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.