Simply slicing knots

TL;DR

This paper provides conditions under which knots in the boundary of a simply-connected 4-manifold can be sliced by locally flat discs with finite cyclic fundamental group complements, including stable cases after connected sums.

Contribution

It establishes new sufficient and necessary conditions for knot slicing in 4-manifolds, extending understanding of knot concordance and 4-manifold topology.

Findings

Identifies sufficient conditions for knot slicing with finite cyclic complement groups.

Provides necessary and sufficient conditions for stable slicing after connected sums.

Advances the understanding of knot slicing in simply-connected 4-manifolds.

Abstract

Given a simply-connected 4-manifold with boundary the 3-sphere, this paper establishes sufficient conditions for a knot in the boundary to be sliced by a locally flat disc in the 4-manifold, whose complement has finite cyclic fundamental group. In addition, necessary and sufficient conditions are described to ensure that such discs exist stably, that is after taking the connected sum of the 4-manifold with copies of $S^2 \times S^2$.