Mirror links have dual odd and generalized Khovanov homology

TL;DR

This paper demonstrates that generalized Khovanov homology admits a specific grading and reveals duality properties linking it to mirror links, unifying even and odd Khovanov homologies within a broader framework.

Contribution

It introduces a grading by ba2a2a2 and shows how generalized Khovanov homology encompasses even and odd variants, establishing duality relations for mirror links.

Findings

Generalized Khovanov homology admits a ba2a2a2 grading.

Switching variables induces a duality between a link and its mirror image.

Specializations recover duality for odd Khovanov homology.

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 $\mathbb{Z}\times\mathbb{Z}_2$, in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring $\mathbb{Z}_{\pi}:=\mathbb{Z}[\pi]/(\pi^2-1)$ (here, setting $\pi$ to $\pm 1$ results either in even or odd Khovanov homology). The generalized homology has $\Bbbk := \mathbb{Z}[X,Y,Z^{\pm 1}]/(X^2=Y^2=1)$ as coefficients, and the above implies that most of automorphisms of $\Bbbk$ fix the isomorphism class of the generalized homology regarded as $\Bbbk$-modules, 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, i.e. 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.