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

Hendricks, Kristen

doi:10.2140/agt.2012.12.2127

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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

DOI: 10.2140/agt.2012.12.2127

Abstract

Given a knot $K$ in $S^{3}$ , let $Σ (K)$ be the double branched cover of $S^{3}$ over $K$ . We show there is a spectral sequence whose $E^{1}$ page is $(\hat{HFK} (Σ (K), K) \otimes V^{\otimes (n - 1)}) \otimes ℤ_{2} ((q))$ , for $V$ a $ℤ_{2}$ –vector space of dimension two, and whose $E^{\infty}$ page is isomorphic to $(\hat{HFK} (S^{3}, K) \otimes V^{\otimes (n - 1)}) \otimes ℤ_{2} ((q))$ , as $ℤ_{2} ((q))$ –modules. As a consequence, we deduce a rank inequality between the knot Floer homologies $\hat{HFK} (Σ (K), K)$ and $\hat{HFK} (S^{3}, K)$ .

Keywords

Heegaard Floer, double branched covers, knot theory, Floer cohomology, localization

Mathematical Subject Classification 2010

Primary: 53D40, 57M25, 57M27, 57R58

References

Publication

Received: 11 July 2011

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/

Volume 12, issue 4 (2012)