A type $A$ structure in Khovanov homology

Abstract

Inspired by bordered Floer homology, we describe a type $A$ structure in Khovanov homology, which complements the type $D$ structure previously defined by the author. 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 found from a pairing. We use this to simplify computations and introduce a modular approach to the computation of Khovanov homologies. Several examples are included, showing in particular how this approach computes the correct torsion summands for the Khovanov homology of connect sums. A lengthy appendix is devoted to establishing the theory of these structures over a characteristic-zero ring.