Injectivity of satellite operators in knot concordance

Abstract

Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite knot of K with pattern P. This defines an operator on the set of knot types and induces a satellite operator P:C--> C on the set of smooth concordance classes of knots. There has been considerable interest in whether certain such functions are injective. For example, it is a famous open problem whether the Whitehead double operator is weakly injective (an operator is called weakly injective if P(K)=P(0) implies K=0 where 0 is the class of the trivial knot). We prove that, modulo the smooth 4-dimensional Poincare Conjecture, any strong winding number one satellite operator is injective on C. More precisely, if P has strong winding number one and P(K)=P(J), then K is smoothly concordant to J in S^3 x [0,1] equipped with a possibly exotic smooth structure. We also prove that any strong winding number one operator is injective on the topological knot concordance group. If P(0) is unknotted then strong winding number one is the same as (ordinary) winding number one. More generally we show that any satellite operator with non-zero winding number n induces an injective function on the set of Z[1/n]-concordance classes of knots. We extend some of our results to links.

16 pages; in second version we have added some results on operators on links and string links and added some references to connections with Mazur manifolds and corks; third version has only very minor changes and will appear in Journal of Topology

References in corpus (6)

• Corks, Plugs and exotic structures
• Primary decomposition and the fractal nature of knot concordance
• Knot concordance and homology cobordism
• Smooth concordance of links topologically concordant to the Hopf link
• Slice knots which bound punctured Klein bottles
• Knot Groups with Many Killers

Cited by in corpus (11)

• Satellites of Infinite Rank in the Smooth Concordance Group
• Satellite operators as group actions on knot concordance
• Satellite operators with distinct iterates in smooth concordance
• The geometry of the knot concordance space
• On the Upsilon invariant and satellite knots
• Shake slice and shake concordant knots
• On rational sliceness of Miyazaki's fibered, $-$amphicheiral knots
• Topological concordance of knots in homology spheres and the solvable filtration
• Slice knots which bound punctured Klein bottles
• Legendrian Satellites and Decomposable Concordances
• Action of the Mazur pattern up to topological concordance