Reflexive polytopes of higher index and the number 12 (Q456328)

Summary: 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.