  We introduce twisted K-theoretic Gromov-Witten invariants - in the frameworks
of both "ordinary" and permutation-equivariant K-theoretic GW theory defined
recently by Givental. We focus on the case when the twisting is given by the
Euler class - we characterize in both cases the range of the $J$-function of
the twisted theory in terms of the untwisted theory. As applications we use the
D_q module structure in the permutation-equivariant case to generalize results
of Givental: we prove a "quantum Lefschetz" type theorem and we relate points
on the cones of the total space with those of the base of a toric fibration.
