Non-orientable genus of knots in punctured Spin 4-manifolds

For a closed 4-manifold X and a knot K in the boundary of punctured X, we define γX0(K) to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured X with boundary K. Note that γS40 is equal to the non-orientable 4-ball genus and hence γX0 is a generalization of the non-orientable 4-ball genus. While it is very likely that for given X, γX0 has no upper bound, it is difficult to show it. In fact, even in the case of γS40, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that for any Spin 4-manifold X, γX0 has no upper bound.