Summary: We consider slice disks for knots in the boundary of a smooth compact 4-manifold \(X^4\). We call a knot \(K\subset\partial X\) deep slice in \(X\) if there is a smooth properly embedded \(2\)-disk in \(X\) with boundary \(K\), but \(K\) is not concordant to the unknot in a collar neighborhood \(\partial X\times{I}\) of the boundary.
We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0- and a nonzero number of 2-handles always has a deep slice knot in the boundary.
We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented \(4\)-manifold \(V\) with spherical boundary such that every knot \(K\subset{S}^3=\partial V\) is slice in \(V\) via a null-homologous disk.