If anyone reads A refinement of Khovanov-Rozansky link homology, a new paper on the arXiv by Emmanuel Wagner, let me know what you make of it!
It uses some new techniques from Khovanov and Rozansky’s latest paper to define a version of homology, which is now triply-graded for each n! He claims that even in the n=2 case, it’s even stronger than the usual Khovanov homology.
I wouldn’t have expected to find such a creature out there, but maybe I’m just scared of having too many gradings at once. I haven’t read the paper closely yet. It seems he leaves out all his proofs — I suspect this paper is a really just an announcement of results from his PhD thesis. If anyone knows what’s going on, I’d love to hear more.
So what are these “new techniques” from Khovanov and Rozansky? Rozansky talked about this a fair bit in Kyoto last month, when introducing a new categorification of the SO(2N) quantum knot polynomials. Essentially, the idea is to extend the matrix factorization definition of link homology to virtual links, and then use this to write the complex associated to a real crossing in a new, easier form. First, the extension to virtual crossings is really easy — Ben could probably say more about this, but you just follow your nose, and associate to a virtual crossing the same sort of object you usually associate to two parallel strands. Second, the complex associated to a real (positive or negative) crossing can then be realised as a “homological deformation” of a virtual crossing. Perhaps for an explanation of this I’ll just point you at Nakajima’s notes (in particular page 3) from Rozansky’s second talk in Kyoto. (The notes from the first talk are there too.) Very roughly, you notice that the matrix factorisation for a “2-edge” is the cone over a morphism between the 2-strand untangle and the 2-strand virtual tangle. The complex of matrix factorisations associated to a postive crossing then has the matrix factorisation for the untangle as the “0-chains”, and the cone from the previous sentence as the “1-chains”.
I’m not exactly sure how this fits with what Wagner’s done, however.