Bordered Floer homology and the spectral sequence of a branched double cover I

TL;DR

This paper develops a combinatorial approach to compute a spectral sequence linking Khovanov homology and Heegaard Floer homology of branched double covers of links using bordered Floer homology techniques.

Contribution

It provides the first explicit calculation of filtered bimodules associated to Dehn twists, enabling a combinatorial construction of the spectral sequence.

Findings

Established a combinatorial spectral sequence from Khovanov to Heegaard Floer homology.

Calculated filtered bimodules for Dehn twists explicitly.

Laid groundwork for verifying the spectral sequence's equivalence in the sequel.

Abstract

Given a link in the three-sphere, Z. Szab\'o and the second author constructed a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double-cover. The aim of this paper and its sequel is to explicitly calculate this spectral sequence, using bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of filtered bimodules associated to Dehn twists and a pairing theorem for polygons. In this paper we give the first ingredient, and so obtain a combinatorial spectral sequence from Khovanov homology to Heegaard Floer homology; in the sequel we show that this spectral sequence agrees with the previously known one.