Mirror links have dual odd and generalized Khovanov homology

We show that the generalized Khovanov homology defined by the second author in the framework of chronological cobordisms admits a grading by the group $ℤ \times ℤ_{2}$ , in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring $ℤ_{π} : = ℤ [π] ∕ (π^{2} - 1)$ . (Here, setting $π$ to $\pm 1$ results either in even or odd Khovanov homology.) The generalized homology has $k : = ℤ [X, Y, Z^{\pm 1}] ∕ (X^{2} = Y^{2} = 1)$ 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.

Keywords

Khovanov homology, odd Khovanov homology, mirror knot

Mathematical Subject Classification 2010

Primary: 55N35, 57M27

References

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

Volume 16, issue 4 (2016)

Krzysztof K Putyra and Wojciech Lubawski

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors