Politopality of 2-orbit maniplexes

TL;DR

This paper investigates 2-orbit maniplexes, demonstrating that some are indeed polytopes and establishing the existence of multiple classes of such polytopes across various ranks, advancing understanding of their symmetry properties.

Contribution

The paper proves the existence of 2-orbit polytopes in all classes where only two types of reflections are forbidden, expanding known classes of symmetric polytopes.

Findings

Existence of 2-orbit polytopes in all classes with two forbidden reflection types

At least n^2 - n + 1 classes of 2-orbit polytopes are non-empty for rank n

Constructed maniplexes satisfy all but possibly one property to be polytopes

Abstract

Abstract polytopes are a combinatorial generalization of convex and skeletal polytopes. Counting how many flag orbits a polytope has under its automorphism group is a way of measuring how symmetric it is. Polytopes with one flag orbit are called regular and are very well known. Polytopes with two flag orbits (called 2-orbit polytopes) are, however, way more elusive. There are $2^n-1$ possible classes of 2-orbit polytopes in rank (dimension) $n$, but for most of those classes, determining whether or not they are empty is still an open problem. In 2019, in their article An existence result on two-orbit maniplexes, Pellicer, Poto\v{c}nik and Toledo constructed 2-orbit maniplexes (objects that generalize abstract polytopes and maps) in all these classes, but the question of whether or not they are also polytopes remained open. In this paper we use the results of a previous paper by the author and Hubard to show that some of these 2-orbit maniplexes are, in fact, polytopes. In particular we prove that there are 2-orbit polytopes in all the classes where exactly two kinds of reflections are forbidden. We use this to show that there are at least $n^2-n+1$ classes of 2-orbit polytopes of rank $n$ that are not empty. We also show that the maniplexes constructed with this method in the remaining classes satisfy all but (possibly) one of the properties necessary to be polytopes, therefore we get closer to proving that there are 2-orbit polytopes in all the classes.