Anna Beliakova 'On the functoriality of the Khovanov homology of tangles'

On the functoriality of the Khovanov homology for tangles

In 2002 Khovanov proved his homology to be projectively functorial with respect to link (or tangle) cobordisms. Shortly after, Jacobson showed that signs do really matter, and Bar-Natan reformulated Khovanov’s construction topologically using a quotient of (an additive closure of) 2-cobordisms called Bar-Natan bicategory. Since then, many approaches were developed to fix the sign issue. All solutions propose to keep track of orientations and actually replace Bar-Natan category by something else. If you want to work over Z, than the replacement is the bicategory of GL(2)-foams. In our recent paper with M. Hogancamp, K. Putyra and S. Wehrli, we proved that Bar-Natan and GL(2)-foams constitute equivalent bicategories (by constructing an explicit bifunctor between them). This gives an explicit way to fix signs inside the Bar-Natan bicategory.

In this talk I will start with a gentle introduction for the subject, explain the proof of our theorem and, if time remains, mention an application of this result, that identifies 3 different models of the colored sl(2) homology in the quantum annular setting.