  Given a closed four-manifold $X$ with an indefinite intersection form, we
consider smoothly embedded surfaces in $X \setminus $int$(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.
