  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 techniques 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.
