## Satellites and concordance of knots in 3–manifolds

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- by Stefan Friedl, Matthias Nagel, Patrick Orson and Mark Powell PDF
- Trans. Amer. Math. Soc.
**371**(2019), 2279-2306 Request permission

## 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.## References

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## Additional Information

**Stefan Friedl**- Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
- MR Author ID: 746949
- Email: stefan.friedl@mathematik.uni-regensburg.de
**Matthias Nagel**- Affiliation: Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
- Email: nagel@cirget.ca
**Patrick Orson**- Affiliation: Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
- MR Author ID: 1157713
- Email: patrick.orson@cirget.ca
**Mark Powell**- Affiliation: Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
- MR Author ID: 975189
- Email: mark@cirget.ca
- Received by editor(s): December 6, 2016
- Received by editor(s) in revised form: June 19, 2017
- Published electronically: September 10, 2018
- Additional Notes: The first author was supported by the SFB 1085 ‘Higher Invariants’ at the University of Regensburg, funded by the Deutsche Forschungsgemeinschaft (DFG). The first author is grateful for the hospitality received at Durham University and wishes to thank Wolfgang Lück for supporting a long stay at the Hausdorff Institute.

The second author was supported by a CIRGET postdoctoral fellowship.

The third author was supported by the EPSRC grant EP/M000389/1 of Andrew Lobb.

The fourth author was supported by an NSERC Discovery Grant.

The second, third, and fourth authors all thank the Hausdorff Institute for Mathematics in Bonn for both support and its outstanding research environment. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 2279-2306 - MSC (2010): Primary 57M27, 57N70
- DOI: https://doi.org/10.1090/tran/7313
- MathSciNet review: 3896081