Algebraic & Geometric Topology

Satellite operators as group actions on knot concordance

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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 945-969.

Dates
Revised: 23 June 2015
Accepted: 5 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841157

Digital Object Identifier
doi:10.2140/agt.2016.16.945

Mathematical Reviews number (MathSciNet)
MR3493412

Zentralblatt MATH identifier
1351.57007

Citation

Davis, Christopher W; Ray, Arunima. Satellite operators as group actions on knot concordance. Algebr. Geom. Topol. 16 (2016), no. 2, 945--969. doi:10.2140/agt.2016.16.945. https://projecteuclid.org/euclid.agt/1510841157

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