Coxeter polytopes with a unique pair of non-intersecting facets

TL;DR

This paper classifies compact hyperbolic Coxeter polytopes with exactly one pair of non-intersecting facets, showing they exist only up to dimension 6 and in dimension 8, and establishing an upper bound on their facets.

Contribution

It proves a maximum facet count for such polytopes and completes their classification in low dimensions.

Findings

Existence only up to dimension 6 and in dimension 8.

Maximum of d+3 facets in d-dimensional space.

Complete classification of these polytopes.

Abstract

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lann\'er, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.