Any knot in a solid torus, called a pattern, induces a function, called a satellite operator, on concordance classes of knots in 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 induce injective satellite operators on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth –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 such that there is a pattern for which is concordant to (topologically as well as smoothly in a potentially exotic ) for all knots ; 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 –dimensional Poincaré conjecture.
"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