On knot Floer homology in double branched covers

Abstract

Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. This paper shows that the knot Floer homology of this lift, with mod 2 coefficients, can be computed from a spectral sequence starting at a type of Khovanov homology already described by Asaeda, Przytycki, and Sikora. We extend the known results about this type of Khovanov homology, and use it to provide a very simple explanation of the case when L is alternating for the obvious projection.Comment: Due to an error in section 8, the results on the contact element have been weakene

On the equivalence of Legendrian and transverse invariants in knot Floer homology

“…The reduced Khovanov skein complex of K A I , as defined by Asaeda, Przytycki and Sikora [1], is the associated graded object of a filtration on the Khovanov complex for K . Roberts proves in [37] that .K/ is characterized with respect to this filtration in the same way that t is with respect to Ᏺ U . His work partially inspired this aspect of our story.…”

Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence

“…When F is an annulus, the topological situation is particularly natural; the thickened annulus, A × I, can be identified with the complement of a standardly-imbedded unknot in S 3 . As observed by L. Roberts [24], the data of the imbedding A × I ⊂ S 3 endows the Khovanov complex associated to L ⊂ S 3 with a filtration, and the resulting invariant of L ⊂ (A × I ⊂ S 3 ) is the filtered chain homotopy type of the complex. The induced spectral sequence converges to Kh(L), the Khovanov homology of L ⊂ S 3 .…”

On Knot Floer Homology for Some Fibered Knots

“…Proof: as in [14] this is proved by induction on the number of crossings in L. In particular, either of the resolutions of a crossing of L results in an alternating link with fewer crossings to which the result should apply. These resolutions correspond to two terms in a surgery exact sequence whose third term is the desired fibered knot.…”

“…Our first goal is to give a proof of the following result, which is a more specific version of the theorem from [14]. A is a round annulus in R 2 to fix the embedding of A × I in S 3…”

“…The other resoolution introduces a pair of critical points into the S 1 -valued Morse function. We can depict this process downstairs by resolving L. From the knot Floer homology surgery sequence applied to each crossing according to the algortihm, we get a surgery spectral sequence, [14].…”