Koszul dual ${\mathcal{A}}_{\mathrm{\infty }}$ 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 ${\mathcal{A}}_{\mathrm{\infty }}$-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 ${\mathcal{A}}_{\mathrm{\infty }}$-structures introduced by Lipshitz, Ozsváth, and Thurston, to construct Koszul dual weighted ${\mathcal{A}}_{\mathrm{\infty }}$-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.