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Politopality of 2-orbit maniplexes

Elías Mochán

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, Potočnik 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.

Politopality of 2-orbit maniplexes

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 possible classes of 2-orbit polytopes in rank (dimension) , 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, Potočnik 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 classes of 2-orbit polytopes of rank 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.
Paper Structure (7 sections, 16 theorems, 9 equations, 4 figures)

This paper contains 7 sections, 16 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Voltage graphs and intersection properties
  4. The construction $\hat{2}^\mathcal{M}$
  5. 2-orbit maniplexes
  6. The voltage of the darts of color $n$
  7. Polytopality

Key Result

Lemma 3.1

IntPropYo Let $\mathcal{X}$ be a premaniplex and $\xi:\Pi(\mathcal{X})\to \Gamma$ a voltage assignment such that $X$ has a spanning tree with trivial voltage on all its darts. Then $\mathcal{X}^\xi$ is a maniplex if and only if the following conditions hold:

Figures (4)

  • Figure 1: The set $S_3$ consisting of the shaded 2-faces is not invariant under any non-trivial symmetry of $\mathcal{M}_3$ and it is not contained in the closure of two 0-faces.
  • Figure 2: The voltage $\xi$ we use on $\mathcal{X}=2^{n+1}_I$.
  • Figure 3: The premaniplex $2^{n+1}_{[1,n]}$ with its voltage assignment.
  • Figure 6: The premaniplex $(2^{n+1}_{[1,n-1]})_{\overline{0}}$ with its voltage assignment.

Theorems & Definitions (19)

  • Lemma 3.1
  • Theorem 3.2
  • Corollary 4.1
  • Lemma 4.2
  • Definition 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Definition 5.1
  • Corollary 5.2
  • Corollary 5.3
  • ...and 9 more