  We show that for ultra-cold neutral bosonic atoms held in a three-dimensional
periodic potential or optical lattice, a Hubbard model with dominant,
attractive three-body interactions can be generated. In fact, we derive that
the effect of pair-wise interactions can be made small or zero starting from
the realization that collisions occur at the zero-point energy of an optical
lattice site and the strength of the interactions is energy dependent from
effective-range contributions. We determine the strength of the two- and
three-body interactions for scattering from van-der-Waals potentials and near
Fano-Feshbach resonances. For van-der-Waals potentials, which for example
describe scattering of alkaline-earth atoms, we find that the pair-wise
interaction can only be turned off for species with a small negative scattering
length, leaving the $^{88}$Sr isotope a possible candidate. Interestingly, for
collisional magnetic Feshbach resonances this restriction does not apply and
there often exist magnetic fields where the two-body interaction is small. We
illustrate this result for several known narrow resonances between alkali-metal
atoms as well as chromium atoms. Finally, we compare the size of the three-body
interaction with hopping rates and describe limits due to three-body
recombination.
