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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Satellites and concordance of knots in 3-manifolds


Authors: Stefan Friedl, Matthias Nagel, Patrick Orson and Mark Powell
Journal: Trans. Amer. Math. Soc. 371 (2019), 2279-2306
MSC (2010): Primary 57M27, 57N70
DOI: https://doi.org/10.1090/tran/7313
Published electronically: September 10, 2018
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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.


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Additional Information

Stefan Friedl
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
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
Email: patrick.orson@cirget.ca

Mark Powell
Affiliation: Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
Email: mark@cirget.ca

DOI: https://doi.org/10.1090/tran/7313
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.
Article copyright: © Copyright 2018 American Mathematical Society

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