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 ∖B^4 , with boundary a knot K ⊂ 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.