Satellite operators as group actions on knot concordance

TL;DR

The paper introduces a group structure on generalized satellite operators acting on knots, proving injectivity and constructing invertible operators that are distinct from connected-sum operators, with implications for knot concordance.

Contribution

It defines a group of generalized satellite operators, characterizes surjectivity and invertibility, and constructs examples of invertible operators with novel properties.

Findings

Satellite operators with strong winding number ±1 are injective on concordance classes.

Constructed infinitely many non-trivial invertible satellite operators.

These operators induce bijections on concordance classes, distinct from connected-sum operators.

Abstract

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ 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 satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.