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Mirror links have dual odd and generalized Khovanov homology

Krzysztof K Putyra and Wojciech Lubawski

Algebraic & Geometric Topology 16 (2016) 2021–2044

We show that the generalized Khovanov homology defined by the second author in the framework of chronological cobordisms admits a grading by the group × 2, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring π := [π](π2 1). (Here, setting π to ± 1 results either in even or odd Khovanov homology.) The generalized homology has k := [X,Y,Z±1](X2=Y 2=1) as coefficients, and the above implies that most automorphisms of k fix the isomorphism class of the generalized homology regarded as a k–module, so that the even and odd Khovanov homology are the only two specializations of the invariant. In particular, switching X with Y induces a derived isomorphism between the generalized Khovanov homology of a link L with its dual version, ie the homology of the mirror image L!, and we compute an explicit formula for this map. When specialized to integers it descends to a duality isomorphism for odd Khovanov homology, which was conjectured by A Shumakovitch.

Khovanov homology, odd Khovanov homology, mirror knot
Mathematical Subject Classification 2010
Primary: 55N35, 57M27
Received: 10 February 2015
Revised: 18 September 2015
Accepted: 29 November 2015
Published: 12 September 2016
Krzysztof K Putyra
Institute for Theoretical Studies
ETH Zürich
Clausiusstrasse 47
CH-8092 Zurich
Wojciech Lubawski
Theoretical Computer Science Department
Faculty of Mathematics and Computer Science
Jagiellonian University
ul. Lojasiewicza 6
30-348 Kraków