The finiteness result for Khovanov homology and localization in monoidal categories

Abstract

In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the homology theory (local system) of finite type and prove the first finiteness result: the Khovanov local system restricted to the subcategory of knots of the crossing number at most n is the theory of type less or equal to n. This result can be further generalized to the categorification of Birman-Lin theorem.