Many Triangulated 3-Spheres

Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres

The aim of this paper is to give a survey of the known results concerning centrally symmetric polytopes, spheres, and manifolds. We further enumerate nearly neighborly centrally symmetric spheres and centrally symmetric products of spheres with dihedral or cyclic symmetry on few vertices, and we present an infinite series of vertex-transitive nearly neighborly centrally symmetric 3-spheres.

The generic rigidity of triangulated spheres with blocks and holes

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal $3$-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from …

Polyhedral Embeddings and Immersions of Many Triangulated 2-Manifolds with Few Vertices

This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one …

关于有限多种半径球体的欧几里得空间紧致堆积

On compact packings of Euclidean space with spheres of finitely many sizes

本文研究欧几里得空间中恰好含 n 种半径的紧致球体堆积，引入异微扰集概念，证明固定异微扰集下所有可能半径配置的集合是有限的。