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

TL;DR

This paper proves the finiteness and explicit characterization of certain circle packings on genus two translation surfaces, using the novel concept of splitting bigons, and extends the results to higher genus surfaces.

Contribution

It introduces the concept of splitting bigons to analyze circle packings on translation surfaces and proves a finiteness and uniqueness result for these packings, generalizing to arbitrary genus.

Findings

Finite number of circle packings with fixed contact graph on genus two surfaces.

Explicit characterization of packings via splitting bigons.

Generalization of the uniqueness theorem to higher genus surfaces.

Abstract

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 $g \geq 2$.