Mixing chiral polytopes
J Algebr Comb
Mixing chiral polytopes
Gabe Cunningham 0
0 G. Cunningham ( ) Department of Mathematics, Northeastern University , Boston, MA 02115 , USA
An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A “mixing” construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.
Abstract regular polytope; Chiral polytope; Chiral maps; Chirality group
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1 Introduction
The study of abstract polytopes is a growing field, uniting combinatorics with
geometry and group theory. At the forefront are the (abstract) regular polytopes, including
the regular convex polytopes, regular tessellations of space-forms, and many new
combinatorial structures with maximal symmetry. Recently, the study of chiral
polytopes has flourished. Chiral polytopes are “half-regular”; the action of the
automorphism group on the flags has two orbits, and adjacent flags belong to distinct orbits.
A chiral polytope occurs in two enantiomorphic (mirror-image) forms, and though
these forms are isomorphic as polytopes, the particular orientation chosen is usually
relevant.
Chiral maps (also called irreflexible maps) have been studied for some time (see
[
8
]), and the study of chiral maps and hypermaps continues to yield interesting
developments (for example, see [
1
]). However, it was only with the introduction of abstract
polytopes that the notion of chirality was defined for polytopes in ranks 4 and higher
[
15
].
Examples of chiral polytopes have been hard to find. A few families have been
found in ranks 3 and 4, but only a handful of examples are known in ranks 5 and
higher. There are several impediments to constructing new examples. Foremost is
that the (n − 2)-faces and the co-faces at the edges of a chiral polytope must be
regular. Therefore, it is not possible to repeatedly extend a chiral polytope to higher
and higher ranks—we need genuinely new examples in each rank. So far, nobody
has found a “nice” family of chiral polytopes of arbitrary rank. Indeed, it was only
recently that Pellicer demonstrated conclusively for the first time that there are chiral
polytopes in every rank [
14
].
Since it is so difficult to extend chiral polytopes to higher ranks, we need another
tactic for building new chiral polytopes. In [
4
], the authors adapted the mixing
technique used in [
12
] to chiral polytopes. Two main difficulties arise. The first is that the
mix of two polytopes is not necessarily polytopal: specifically, the resulting group
does not necessarily have the required intersection property. However, under some
fairly mild conditions on the polytopes being mixed, we can ensure that the mix is, in
fact, polytopal. The second difficulty is that we need a way of determining whether
the mix is chiral or not, which can be difficult to do directly. We would like to have
simple combinatorial criteria which will tell us when the mix is chiral.
By using the idea of the chirality group of a polytope, introduced in [
3
] for
hypermaps and in [
4
] for polytopes, we are able to outline such criteria. We then apply these
results to construct new examples of chiral 5-polytopes. We also see how to construct
infinitely many chiral n-polytopes given a single chiral n-polytope satisfying some
mild conditions.
We start by giving some background information on regular and chiral abstract
polytopes in Sect. 2. In Sect. 3, we introduce the mixing operation for chiral and
directly regular polytopes, and we give a few results (including one new one) for
when the mix of two polytopes is again a polytope. In Sect. 4, we define the chirality
group of a polytope, which we then use to give several simple criteria for when the
mix of two or more polytopes is chiral. Finally, in Sect. 5 we highlight the main
results with several constructions that build new chiral polytopes.
2 Polytopes
General background information on abstract polytopes can be found in [12,
Chaps. 2, 3], and information on chiral polytopes specifically can be found in [
10, 15
].
Here we review the concepts essential for this paper.
2.1 Definition of a polytope
Let P be a ranked partially ordered set whose elements will be called faces. The
faces of P will range in rank from −1 to n, and a face of rank j is called a j -face.
The 0-faces, 1-faces, and (n − 1)-faces are also called vertices, edges, and facets,
respectively. A flag of P is a maximal chain. We say that two flags are adjacent (j
adjacent) if they differ in exactly one face (their j -face, respectively). If F and G are
faces of P such that F < G, then the section G/F consists of those faces H such that
F ≤ H ≤ G.
We say that P is an (abstract) polytope of rank n, also called an n-polytope, if it
satisfies t (...truncated)