  Cobordisms are naturally bigraded and we show that this grading extends to
Khovanov homology, making it a triply graded theory. Although the new grading
does not make the homology a stronger invariant, it can be used to show that
odd Khovanov homology is multiplicative with respect to disjoint unions and
connected sums of links; same results hold for the generalized Khovanov
homology defined by the author in his previous work. We also examine the module
structure on both odd and even Khovanov homology, in particular computing the
effect of sliding a basepoint through a crossing on the integral homology.
