Injectivity of satellite operators in knot concordance

Let P be a knot in a solid torus, K be a knot in S3 and P(K) be the satellite knot of K with pattern P. This defines an operator P:K→K 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 four‐dimensional Poincaré Conjecture, any strong winding number 1 satellite operator is injective on C. More precisely, if P has strong winding number 1 and P(K)=P(J), then K is smoothly concordant to J in S3×[0,1] equipped with a possibly exotic smooth structure. We also prove that any strong winding number 1 operator is injective on the topological knot concordance group. If P(0) is unknotted, then strong winding number 1 is the same as (ordinary) winding number 1. 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 deduce some analogous results for links.