On spectral sequences from Khovanov homology
Abstract
There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1handle 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. In this paper we focus on the spectral sequences due to KronheimerMrowka from Khovanov homology to instanton knot Floer homology, and on that due to OzsváthSzabó to the HeegaardFloer homology of the branched double cover. For example, we use the 1handle morphisms to give new information about the filtrations on the instanton knot Floer homology of the (4,5) torus knot, determining these up to an ambiguity in a pair of degrees; to determine the OzsváthSzabó spectral sequence for an infinite class of prime knots; and to show that higher differentials of both the KronheimerMrowka and the OzsváthSzabó spectral sequences necessarily lower the delta grading for all pretzel knots.
 Publication:

arXiv eprints
 Pub Date:
 October 2013
 arXiv:
 arXiv:1310.7909
 Bibcode:
 2013arXiv1310.7909L
 Keywords:

 Mathematics  Geometric Topology;
 57M25
 EPrint:
 24 pages, 2 figures