We show that by coupling two hyperfine states of an atom in an optical
lattice one can independently control two-, three-, and four-body on-site
interactions in a non-perturbative manner. In particular, under typical
conditions of current experiments one can have a purely three- or four-body
interacting gas of 39K atoms characterized by on-site interaction shifts
of several 100Hz.Comment: 6 pages, 3 figure