
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 ChernSimons 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 3manifolds.


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


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
10_{100} 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
4dimensional invariance is proven:
