Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds

TL;DR

This paper introduces a rapid algorithm for enumerating combinatorial 2- and 3-manifolds, successfully listing all triangulations with up to 12 vertices and specific equivelar maps on certain surfaces.

Contribution

It presents a novel, efficient enumeration method that avoids isomorphisms, enabling comprehensive classification of small triangulated surfaces and 3-manifolds.

Findings

Enumerated all triangulated surfaces with 11 and 12 vertices.

Enumerated all triangulated 3-manifolds with 11 vertices.

Determined all equivelar polyhedral maps on specific non-orientable surfaces.

Abstract

We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar polyhedral maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6.