Several examples of neigbourly polyhedra in co-dimension 4

Abstract

In the article, a series of neigbourly polyhedra is constructed. They have $N=2d+4$ vertices and are embedded in ${\mathbb{R}}^{2d}$. Their (affine) Gale diagrams in ${\mathbb{R}}^2$ have $d+3$ black points that form a convex polygon. These Gale diagams can be enumerated using 3-trees (trees with some additional structure). Given $d$ and $m$, each of the constructed polyhedra in ${\mathbb{R}}^{2d}$ has a fixed number of faces of dimension $m$ that contain a vertex $A$. (This number depends on $d$ and $m$ does not depend on the polyhedron and the vertex $A$).

arXiv commentIn russian, 25 pages, 16 figures