Bordered Heegaard Floer Modules for Satellite Operations using Planar Graphs

Abstract

Lipshitz, Ozsváth, and Thurston extend the theory of bordered Heegaard Floer homology to compute CF minus. Like with the hat theory, their minus invariants provide a recipe to compute knot invariants associated to satellite knots. We combinatorially construct the weighted A infinity-modules associated to the (p, 1)-cable. The operations on these modules count certain classes of inductively constructed decorated planar graphs. This description of the weighted A infinity-modules provides a combinatorial proof of the A infinity structure relations for the modules. We further prove a uniqueness property for the modules we construct: any weighted extensions of the unweighted U = 0 modules have isomorphic associated type D modules.