Bounding the Kreuzer‐Skarke Landscape.

We study Calabi‐Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, we show that the number of fine, regular, star triangulations of polytopes in the Kreuzer‐Skarke list is bounded above by 14,111494≈10928. Adapting a result of Anclin on triangulations of lattice polygons, we obtain a bound on the number of triangulations of each 2‐face of each polytope in the list. In this way we prove that the number of topologically inequivalent Calabi‐Yau hypersurfaces arising from the Kreuzer‐Skarke list is bounded above by 10428. We introduce efficient algorithms for constructing representative ensembles of Calabi‐Yau hypersurfaces, including the extremal case h1,1=491, and we study the distributions of topological and physical data therein. Finally, we demonstrate that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope. The authors study Calabi‐Yau threefolds with large Hodge numbers by constructing and counting triangulations of reflexive polytopes. By counting points in the associated secondary polytopes, it is shown that the number of fine, regular, star triangulations of polytopes in the Kreuzer‐Skarke list is bounded above. Adapting a result of Anclin on triangulations of lattice polygons, one obtains a bound on the number of triangulations of each 2‐face of each polytope in the list. In this way it is proven that the number of topologically inequivalent Calabi‐Yau hypersurfaces arising from the Kreuzer‐Skarke list is bounded above. Efficient algorithms for constructing representative ensembles of Calabi‐Yau hypersurfaces are introduced, including the extremal case h1,1=491. Finally, it will be demonstrated that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope. [ABSTRACT FROM AUTHOR]