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 ${\mathbf{CF}}^{-}$. Like with the hat theory, their minus invariants provide a recipe to compute knot invariants associated to satellite knots. We combinatorially construct the weighted $A_{\infty }$-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 $A_{\infty }$-modules provides a combinatorial proof of the $A_{\infty }$ 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.

55 pages, 32 figures