Toric Calabi-Yau hypersurfaces fibered by weighted K3 hypersurfaces

In response to a question of Reid, we find all anti-canonical Calabi-Yau hypersurfaces X in toric weighted P(3)-bundles over P(1) where the general fiber of X over P(1) is a weighted K3 hypersurface. This gives a direct generalization of Reid's discovery of the 95 families of weighted K3 hypersurfaces in [18]. We also treat the case where X is fibered over P(2) with general fiber a genus one curve in a weighted projective plane.