Mixed invariants in Khovanov homology for unorientable cobordisms
Using Bar-Natan's and Lee's deformations of Khovanov homology of links, we define minus, plus, and infinity versions of Khovanov homology. Given an unorientable cobordism in [0,1]\times S^3 from a link L_0 to a link L_1, we define a mixed invariant as a map from the minus version of the Khovanov homology of L_0 to the plus version of the Khovanov homology of L_1. The construction is similar to the mixed invariant in Heegaard Floer homology. This invariant can be used to distinguish exotic cobordisms, that is, two cobordisms which are topologically isotopic but not smoothly isotopic. This is joint with Robert Lipshitz.