We show that the generalized Khovanov homology defined by the second author in the framework of chronological cobordisms admits a grading by the group , in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring . (Here, setting to results either in even or odd Khovanov homology.) The generalized homology has as coefficients, and the above implies that most automorphisms of fix the isomorphism class of the generalized homology regarded as a –module, so that the even and odd Khovanov homology are the only two specializations of the invariant. In particular, switching with induces a derived isomorphism between the generalized Khovanov homology of a link with its dual version, ie the homology of the mirror image , 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.
"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