Reflexive polytopes of higher index and the number 12

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Kasprzyk, Alexander M. and Nill, Benjamin (2012) Reflexive polytopes of higher index and the number 12. Electronic Journal of Combinatorics, 19 (3). P9/1-P9/18. ISSN 1077-8926

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Official URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p9

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.

Item Type:

Article

RIS ID:

https://nottingham-repository.worktribe.com/output/710639

Keywords:

Convex lattice polygons; reflexive polytopes

Schools/Departments:

University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences

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