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 $\mathbb{Z}\times\mathbb{Z}_2$, 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.