Volume 16, issue 6 (2016)

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A type $A$ structure in Khovanov homology

Lawrence P Roberts

Algebraic & Geometric Topology 16 (2016) 3653–3719

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.

Khovanov homology, bordered theory, tangle invariant
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 55N35
Received: 1 April 2016
Accepted: 18 April 2016
Published: 15 December 2016
Lawrence P Roberts
Department of Mathematics
University of Alabama
Box 870350
Tuscaloosa, AL 35487-0350
United States