Lattice Surfaces and smallest triangles

TL;DR

This paper computes the minimal triangle areas on various lattice surfaces, establishing a complete list of surfaces with virtual triangle area greater than 0.05, and improves existing algorithms for such calculations.

Contribution

It provides a comprehensive list of lattice surfaces with large virtual triangle areas and enhances the algorithm for calculating these areas.

Findings

No new lattice surfaces have a virtual triangle area greater than 0.05.

The list of such lattice surfaces is complete.

The method improves upon previous algorithms for area calculation.

Abstract

We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than .05 is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is greater than .05. Our method follows an algorithm described by Smillie and Weiss and improves on it in certain respects.