Holomorphic polygons and the bordered Heegaard Floer homology of link complements

Abstract

We describe the construction of an ${\mathcal{A}}_{\infty }$ multi-module in terms of counts of holomorphic polygons in a series Heegaard multi-diagrams. We show that this is quasi-isomorphic to the type-A bordered-sutured invariant of a link complement with a view to calculating, in the sequel, these invariants in terms of the link Floer homology of the corresponding link.

70 pages, 19 figures. Comments and queries very welcome