Combinatorial 3-manifolds with 10 vertices

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.Comment: 9 pages, minor revisions, to appear in Beitr. Algebra Geo

There are 174 Subdivisions of the Hexahedron into Tetrahedra

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian

Combinatorial 3-manifolds with few vertices

AbstractIt is proved that every combinatorial 3-manifold with at most eight vertices is a combinatorial sphere