There are a number of homological knot invariants, each satisfying an unoriented
skein exact sequence, which can be realised as the limit page of a spectral sequence
starting at a version of the Khovanov chain complex. Compositions of elementary
–handle
movie moves induce a morphism of spectral sequences. These morphisms remain
unexploited in the literature, perhaps because there is still an open question
concerning the naturality of maps induced by general movies.
Here we focus on the spectral sequence due to Kronheimer and Mrowka from Khovanov
homology to instanton knot Floer homology, and on that due to Ozsváth and Szabó to
the Heegaard Floer homology of the branched double cover. For example, we use the
–handle
morphisms to give new information about the filtrations on the instanton knot Floer homology
of the
–torus
knot, determining these up to an ambiguity in a pair of degrees; to determine the
Ozsváth–Szabó spectral sequence for an infinite class of prime knots; and
to show that higher differentials of both the Kronheimer–Mrowka and the
Ozsváth–Szabó spectral sequences necessarily lower the delta grading for all
pretzel knots.
