Koszul dual $\mathcal{A}_{\infty}$ algebras for star-shaped diagrams -- Part 1

By slicing the Heegaard diagram for a given $3$-manifold in a particular way, it is possible to construct $\mathcal{A}_{\infty}$-bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. Here, we use the graphical calculus for $\mathcal{A}_{\infty}$-structures introduced by Lipshitz, Ozsv\'ath, and Thurston, to construct Koszul dual weighted $\mathcal{A}_{\infty}$-algebras $\mathcal{A}$ and $\mathcal{B}$, and dualizing bimodules for a particular star-shaped class of slice. The duality result is then proved in the sequel.