By slicing a Heegaard diagram for a knot K in $S^3$, it is possible to retrieve the knot Floer homology of K as a tensor product of bimodules over an $\A_{\infty}$ algebra corresponding to the slice. The first step in this process is to assign an $\mathcal{A}_{\infty}$ algebra to this slice, which can also be written as a planar graph. In this talk, we construct a pair of Koszul dual $\mathcal{A}_{\infty}$ algebras corresponding to planar graphs arising as slices of these Heegaard diagrams, and discuss how to verify the Koszul duality relation.