A bstract We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with h _(1,1)≥ 140 or h _(2,1)≥ 140 that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small h _(1,1)the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h _(1,1)increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.
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