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

Abstract

Given a knot $K$ in $S^3$, let $\Sigma \left(K\right)$ be the double branched cover of $S^3$ over $K$. We show there is a spectral sequence whose $E^1$ page is $\left(\widehat{\mathit{HFK}}\left(\Sigma \left(K\right),K\right)\otimes V^{\otimes \left(n-1\right)}\right)\otimes {\mathbb{Z}}_2\left(\left(q\right)\right)$, for $V$ a ${\mathbb{Z}}_2$–vector space of dimension two, and whose $E^{\infty }$ page is isomorphic to $\left(\widehat{\mathit{HFK}}\left(S^3,K\right)\otimes V^{\otimes \left(n-1\right)}\right)\otimes {\mathbb{Z}}_2\left(\left(q\right)\right)$, as ${\mathbb{Z}}_2\left(\left(q\right)\right)$–modules. As a consequence, we deduce a rank inequality between the knot Floer homologies $\widehat{\mathit{HFK}}\left(\Sigma \left(K\right),K\right)$ and $\widehat{\mathit{HFK}}\left(S^3,K\right)$.