#### Volume 18, issue 5 (2018)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Other MSP Journals
A note on knot concordance

### Eylem Zeliha Yildiz

Algebraic & Geometric Topology 18 (2018) 3119–3128
##### 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 ${S}^{1}×pt$ in ${S}^{1}×{S}^{2}$ are smoothly concordant.

##### Keywords
knot concordance, homology sphere, $S^1\times S^2$, almost concordance, singular concordance, PL concordance
##### Mathematical Subject Classification 2010
Primary: 57M27, 57Q60
##### Publication
Revised: 11 May 2018
Accepted: 25 May 2018
Published: 22 August 2018
##### Authors
 Eylem Zeliha Yildiz Department of Mathematics Michigan State University East Lansing United States