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

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.