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Reflexive polytopes of higher index and the number 12

Kasprzyk, Alexander M.; Nill, Benjamin

Authors

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.

Citation

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

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 http://eprints.nottingham.ac.uk/id/eprint/30721
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.nottingham.ac.uk/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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