Deep and shallow slice knots in 4-manifolds

TL;DR

This paper investigates the concept of deep slice knots in 4-manifolds, providing criteria for their nonexistence, and demonstrates their existence in certain 4-manifolds using topological invariants, while also exploring limitations on universal sliceness.

Contribution

It introduces the notion of deep slice knots in 4-manifolds, relates it to conjectures, and proves their existence in manifolds with 2-handles, also establishing nonexistence results for universal sliceness.

Findings

Deep slice knots exist in 4-manifolds with 2-handles.

Criteria for the nonexistence of deep slice knots are provided.

Not all knots in the boundary of a 4-manifold can be slice in the interior.

Abstract

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.