A note on knot concordance

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 $Y^3\ne S^3$, for any nontrivial element $g\in {\pi }_1\left(Y\right)$ 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.