Greene's residue criterion for the breakup of invariant tori of volume-preserving maps

► We study codimension-one tori in three-dimensional, volume-preserving maps. ► The generalized Farey Tree of Kim and Ostlund is used to construct periodic orbits. ► These sequences of periodic orbits converge to a given incommensurate torus. ► Growth of the residue of nearby periodic orbits predicts the destruction of tori. ► The critical function is computed and the most robust torus is located.