The Tri-Pants Graph of the Twice-Punctured Torus

TL;DR

This paper studies the tri-pants graph of the twice-punctured torus, proving it is connected and has infinite diameter by relating it to the Farey complex's dual.

Contribution

It establishes the connectivity and infinite diameter of the tri-pants graph, linking it to the dual of the Farey complex, a novel structural insight.

Findings

The tri-pants graph is connected.

The tri-pants graph has infinite diameter.

A relationship with the dual of the Farey complex is established.

Abstract

We investigate the structure of the tri-pants graph, a simplicial graph introduced by Maloni and Palesi, whose vertices correspond to particular collections of homotopy classes of simple closed curves of the twice-punctured torus, called tri-pants, and whose edges connect two vertices whenever the corresponding pants differ by suitable elementary moves. In particular, by examining the relationship between the tri-pants graph and the dual of the Farey complex, we prove that the tri-pants graph is connected and has infinite diameter.