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2016 Mirror links have dual odd and generalized Khovanov homology
Krzysztof Putyra, Wojciech Lubawski
Algebr. Geom. Topol. 16(4): 2021-2044 (2016). DOI: 10.2140/agt.2016.16.2021

Abstract

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.

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Krzysztof Putyra. Wojciech Lubawski. "Mirror links have dual odd and generalized Khovanov homology." Algebr. Geom. Topol. 16 (4) 2021 - 2044, 2016. https://doi.org/10.2140/agt.2016.16.2021

Information

Received: 10 February 2015; Revised: 18 September 2015; Accepted: 29 November 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 06627567
MathSciNet: MR3546457
Digital Object Identifier: 10.2140/agt.2016.16.2021

Subjects:
Primary: 55N35, 57M27

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.16 • No. 4 • 2016
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