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.
Suyama,Fano generalized Bott manifolds.Manuscripta Mathematica163(2020), 427–435
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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.