On the Uniqueness of Certain Types of Circle Packings on Translation Surfaces

Consider a collection of finitely many polygons in $\mathbb C$, such that for each side of each polygon, there exists another side of some polygon in the collection (possibly the same) that is parallel and of equal length. A translation surface is the surface formed by identifying these opposite sides with one another. The $H(1, 1)$ stratum consists of genus two translation surfaces with two singularities of order one. A circle packing corresponding to a graph $G$ is a configuration of disjoint disks such that each vertex of $G$ corresponds to a circle, two disks are externally tangent if and only if their vertices are connected by an edge in $G$, and $G$ is a triangulation of the surface. It is proven that for certain circle packings on $H(1, 1)$ translation surfaces, there are only a finite number of ways the packing can vary without changing the contacts graph, if two disks along the slit are fixed in place. These variations can be explicitly characterized using a new concept known as splitting bigons. Finally, the uniqueness theorem is generalized to a specific type of translation surfaces with arbitrary genus $g \geq 2$.