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 $\binom{14,111}{494} \approx 10^{928}$. 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 $10^{428}$. We introduce efficient algorithms for constructing representative ensembles of Calabi-Yau hypersurfaces, including the extremal case $h^{1,1}=491$, 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.