Counting Calabi-Yau Threefolds

We enumerate topologically-inequivalent compact Calabi-Yau threefold hypersurfaces. By computing arithmetic and algebraic invariants and the Gopakumar-Vafa invariants of curves, we prove that the number of distinct simply connected Calabi-Yau threefold hypersurfaces resulting from triangulations of four-dimensional reflexive polytopes is 4, 27, 183, 1,184 and 8,036 at $h^{1,1}$ = 1, 2, 3, 4, and 5, respectively. We also establish that there are ten equivalence classes of Wall data of non-simply connected Calabi-Yau threefolds from the Kreuzer-Skarke list. Finally, we give a provisional count of threefolds obtained by enumerating non-toric flops at $h^{1,1} =2$.