Comparing the nonorientable three and four genus of torus knots
We compare the values of the nonorientable three genus (or crosscap number) and the nonorientable four genus of torus knots. In particular, we show that for any fixed odd q, the difference between these two invariants on T(p,q) can be arbitrarily large. The primary tool that enables this result is a connection between continued fractions and torus knots, which we will describe. This result contrasts with the orientable case: Seifert proved in 1935 that the orientable three genus of the torus knot T(p, q) is (p-1)(q-1)/2, and subsequently in 1993, Kronheimer and Mrowka proved that the orientable four genus of T(p, q) is also this value. This is joint work with Slaven Jabuka.