# Comparing nonorientable three genus and nonorientable four genus of torus knots

## TL;DR

This paper investigates the relationship between nonorientable three genus and four genus of torus knots, revealing that their difference can grow arbitrarily large, contrasting with the orientable case where they are equal.

## Contribution

It establishes a lower bound on the difference between nonorientable three and four genus for certain torus knots, showing this difference can be arbitrarily large.

## Key findings

- Difference between nonorientable three and four genus grows with p
- Contrast with orientable case where these invariants are equal
- Difference can be arbitrarily large for fixed q as p varies

## 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.