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 Pinz\'{o}n-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.