Injectivity of satellite operators in knot concordance

TL;DR

This paper proves that certain satellite operators with strong winding number one are injective on knot concordance classes, assuming the smooth 4-dimensional Poincare Conjecture, advancing understanding of knot invariants and satellite operations.

Contribution

It establishes the injectivity of strong winding number one satellite operators on smooth and topological concordance classes, linking satellite structure to knot equivalence.

Findings

Strong winding number one operators are injective on smooth concordance classes (modulo Poincare Conjecture).

Such operators are also injective on topological concordance groups.

Injectivity extends to operators with non-zero winding number n on Z[1/n]-concordance classes.

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.