Reflexive polytopes of higher index and the number 12
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...
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| Format: | Article |
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Electronic Journal of Combinatorics
2012
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| Online Access: | https://eprints.nottingham.ac.uk/30721/ |
| _version_ | 1848794044217425920 |
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| author | Kasprzyk, Alexander M. Nill, Benjamin |
| author_facet | Kasprzyk, Alexander M. Nill, Benjamin |
| author_sort | Kasprzyk, Alexander M. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:09:56Z |
| format | Article |
| id | nottingham-30721 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:09:56Z |
| publishDate | 2012 |
| publisher | Electronic Journal of Combinatorics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-307212020-05-04T16:33:32Z https://eprints.nottingham.ac.uk/30721/ Reflexive polytopes of higher index and the number 12 Kasprzyk, Alexander M. Nill, Benjamin 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. Electronic Journal of Combinatorics 2012-07-19 Article PeerReviewed 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 Convex lattice polygons; reflexive polytopes http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p9 |
| spellingShingle | Convex lattice polygons; reflexive polytopes Kasprzyk, Alexander M. Nill, Benjamin Reflexive polytopes of higher index and the number 12 |
| title | Reflexive polytopes of higher index and the number 12 |
| title_full | Reflexive polytopes of higher index and the number 12 |
| title_fullStr | Reflexive polytopes of higher index and the number 12 |
| title_full_unstemmed | Reflexive polytopes of higher index and the number 12 |
| title_short | Reflexive polytopes of higher index and the number 12 |
| title_sort | reflexive polytopes of higher index and the number 12 |
| topic | Convex lattice polygons; reflexive polytopes |
| url | https://eprints.nottingham.ac.uk/30721/ https://eprints.nottingham.ac.uk/30721/ |