Near Classification of Compact Hyperbolic Coxeter $d$-Polytopes with $d+4$ Facets and Related Dimension Bounds

Abstract

We complete the classification of compact hyperbolic Coxeter d-polytopes with d + 4 facets for d = 4 and 5. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is d = 6. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions 4 and 5, there are 348 and 50 polytopes, respectively, yielding many new examples for further study.We furthermore provide new upper bounds on the dimension d of compact hyperbolic Coxeter polytopes with d + k facets for k ≤ 10. It was shown by Vinberg in 1985 that for any k, we have d ≤ 29, and no better bounds have previously been published for k ≥ 5. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.

Lannér diagrams and combinatorial properties of compact hyperbolic Coxeter polytopes

“…All cubes were classified by Jacquemet and Tschantz [JT18]. Very recently and independently, Ma & Zheng [MZ22] and Burcroff [Bur22] listed all compact Coxeter polytopes in H d with d + 4 facets for d = 4, 5.…”