Reflexive polytopes of higher index and the number 12

Kasprzyk, Alexander M.; Nill, Benjamin

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.

Citation

Kasprzyk, A. M., & Nill, B. (2012). Reflexive polytopes of higher index and the number 12. Electronic Journal of Combinatorics, 19(3), Article P9

• Journal Article Type: Article
• Publication Date: Jul 19, 2012
• Deposit Date: Nov 12, 2015
• Publicly Available Date: Nov 12, 2015
• Journal: Electronic Journal of Combinatorics
• Electronic ISSN: 1077-8926
• Publisher: Electronic Journal of Combinatorics
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 3
• Article Number: P9
• Keywords: Convex lattice polygons; reflexive polytopes
• Public URL: https://nottingham-repository.worktribe.com/output/710639
• Publisher URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p9
• Related Public URLs: http://www.combinatorics.org/ojs/index.php/eljc/index