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Reflexive polytopes of higher index and the number 12

Kasprzyk, Alexander M.; Nill, Benjamin

Authors

Alexander M. Kasprzyk

Benjamin Nill

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.

Journal Article Type Article
Publication Date Jul 19, 2012
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
Institution Citation Kasprzyk, A. M., & Nill, B. (2012). Reflexive polytopes of higher index and the number 12. Electronic Journal of Combinatorics, 19(3),
Keywords Convex lattice polygons; reflexive polytopes
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
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf




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