[Paper Review] A Classification of Non-Compact Coxeter Polytopes with $n+3$ Facets and One Non-Simple Vertex

Mike Roberts|arXiv (Cornell University)|Nov 26, 2015

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper provides a complete classification of non-compact, non-pyramidal Coxeter polytopes in hyperbolic $n$-space with $n+3$ facets and exactly one non-simple vertex—i.e., one vertex formed by $n+1$ hyperplanes. Using Gale diagrams, Gram matrices, and Vinberg’s signature theorem, the author identifies all such polytopes up to isometry, showing no examples exist in dimensions 11 and above.

ABSTRACT

In this paper we state a full classification for Coxeter polytopes in $\mathbb{H}^{n}$ with $n+3$ facets which are non-compact and have precisely one non-simple vertex.

Motivation & Objective

• To classify non-compact, non-pyramidal Coxeter polytopes in $\mathbb{H}^n$ with $n+3$ facets and exactly one non-simple vertex.
• To address the gap in classification for non-compact Coxeter polytopes with $n+3$ facets, particularly in dimensions 4 to 16.
• To extend prior work on compact and pyramidal cases by focusing on non-pyramidal, non-compact polytopes with a single non-simple vertex.
• To provide a systematic method using Gale diagrams, Gram matrices, and combinatorial constraints to enumerate all such polytopes.

Proposed method

• Construct Gale diagrams for $n+3$-faceted polytopes, representing them as labeled points on a circle with integer weights summing to $n+3$.
• Apply Tumarkin’s lemmas to restrict possible Coxeter subdiagrams, especially those that must be parabolic, quasi-Lannér, or Lannér.
• Use the Gale diagram to derive combinatorial constraints, such as non-zero labels on opposite nodes and non-zero sums in open halfspaces.
• Construct the Gram matrix from the Coxeter diagram, ensuring determinant zero for $ (n+3) \times (n+3) $ and all $ (n+2) \times (n+2) $ minors.
• Verify the signature of the Gram matrix using Vinberg’s Theorem 11: only matrices of signature $ (n,1) $ correspond to valid hyperbolic polytopes.
• Systematically enumerate all valid configurations up to congruence and isometry, focusing on those with exactly one pair of opposite nodes of weight 1.

Experimental results

Research questions

• RQ1 Which non-compact, non-pyramidal Coxeter polytopes in $\mathbb{H}^n$ with $n+3$ facets have exactly one non-simple vertex?
• RQ2 What combinatorial and geometric constraints arise from the presence of a single non-simple vertex in such polytopes?
• RQ3 For which dimensions $n$ do such polytopes exist, and are there any upper bounds on $n$?
• RQ4 How can Gale diagrams and Gram matrices be used to systematically classify these polytopes?
• RQ5 What is the complete list of such polytopes up to isometry, and what is their structural form?

Key findings

• A full classification of non-compact, non-pyramidal Coxeter polytopes with $n+3$ facets and exactly one non-simple vertex is provided in Appendix A.
• No such polytopes exist in dimensions 11 and above, establishing an upper bound of $n=10$ for their existence.
• All valid polytopes are constructed via Gale diagrams with exactly one pair of opposite nodes of weight 1, and all other nodes have weight 0 or 1.
• The method successfully identifies all such polytopes by combining combinatorial constraints from Tumarkin’s lemmas with signature verification via Vinberg’s theorem.
• The classification confirms that the only possible configurations are those where the non-simple vertex arises from a single pair of opposite hyperplanes with weight 1, and the rest of the structure satisfies parabolic and quasi-Lannér conditions.
• The results complete a key missing piece in the classification of $n+3$-faceted Coxeter polytopes, particularly in the non-compact, non-pyramidal, and non-simple case.