On knot Floer homology in double branched covers

TL;DR

This paper establishes a spectral sequence relating the knot Floer homology of a lifted knot in a double branched cover to a specific type of Khovanov homology, extending known results and simplifying cases for alternating links.

Contribution

It introduces a spectral sequence connecting knot Floer homology in double branched covers to a particular Khovanov homology, expanding the understanding of these invariants.

Findings

Knot Floer homology can be computed from a spectral sequence starting at a specific Khovanov homology.

Extended results about this type of Khovanov homology are provided.

Simplified explanation for the case when the link is alternating is given.

Abstract

Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. This paper shows that the knot Floer homology of this lift, with mod 2 coefficients, can be computed from a spectral sequence starting at a type of Khovanov homology already described by Asaeda, Przytycki, and Sikora. We extend the known results about this type of Khovanov homology, and use it to provide a very simple explanation of the case when L is alternating for the obvious projection.