  We compare the values of the nonorientable three genus (or, crosscap number)
and the nonorientable four genus of torus knots. In particular, let T(p,q) be
any torus knot with p even and q odd. The difference between these two
invariants on T(p,q) is at least k/2, where p = qk + a and 0 < a < q and $k\geq
0$. Hence, the difference between the two invariants on torus knots T(p,q)
grows arbitrarily large for any fixed odd q, as p ranges over values of a fixed
congruence class modulo q. This contrasts with the orientable setting. Seifert
proved that the orientable three genus of the torus knot T(p,q) is
(p-1)(q-1)/2, and Kronheimer and Mrowka later proved that the orientable four
genus of T(p,q) is also this same value.
