On knot Floer homology in double branched covers

Abstract

We define a link surgery spectral sequence for each knot Floer homology group for a knot, $K$, in a three manifold, $Y$. When $K$ arises as the double cover of an unknot in $S^3$, and $Y$ is the double cover of $S^3$ branched over a link, we relate the $E^2$–page to a version of Khovanov homology for links in an annulus defined by Asaeda, Przytycki and Sikora. Finally we examine the specific cases when the branch locus is a braid, and when it is alternating.