Tight Chiral Polytopes

TL;DR

This paper investigates the existence and classification of tight chiral polytopes, proving non-existence in certain dimensions and describing specific families in others, advancing understanding of their structural properties.

Contribution

It proves the non-existence of tight chiral 5-polytopes, classifies 11 families of tight chiral 4-polytopes, and shows all such 4-polytopes cover these families.

Findings

No tight chiral 5-polytopes exist.

11 families of tight chiral 4-polytopes are described.

Every tight chiral 4-polytope covers a polytope from these families.

Abstract

A chiral polytope with Schl\"{a}fli symbol $\{p_1, \ldots, p_{n-1}\}$ has at least $2p_1 \cdots p_{n-1}$ flags, and it is called \emph{tight} if the number of flags meets this lower bound. The Schl\"{a}fli symbols of tight chiral polyhedra were classified in an earlier paper, and another paper proved that there are no tight chiral $n$-polytopes with $n \geq 6$. Here we prove that there are no tight chiral $5$-polytopes, describe 11 families of tight chiral $4$-polytopes, and show that every tight chiral $4$-polytope covers a polytope from one of those families.