Gorenstein toric contact manifolds show flexibility where multiple toric diagrams share Ehrhart polynomials matching the cross-polytope, but rigidity where the small cross-polytope is the unique diagram for its polynomial, determining the manifold from contact Betti numbers.
Delzant,Hamiltoniens périodiques et images convexes de l’application moment.Bulletin de la Société Mathématique de France116(1988), 315–339
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Contact flexibility and rigidity for toric Gorenstein prequantizations and Ehrhart theory of toric diagrams
Gorenstein toric contact manifolds show flexibility where multiple toric diagrams share Ehrhart polynomials matching the cross-polytope, but rigidity where the small cross-polytope is the unique diagram for its polynomial, determining the manifold from contact Betti numbers.