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?



Today's world record on the Betti numbers of Calabi-Yau three-folds.
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](http://arxiv.org/abs/hep-th/0002240) 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.


See [this recent paper](http://arxiv.org/abs/arXiv:1205.0952) and [this one](http://arxiv.org/abs/arXiv:1207.4792) for nice plots and a discussion of these Calabi-Yau threefolds and others with large Hodge numbers.


