On knot Floer homology in double branched covers

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.

Due to an error in section 8, the results on the contact element have been weakened

References in corpus (3)

• The Jones polynomial and graphs on surfaces
• Categorification of the Kauffman bracket skein module of I-bundles over surfaces
• Knot Floer homology in cyclic branched covers

Cited by in corpus (17)

• Khovanov module and the detection of unlinks
• On the equivalence of Legendrian and transverse invariants in knot Floer homology
• Annular Khovanov homology and knotted Schur-Weyl representations
• Transverse invariants and right-veering
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• Relative cellular algebras
• An annular refinement of the transverse element in Khovanov homology
• Chen-Khovanov spectra for tangles
• A rank inequality for the annular Khovanov homology of 2-periodic links
• Bordered Floer homology and the spectral sequence of a branched double cover I
• Anchored foams and annular homology
• Stable homotopy refinement of quantum annular homology
• Knot Floer homology, link Floer homology and link detection
• Categorical lifting of the Jones polynomial: a survey
• Localization in Khovanov homology
• Annular Khovanov homology and augmented links
• A deformation of Asaeda-Przytycki-Sikora homology