Number of Systoles of Once-Punctured Torus and Four-Punctured Sphere

TL;DR

This paper calculates the number of shortest and second shortest simple closed geodesics, known as systoles and 2-systoles, on hyperbolic surfaces of specific topologies, namely once-punctured torus and four-punctured sphere.

Contribution

It provides explicit counts of systoles and 2-systoles for these hyperbolic surfaces, advancing understanding of their geometric properties.

Findings

Number of systoles on once-punctured torus computed

Number of systoles on four-punctured sphere computed

Explicit enumeration of 2-systoles provided

Abstract

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.