Combinatorial 3-manifolds with 10 vertices

TL;DR

This paper provides a complete enumeration of all combinatorial 3-manifolds with 10 vertices, including spheres and sphere products, revealing their shellability and non-shellability properties.

Contribution

It offers the first exhaustive classification of 3-manifolds with 10 vertices, including triangulations of spheres and sphere products, and analyzes their shellability.

Findings

247,882 triangulated 3-spheres with 10 vertices

518 vertex-minimal triangulations of S^2×S^1

615 triangulations of the twisted sphere product S^2×_S^1

Abstract

We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product $S^2\times S^1$ and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertex-minimal non-shellable 3-balls with 9 vertices.