2019/07/30 by Jabuka, Stanislav, Van Cott, Cornelia A.
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1907.12970
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≥ 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.