Deep and shallow slice knots in 4-manifolds

Abstract

We consider slice disks for knots in the boundary of a smooth compact 4-manifold X 4 . We call a knot K ⊂ ∂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 ∂X × 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 ⊂ S 3 = ∂V is slice in V via a null-homologous disk.