# Today's world record on the Betti numbers of Calabi-Yau three-folds.

What are largest betti numbers $b_2$ and $b_3$ of three-dimensional Calabi-Yau manifolds that are discovered for today?

Is there some nice reference?

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Since mirror symmetry exchanges the Hodge numbers $h^{1,1} = b_2$ and $h^{2,1} = \frac{1}{2}(b_3 - 1)$, it is perhaps more natural (and of course equivalent) to discuss these. The record-holders all come from the list of hypersurfaces in toric fourfolds, constructed by Kreuzer and Skarke. The largest value of $h^{1,1}$ is $491$, and the same for $h^{2,1}$. These two manifolds (which are mirror) also hold the record for largest $h^{1,1} + h^{2,1}$, which is 502. There is a third manifold which shares this record; its Hodge numbers are $(251,251)$, and it is also in the Kreuzer-Skarke list.