a "physical" explanation of categorification (like Witten's Chern-Simons explanation of many other knot invariants).
	We don't know how to repeat the story in the case of other knot polynomials (though we have high expectations in some cases).
	We don't know to generalize categorification to the case of knots inside other 3-manifolds.
	We don't know if the story generalizes to the case of invariants of 3-manifolds.
	Categorification doesn't seem to generalize to virtual knots.
	We don't understand why the rational homology for all the knots for which it was computed always decomposes as a sum of many "knight moves" and a single "pawn move" at height 0. At the right are the dimensions of the rational homology of the knot 10₁₀₀ at height r and degree m.
	Khovanov's homology is a functor from the category of knots with cobordisms to the category of vector spaces! Here's how 4-dimensional invariance is proven: