Bordered Heegaard Floer modules for satellite operations using planar graphs

TL;DR

This paper develops a combinatorial approach to bordered Heegaard Floer modules for satellite knots, specifically for the (p,1)-cable, using decorated planar graphs to establish new computational and structural results.

Contribution

It introduces a combinatorial construction of weighted $A_$-modules for satellite operations, providing a new proof of their $A_$-structure relations and a uniqueness property.

Findings

Constructed weighted $A_$-modules for (p,1)-cable satellites.

Provided a combinatorial proof of $A_$-structure relations.

Proved a uniqueness property for the modules.

Abstract

Lipshitz, Ozsv\'ath, 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.