Dror Bar-Natan: Talks: HUJI-020613:

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

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.

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.
Dimensions of the rational homology of the knot 10(100) at height r and degree m

We know a reason why the dimensions of the rational homology are so much smaller than the dimensions of the corresponding chain spaces, but perhaps it isn't the right reason. Is there a much smaller complex that computes the same homology? Perhaps based on maximal trees in the checkerboard graph of the knot?
(dim homology)/(dim chains) for the knot 10(100) at height r and degree m

The Thistlethwaite link, it's checkerboard coloring, it's graph, and a maximal tree in it.

	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.
	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.	Dimensions of the rational homology of the knot 10(100) at height r and degree m
	We know a reason why the dimensions of the rational homology are so much smaller than the dimensions of the corresponding chain spaces, but perhaps it isn't the right reason. Is there a much smaller complex that computes the same homology? Perhaps based on maximal trees in the checkerboard graph of the knot?	(dim homology)/(dim chains) for the knot 10(100) at height r and degree m
		The Thistlethwaite link, it's checkerboard coloring, it's graph, and a maximal tree in it.