A rank inequality for the knot Floer homology of double branched covers

Given a knot K in S-3, let Sigma(K) be the double branched cover of S-3 over K. We show there is a spectral sequence whose E-1 page is ((HFK) over cap(Sigma(K), K) circle times V circle times(n-1)) circle times Z(2)((q)), for V a Z(2)-vector space of dimension two, and whose E-infinity page is isomorphic to ((HFK) over cap (S-3, K) circle times V circle times(n-1)) circle times Z(2)((q)), as Z(2)((q))-modules. As a consequence, we deduce a rank inequality between the knot Floer homologies (HFK) over cap(Sigma(K), K) and (HFK) over cap (S-3, K).