Volume 16, issue 4 (2016)

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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
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 $ℤ×{ℤ}_{2}$, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring ${ℤ}_{\pi }:=ℤ\left[\pi \right]∕\left({\pi }^{2}-1\right)$. (Here, setting $\pi$ to $±1$ results either in even or odd Khovanov homology.) The generalized homology has $\mathbb{k}:=ℤ\left[X,Y,{Z}^{±1}\right]∕\left({X}^{2}={Y}^{2}=1\right)$ as coefficients, and the above implies that most automorphisms of $\mathbb{k}$ fix the isomorphism class of the generalized homology regarded as a $\mathbb{k}$–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.

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