The finiteness result for Khovanov homology and localization in monoidal categories

TL;DR

This paper proves a finiteness property of Khovanov homology, showing that the local system of complexes is of finite type when restricted to knots with bounded crossing number, and extends this to a categorification of a classical theorem.

Contribution

It introduces the concept of finite type for Khovanov homology and proves a key finiteness result, extending the theory to categorify the Birman-Lin theorem.

Findings

Khovanov local system is of finite type for knots with bounded crossing number

The finiteness result generalizes to a categorification of Birman-Lin theorem

Provides a new perspective on the structure of Khovanov homology in relation to knot complexity

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.