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2016 Satellite operators as group actions on knot concordance
Christopher W Davis, Arunima Ray
Algebr. Geom. Topol. 16(2): 945-969 (2016). DOI: 10.2140/agt.2016.16.945

Abstract

Any knot in a solid torus, called a pattern, induces a function, called a satellite operator, on concordance classes of knots in S3 via the satellite construction. We introduce a generalization of patterns that form a group (unlike traditional patterns), modulo a generalization of concordance. Generalized patterns induce functions, called generalized satellite operators, on concordance classes of knots in homology spheres; using this we recover the recent result of Cochran and the authors that patterns with strong winding number  ± 1 induce injective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4–dimensional Poincaré conjecture. We also obtain a characterization of patterns inducing surjective satellite operators, as well as a sufficient condition for a generalized pattern to have an inverse. As a consequence, we are able to construct infinitely many nontrivial patterns P such that there is a pattern P¯ for which P¯(P(K)) is concordant to K (topologically as well as smoothly in a potentially exotic S3 × [0,1]) for all knots K; we show that these patterns are distinct from all connected-sum patterns, even up to concordance, and that they induce bijective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4–dimensional Poincaré conjecture.

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Christopher W Davis. Arunima Ray. "Satellite operators as group actions on knot concordance." Algebr. Geom. Topol. 16 (2) 945 - 969, 2016. https://doi.org/10.2140/agt.2016.16.945

Information

Received: 24 October 2013; Revised: 23 June 2015; Accepted: 5 July 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1351.57007
MathSciNet: MR3493412
Digital Object Identifier: 10.2140/agt.2016.16.945

Subjects:
Primary: 57M25

Rights: Copyright © 2016 Mathematical Sciences Publishers

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