Iterated satellite operators on the knot concordance group

Abstract

We show that for a winding number zero satellite operator P on the knot concordance group, if the axis of P has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration $P^n$ generates an infinite rank subgroup for each n. Furthermore, the graded quotients of the filtration of the knot concordance group associated with P have infinite rank at all levels. This gives an affirmative answer to a question of Hedden and Pinzón-Caicedo in many cases. We also show that under the same hypotheses, $P^n$ is not a homomorphism on the knot concordance group for each n. We use amenable $L^2$-signatures to prove these results.