The Ozsv\'ath-Szab\'o spectral sequence and combinatorial link homology

TL;DR

This paper constructs a simplified spectral sequence linking Khovanov homology and Heegaard Floer homology, reveals a discrepancy with Szabó's combinatorial sequence, and introduces a new link invariant in an annular setting.

Contribution

It introduces an isomorphic spectral sequence with minimal complexity and defines a new link invariant in a thickened annulus, challenging previous conjectures.

Findings

Constructed a simplified spectral sequence with the same rank as the Khovanov chain group.

Proved the spectral sequence is not isomorphic to Szabó's combinatorial sequence.

Proposed a refined conjecture for the new annular link theory.

Abstract

The Khovanov homology of a link in $S^3$ and the Heegaard Floer homology of its branched double cover are related through a spectral sequence constructed by Ozsv\'ath and Szab\'o. This spectral sequence has topological applications but is difficult to compute. We build an isomorphic spectral sequence whose underlying filtered complex is as simple as possible: it has the same rank as the Khovanov chain group. We show that this spectral sequence is not isomorphic to Szab\'o's combinatorial spectral sequence, which Seed and Szab\'o conjectured to be equivalent to Ozsv\'ath-\Szabo's. The discrepancy leads us to define a variation of Szab\'o's theory for links embedded in a thickened annulus. We conclude with a refinement of Seed and Szab\'o's conjecture for the new theory.