A characterization of triangulations of closed surfaces

TL;DR

This paper proves that the intersection matrix uniquely determines a finite triangulation of a connected closed surface, providing a complete characterization of such triangulations.

Contribution

It introduces and proves that the intersection matrix fully characterizes finite triangulations of connected closed surfaces, a novel theoretical result.

Findings

Intersection matrix uniquely determines triangulation

Complete characterization of surface triangulations

Theoretical foundation for surface triangulation analysis

Abstract

In this paper we prove that a finite triangulation of a connected closed surface is completely determined by its intersection matrix. The \emph{intersection matrix} of a finite triangulation, $K$, is defined as $M_{K}=(dim(s_{i}\cap s_{j}))_{0\leq i,0\leq j}^{n-1}$, where $K_{2}=\{s_{0}, \ldots s_{n-1}\}$ is a labelling of the triangles of $K$.