Satellites and concordance of knots in 3-manifolds

TL;DR

This paper explores the classification of knots in 3-manifolds up to concordance, revealing that the action of the concordance group is often non-transitive, with specific results for different manifolds.

Contribution

It introduces new techniques using Reidemeister torsion and linking numbers to analyze concordance classes of knots in 3-manifolds, and characterizes the structure of these classes in various cases.

Findings

The action of the concordance group is not transitive in many cases.

For non-spherical 3-manifolds, the set of orbits for the trivial homotopy class is infinite.

All knots with winding number one in $S^1 \times S^2$ are concordant.

Abstract

Given a 3-manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the 3-sphere acts on this set. We show in many cases that the action is not transitive, using two techniques. Our first technique uses Reidemeister torsion invariants, and the second uses linking numbers in covering spaces. In particular, we show using covering links that for the trivial homotopy class, and for any 3-manifold that is not the 3-sphere, the set of orbits is infinite. On the other hand, for the case that $Y=S^1 \times S^2$, we apply topological surgery theory to show that all knots with winding number one are concordant.