A type A structure in Khovanov Homology

TL;DR

This paper introduces a new type A structure on Khovanov homology for tangles, inspired by bordered Floer homology, enabling modular computations and exact calculations of Khovanov homology for connect sums.

Contribution

It defines a homotopy-invariant type A structure that pairs with a type D structure to simplify and modularize Khovanov homology computations, especially for connect sums.

Findings

Homotopy type of the type A structure is a tangle invariant.

The approach allows exact computation of Khovanov homology from tangle structures.

Examples demonstrate correct torsion summands in connect sum cases.

Abstract

Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle, which complements the type D structure in a previous paper. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. The homotopy type of the type A structure is a tangle invariant, and homotopy equivalences of the type A structure result in chain homotopy equivalences on the Khovanov chain complex. We can use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. This approach adds to the literature even in the case of a connect sum, where the techniques here will allow an exact computation of Khovanov homology from the structures for two tangles coming from the summands. Several examples are included, showing in particular how we can compute the correct torsion summands for the Khovanov homology of the connect sum. A lengthy appendix is devoted to establishing the theory of these structures over a characterstic zero ring.