Satellites and concordance of knots in 3-manifolds

Friedl, Stefan; Nagel, Matthias; Orson, Patrick; Powell, Mark

doi:10.1090/tran/7313

Satellites and concordance of knots in 3-manifolds

Friedl, Stefan; Nagel, Matthias; Orson, Patrick; Powell, Mark

Authors

Stefan Friedl

Matthias Nagel

Patrick Orson

Mark Powell

Abstract

Given a 3–manifold Y and a free homotopy class in [S1, Y ], we investigate the set of topological concordance classes of knots in Y × [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 = S1 × S2, we apply topological surgery theory to show that all knots with winding number one are concordant.

Citation

Friedl, S., Nagel, M., Orson, P., & Powell, M. (2019). Satellites and concordance of knots in 3-manifolds. Transactions of the American Mathematical Society, 371(4), 2279-2306. https://doi.org/10.1090/tran/7313

Journal Article Type	Article
Online Publication Date	Sep 10, 2018
Publication Date	Feb 15, 2019
Deposit Date	Oct 3, 2017
Publicly Available Date	Oct 3, 2017
Journal	Transactions of the American Mathematical Society
Print ISSN	0002-9947
Electronic ISSN	1088-6850
Publisher	American Mathematical Society
Peer Reviewed	Peer Reviewed
Volume	371
Issue	4
Pages	2279-2306
DOI	https://doi.org/10.1090/tran/7313
Related Public URLs	https://arxiv.org/abs/1611.09114