The Ozsváth-Szabó spectral sequence and combinatorial link homology

Adam Saltz

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arXiv · 1510.02819

The Ozsváth-Szabó spectral sequence and combinatorial link homology

Adam Saltz

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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 Ozsváth and Szabó. 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 Szabó's combinatorial spectral sequence, which Seed and Szabó conjectured to be equivalent to Ozsváth-\Szabo's. The discrepancy leads us to define a variation of Szabó's theory for links embedded in a thickened annulus. We conclude with a refinement of Seed and Szabó's conjecture for the new theory.

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Adam Saltz. 2017-07-13. The Ozsváth-Szabó spectral sequence and combinatorial link homology. https://arxiv.org/abs/1510.02819

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