An algorithm that constructs irreducible triangulations of once-punctured surfaces

TL;DR

This paper presents an algorithm to generate all irreducible triangulations of surfaces with one boundary component, successfully identifying 297 unique types on the once-punctured torus through computational implementation.

Contribution

The paper introduces a novel algorithm for constructing irreducible triangulations of surfaces with a boundary, and provides the first complete enumeration for the once-punctured torus.

Findings

Identified 297 nonisomorphic irreducible triangulations on the once-punctured torus.

Developed an algorithm applicable to any surface with one boundary component.

Successfully implemented the algorithm to enumerate all such triangulations.

Abstract

A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with precisely one boundary component. By implementing the algorithm on computer, we have found a list of 297 nonisomorphic combinatorial types of irreducible triangulations on the once-punctured torus.