On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology

TL;DR

This paper proves the naturality of a spectral sequence connecting Khovanov homology and Heegaard Floer homology under elementary TQFT operations, broadening understanding of their relationship.

Contribution

It demonstrates the naturality of the spectral sequence using a generalized surface decomposition theorem, linking Khovanov and Heegaard Floer theories more robustly.

Findings

Proves naturality of the spectral sequence under TQFT operations

Generalizes Juhasz's surface decomposition theorem

Establishes a broader connection between Khovanov and Heegaard Floer homologies

Abstract

Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz. In the present work we prove the naturality of the spectral sequence under certain elementary TQFT operations, using a generalization of Juhasz's surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.