  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.
