Decomposition and Enumeration of Triangulated Surfaces

TL;DR

This paper develops a theoretical framework for triangulations of surfaces, introducing concepts like roots and genus-surfaces, and applies it to enumerate all triangulations of closed surfaces with up to 11 vertices.

Contribution

It presents a new theory on roots and decompositions of surface triangulations and an algorithm for listing all such triangulations with a limited number of vertices.

Findings

Enumerated all triangulations of closed surfaces with up to 11 vertices.

Established theoretical restrictions on genus-surfaces for small vertex counts.

Developed an algorithm for systematic enumeration of surface triangulations.

Abstract

We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory for the case where the number of vertices is at most 11 and we get theoretical restrictions on genus-surfaces allowing us to get the list of triangulations of closed surfaces with at most 11 vertices.