12, 24 and Beyond

TL;DR

This paper extends classical theorems on reflexive polytopes to higher dimensions and broader categories, linking combinatorial geometry with symplectic topology and providing bounds on topological invariants.

Contribution

It generalizes the 12 and 24 Theorems to all smooth reflexive polytopes and introduces reflexive GKM graphs applicable to non-toric symplectic manifolds.

Findings

Bounds on Betti numbers depending on minimal Chern number

Extension of classical theorems to higher dimensions

Application to monotone Hamiltonian spaces

Abstract

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.