The Pollicott-Ruelle resonances for the sawtooth map are investigated. We turn our attention to the parametric dependence of them with respect to the bifurcation parameter K. It is numerically shown that the resonances move in an erratic way if the bifurcation parameter K is supposed to be time. At certain rational values of K, it is observed that some resonances shrink to z = 0. In particular, at positive integer values of K which correspond to the Arnold cat map, all resonances except z = 1 (i.e., the equilibrium state) shrink to z = 0. This peculiar behavior is rigorously proved in the Appendix. In addition, the diffusion coefficient of this map is numerically calculated in a very accurate way by evaluating the leading resonance.