Non-orientable Genus of a Knot in Punctured $\mathbf{C}P^2$

Abstract

For a closed 4-manifold $X$, any knot $K$ in the boundary of punctured $X$ bounds a non-orientable and null-homologous embedded surface in punctured $X$. Thus we can define an invariant $\gamma_X^0(K)$ to be the smallest first Betti number of such surfaces. Note that $\gamma^0_{S^4}$ is equal to the non-orientable 4-ball genus. While it is very likely that for a given $X$, $\gamma^0_X$ has no upper bound, it is difficult to show it. Recently, Batson showed that $\gamma^0_{S^4}$ has no upper bound. In this paper we show that for any positive integer $n$, $\gamma^0_{n\mathbf{C}P^2}$ has no upper bound.