Reflexive polytopes of higher index and the number 12

TL;DR

This paper introduces a generalization of reflexive polytopes called l-reflexive polytopes, classifies them in two dimensions, and demonstrates that all such polygons satisfy the 'number 12' property, extending known results to an infinite class.

Contribution

It defines and classifies l-reflexive polygons, proves they satisfy the 'number 12' property, and explores higher-dimensional cases and open questions.

Findings

Classified all l-reflexive polygons up to index 200.

Proved all l-reflexive polygons satisfy the 'number 12' property.

Extended the 'number 12' property to non-convex and self-intersecting polygons.

Abstract

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexive polygons up to index 200. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number 12" property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances. The number 12 property also holds more generally for l-reflexive non-convex or self-intersecting polygonal loops. We conclude by discussing higher-dimensional examples and open questions.