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