Comparing nonorientable three genus and nonorientable four genus of torus knots

Abstract

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.