More indecomposable polyhedra

TL;DR

This paper introduces combinatorial techniques to classify polytopes based on Minkowski decomposability, providing new insights into the structure of indecomposable polyhedra across various dimensions.

Contribution

It develops combinatorial methods to determine polytope indecomposability, surpassing traditional face-based approaches, and completes classifications for specific edge counts.

Findings

In dimensions other than 2, only one polytope with ≤ d^2 + d/2 edges is decomposable.

Complete classification of 3D polyhedra with 15 or fewer edges.

New criteria for polytope indecomposability based on skeleton properties.

Abstract

We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension $d\neq 2$, we show that of all the polytopes with $d^2+\frac{d}{2}$ or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.