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Existence of Circle Packings on Translation Surfaces

Anton Levonian

Abstract

A translation surface is a surface formed by identifying edges of a collection of polygons in the complex plane that are parallel and of equal length using only translations. We determined that the same circle packing can be realized on varying translation surfaces in a certain stratum. We also determined possible complexities of contacts graphs and provide a bound on this complexity in some low-genus strata. Finally, we established the possibility of certain contacts graphs' complexities in strata with genus greater than $2$.

Existence of Circle Packings on Translation Surfaces

Abstract

A translation surface is a surface formed by identifying edges of a collection of polygons in the complex plane that are parallel and of equal length using only translations. We determined that the same circle packing can be realized on varying translation surfaces in a certain stratum. We also determined possible complexities of contacts graphs and provide a bound on this complexity in some low-genus strata. Finally, we established the possibility of certain contacts graphs' complexities in strata with genus greater than .
Paper Structure (9 sections, 10 theorems, 2 equations, 13 figures)

This paper contains 9 sections, 10 theorems, 2 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Definition and Basic Properties of Translation Surfaces
  4. Genus and Stratum of Translation Surfaces
  5. Circle Packings
  6. Existence of Equivalent Circle Packings on Distinct Surfaces
  7. Simplest Achievable Contacts Graphs
  8. Research Directions
  9. Acknowledgements

Key Result

Theorem 2.1

(Gauss-Bonnet Theorem). Let $X$ be a translation surface with $k$ singular points $v_i$ each of order $\delta(v_i)$, and let $\chi(X)$ be the Euler characteristic of $X.$ Then the following is true:

Figures (13)

  • Figure 1: Genus $2$ 3-squared surface.
  • Figure 2: Translation surface in $\mathcal{H}(1,1).$
  • Figure 3: The 2 polygons above are identified to the same $\mathcal{H}(2)$ surface.
  • Figure 4: A circle packing $C_3$ on a surface in the complex plane.
  • Figure 5: Contacts graph for $C_3.$
  • ...and 8 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Theorem 2.1
  • Example 2.8
  • Definition 2.9
  • ...and 29 more