#### Volume 12, issue 4 (2012)

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

### Kristen Hendricks

Algebraic & Geometric Topology 12 (2012) 2127–2178
##### 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 $\left(\stackrel{̂}{\mathit{HFK}}\left(\Sigma \left(K\right),K\right)\otimes {V}^{\otimes \left(n-1\right)}\right)\otimes {ℤ}_{2}\left(\phantom{\rule{0.3em}{0ex}}\left(q\right)\phantom{\rule{0.3em}{0ex}}\right)$, for $V$ a ${ℤ}_{2}$–vector space of dimension two, and whose ${E}^{\infty }$ page is isomorphic to $\left(\stackrel{̂}{\mathit{HFK}}\left({S}^{3},K\right)\otimes {V}^{\otimes \left(n-1\right)}\right)\otimes {ℤ}_{2}\left(\phantom{\rule{0.3em}{0ex}}\left(q\right)\phantom{\rule{0.3em}{0ex}}\right)$, as ${ℤ}_{2}\left(\phantom{\rule{0.3em}{0ex}}\left(q\right)\phantom{\rule{0.3em}{0ex}}\right)$–modules. As a consequence, we deduce a rank inequality between the knot Floer homologies $\stackrel{̂}{\mathit{HFK}}\left(\Sigma \left(K\right),K\right)$ and $\stackrel{̂}{\mathit{HFK}}\left({S}^{3},K\right)$.

##### Keywords
Heegaard Floer, double branched covers, knot theory, Floer cohomology, localization
##### Mathematical Subject Classification 2010
Primary: 53D40, 57M25, 57M27, 57R58
##### Publication
Revised: 18 June 2012
Accepted: 12 July 2012
Published: 26 December 2012
##### Authors
 Kristen Hendricks Department of Mathematics Columbia University 2990 Broadway New York NY 10027 USA http://math.columbia.edu/~hendricks/