  Recent results on duality between string theories and connectedness of their
moduli spaces seem to go a long way toward establishing the uniqueness of an
underlying theory. For the large class of Calabi-Yau 3-folds that can be
embedded as hypersurfaces in toric varieties the proof of mathematical
connectedness via singular limits is greatly simplified by using polytopes that
are maximal with respect to certain single or multiple weight systems. We
identify the multiple weight systems occurring in this approach. We show that
all of the corresponding Calabi-Yau manifolds are connected among themselves
and to the web of CICY's. This almost completes the proof of connectedness for
toric Calabi-Yau hypersurfaces.
