A Note on Knot Concordance

TL;DR

This paper investigates the almost-concordance classes of knots in various 3-manifolds, revealing infinite classes in non-spherical manifolds and uniform concordance in certain product manifolds, using classical techniques.

Contribution

It demonstrates the existence of infinitely many smooth almost-concordance classes in non-spherical 3-manifolds and establishes uniform concordance for knots in S^1×S^2.

Findings

Infinite almost-concordance classes in non-spherical manifolds.

All knots in the free homotopy class of S^1 in S^1×S^2 are smoothly concordant.

Constructed classes are topologically slice but do not bound PL-disks in Mazur manifolds.

Abstract

We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed $3$-manifolds. We first prove that, given $Y^3 \neq S^3$, for any non-trivial element $g\in \pi_1(Y)$ there are infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot. In particular we consider these distinct smooth almost-concordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PL-disk in the Mazur manifold, but all the representatives we construct are topologically slice. We also prove that all knots in the free homotopy class of $S^1 \times pt$ in $S^1 \times S^2$ are smoothly concordant.