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

Given a link in the three-sphere, Z. Szabó 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.