Characterization of local knots by satellite operators

Let $Y$ be an integral homology sphere and let $K\in\mathcal{K}(Y)$ be a knot. A knot is local if it lies in a ball in $Y$. Local-knot characterization. The almost-concordance class of $K$ is preserved by all satellite operators if and only if $K$ is local. This gives a proposed converse to the preceding lemma, characterizing local knots through the behavior of their almost-concordance classes under satellite operations.