Satellite operators as group actions on knot concordance

Davis, Christopher W; Ray, Arunima

doi:10.2140/agt.2016.16.945

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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

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 $S^{3}$ 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 $\pm 1$ 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 $\bar{P}$ for which $\bar{P} (P (K))$ is concordant to $K$ (topologically as well as smoothly in a potentially exotic $S^{3} \times [0, 1]$ ) 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.

Keywords

knot, satellite knot, knot concordance, group action, satellite operator, homology cylinder

Mathematical Subject Classification 2010

Primary: 57M25

References

Publication

Received: 24 October 2013

Revised: 23 June 2015

Accepted: 5 July 2015

Published: 26 April 2016

Authors

Christopher W Davis
Department of Mathematics The University of Wisconsin at Eau Claire 533 Hibbard Humanities Hall Eau Claire, WI 54702 USA
http://people.uwec.edu/daviscw/
Arunima Ray
Department of Mathematics Brandeis University MS-050 415 South St Waltham, MA 02453 USA
http://people.brandeis.edu/~aruray/

Volume 16, issue 2 (2016)