We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let [Formula: see text] be any torus knot with [Formula: see text] even and [Formula: see text] odd. The difference between these two invariants on [Formula: see text] is at least [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text]. Hence, the difference between the two invariants on torus knots [Formula: see text] grows arbitrarily large for any fixed odd [Formula: see text], as [Formula: see text] ranges over values of a fixed congruence class modulo [Formula: see text]. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot [Formula: see text] is [Formula: see text], and Kronheimer and Mrowka later proved that the orientable four genus of [Formula: see text] is also this same value.
Comparing nonorientable three genus and nonorientable four genus of torus knots
