Generating irreducible triangulations of surfaces

TL;DR

This paper presents an algorithm to generate all triangulations of a fixed surface from its irreducible triangulations, expanding the known sets for several surfaces and providing a computational method for their enumeration.

Contribution

The paper introduces a new algorithm for generating irreducible triangulations of surfaces using existing triangulations of other surfaces, and implements it to expand known sets for multiple surfaces.

Findings

Complete sets of irreducible triangulations for S_2, N_3, N_4 are now known.

The algorithm terminates successfully for multiple surfaces.

Cardinalities of the generated sets are provided.

Abstract

Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined for the surfaces S_0, S_1, N_1,and N_2. An algorithm is presented for generating the irreducible triangulations of a fixed surface using triangulations of other surfaces. This algorithm has been implemented as a computer program which terminates for S_1, S_2, N_1, N_2, N_3, and N_4. Thus the complete sets irreducible triangulations are now also known for S_2, N_3, and N_4, with respective cardinalities 396784, 9708, and 6297982.