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Satellite operators as group actions on knot concordance

Christopher W Davis and Arunima Ray

Algebraic & Geometric Topology 16 (2016) 945–969

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.

knot, satellite knot, knot concordance, group action, satellite operator, homology cylinder
Mathematical Subject Classification 2010
Primary: 57M25
Received: 24 October 2013
Revised: 23 June 2015
Accepted: 5 July 2015
Published: 26 April 2016
Christopher W Davis
Department of Mathematics
The University of Wisconsin at Eau Claire
533 Hibbard Humanities Hall
Eau Claire, WI 54702
Arunima Ray
Department of Mathematics
Brandeis University
415 South St
Waltham, MA 02453