A note on knot concordance

Yildiz, Eylem

doi:10.2140/agt.2018.18.3119

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A note on knot concordance

Eylem Zeliha Yildiz

Algebraic & Geometric Topology 18 (2018) 3119–3128

DOI: 10.2140/agt.2018.18.3119

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 nontrivial element $g \in π_{1} (Y)$ 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.

Keywords

knot concordance, homology sphere, $S^1\times S^2$, almost concordance, singular concordance, PL concordance

Mathematical Subject Classification 2010

Primary: 57M27, 57Q60

References

Publication

Received: 18 March 2018

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

Volume 18, issue 5 (2018)