Concordance of knots in $S^1\times S^2$

TL;DR

This paper investigates the smooth and topological concordance of knots in $S^1\times S^2$, establishing uniqueness results for certain winding numbers, and constructing examples that distinguish smooth and topological categories.

Contribution

It proves the uniqueness of smooth concordance classes for knots with winding number one and constructs infinite families of knots illustrating differences between smooth and topological concordance.

Findings

Unique smooth concordance class for winding number one

Infinitely many topological concordance classes of non-slice knots

Existence of knots that are topologically but not smoothly concordant

Abstract

We establish a number of results about smooth and topological concordance of knots in $S^1\times S^2$. The winding number of a knot in $S^1\times S^2$ is defined to be its class in $H_1(S^1\times S^2;\mathbb{Z})\cong \mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in $S^1\times S^2$ is slice (resp. topologically slice) if it bounds a smooth (resp. locally flat) disk in $D^2\times S^2$. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than $\pm 1$, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and topologically concordant, but not smoothly concordant.