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.
@article{10_37236_2366,
author = {Alexander M. Kasprzyk and Benjamin Nill},
title = {Reflexive polytopes of higher index and the number 12},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {3},
doi = {10.37236/2366},
zbl = {1258.52008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2366/}
}
TY - JOUR
AU - Alexander M. Kasprzyk
AU - Benjamin Nill
TI - Reflexive polytopes of higher index and the number 12
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/2366/
DO - 10.37236/2366
ID - 10_37236_2366
ER -
%0 Journal Article
%A Alexander M. Kasprzyk
%A Benjamin Nill
%T Reflexive polytopes of higher index and the number 12
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/2366/
%R 10.37236/2366
%F 10_37236_2366
Alexander M. Kasprzyk; Benjamin Nill. Reflexive polytopes of higher index and the number 12. The electronic journal of combinatorics, Tome 19 (2012) no. 3. doi: 10.37236/2366
