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The diffeomorphism class of simply-connected smooth Calabi-Yau threefoldswith torsion-free cohomology is determined via certain basic topologicalinvariants: the Hodge numbers, the triple intersection form, and the secondChern class. In the present paper, we shed some light on this classification byplacing bounds on the number of diffeomorphism classes present in the set ofsmooth Calabi-Yau threefolds constructed from the Kreuzer-Skarke list ofreflexive polytopes up to Picard number six. The main difficulty arises fromthe comparison of triple intersection numbers and divisor integrals of thesecond Chern class up to basis transformations. By using certainbasis-independent invariants, some of which appear here for the first time, weare able to place lower bounds on the number of classes. Upper bounds areobtained by explicitly identifying basis transformations, using constraintsrelated to the index of line bundles. Extrapolating our results, we conjecturethat the favourable entries of the Kreuzer-Skarke list of reflexive polytopesleads to some $10^{400}$ diffeomorphically distinct Calabi-Yau threefolds.

Enumerating Calabi-Yau Manifolds: Placing bounds on the number of diffeomorphism classes in the Kreuzer-Skarke list