Systolic geometry of translation surfaces

TL;DR

This paper explores the systolic geometry of translation surfaces, analyzing their properties, relationships with saddle connection graphs, and computing maximal systolic ratios for origamis in a specific stratum.

Contribution

It introduces an algorithm to compute systolic ratios of origamis and provides extensive computations supporting a conjecture about maximal ratios.

Findings

Developed an algorithm for systolic ratio computation

Computed maximal systolic ratios for origamis with up to 67 squares

Supported a conjecture on maximal systolic ratios in the stratum ,1

Abstract

In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic ratio of origamis in the stratum $\mathcal{H}(1,1)$. We compute the maximal systolic ratio of all origamis in $\mathcal{H}(1,1)$ with up to 67 squares. These computations support a conjecture of Judge and Parlier about the maximal systolic ratio in $\mathcal{H}(1,1)$.