Mixing Chiral Polytopes

TL;DR

This paper introduces a mixing construction for abstract polytopes that helps generate new chiral polytopes and provides criteria to determine when the mix of two polytopes results in a chiral structure.

Contribution

It presents a novel mixing method for polytopes and uses the chirality group to establish simple criteria for chiral outcomes.

Findings

New mixing construction for polytopes

Criteria for chiral polytope formation

Enhanced ability to generate chiral examples

Abstract

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.