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

TL;DR

This paper establishes a spectral sequence relating the knot Floer homology of a knot in S^3 to that of its double branched cover, leading to a new rank inequality between these homologies.

Contribution

It introduces a spectral sequence connecting the knot Floer homology of a knot and its double branched cover, revealing a new rank inequality.

Findings

Spectral sequence from (\u00b5(K), K) to (S^3, K)

Rank inequality between (\u00b5(K), K) and (S^3, K)

Isomorphism of ((K), K) and (S^3, K) as ()-modules

Abstract

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 (\hat{HFK}(\Sigma(K), K) \otimes V^{n-1}) \otimes \mathbb Z_2((q)), for V a \mathbb Z_2-vector space of dimension two, and whose E^{\infty} page is isomorphic to (\hat{HFK}(S^3, K) \otimes V^{n-1}) \otimes \mathbb Z_2((q)), as \mathbb Z_2((q))-modules. As a consequence, we deduce a rank inequality between the knot Floer homologies \hat{HFK}(\Sigma(K), K) and \hat{HFK}(S^3, K).