809 research outputs found
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 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
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 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.Comment: Dedicated to the memory of Maximilian Kreuzer. 23 pages, 4 figures, 4
tables, an appendix containing Magma source cod
12, 24 and Beyond
We generalize the well-known "12" and "24" Theorems for reflexive polytopes
of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a
wider category of objects, here called reflexive GKM graphs, that are
associated with certain monotone symplectic manifolds which do not necessarily
admit a toric action. As an application, we provide bounds on the Betti numbers
for certain monotone Hamiltonian spaces which depend on the minimal Chern
number of the manifold.Comment: 39 pages, 4 figure
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