Irreducible triangulations of the once-punctured torus

TL;DR

This paper provides a complete classification of irreducible triangulations of the once-punctured torus, identifying exactly 297 non-isomorphic structures through manual enumeration.

Contribution

It offers the first comprehensive list of all irreducible triangulations for the once-punctured torus, a fundamental topological surface.

Findings

297 non-isomorphic irreducible triangulations identified

Complete manual enumeration of structures achieved

Provides foundational data for topological surface studies

Abstract

A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations.