A type A structure in Khovanov homology

Roberts, Lawrence

doi:10.2140/agt.2016.16.3653

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A type $A$ structure in Khovanov homology

Lawrence P Roberts

Algebraic & Geometric Topology 16 (2016) 3653–3719

DOI: 10.2140/agt.2016.16.3653

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.

Keywords

Khovanov homology, bordered theory, tangle invariant

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 55N35

References

Publication

Received: 1 April 2016

Accepted: 18 April 2016

Published: 15 December 2016

Authors

Lawrence P Roberts
Department of Mathematics University of Alabama Box 870350 Tuscaloosa, AL 35487-0350 United States

Volume 16, issue 6 (2016)