Algebraic & Geometric Topology

A type $A$ structure in Khovanov homology

Lawrence Roberts

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3653-3719.

Received: 1 April 2016
Accepted: 18 April 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 55N35: Other homology theories

Khovanov homology bordered theory tangle invariant


Roberts, Lawrence. A type $A$ structure in Khovanov homology. Algebr. Geom. Topol. 16 (2016), no. 6, 3653--3719. doi:10.2140/agt.2016.16.3653.

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