Reflexive polytopes of higher index and the number 12

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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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
Keywords: Convex lattice polygons; reflexive polytopes
Schools/Departments: University of Nottingham UK Campus > Faculty of Science > School of Mathematical Sciences
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Depositing User: Kasprzyk, Dr Alexander
Date Deposited: 12 Nov 2015 10:28
Last Modified: 14 Sep 2016 06:52

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