Relative genus bounds in indefinite four-manifolds

Given a closed four-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in \(X {\setminus } \smash {\mathring{B}^4}\), with boundary a knot \(K \subset S^3\). We give several methods to bound the genus of such surfaces in a fixed homology class. Our tools include adjunction inequalities and the \(10/8 + 4\) theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.