Satellite operators with distinct iterates in smooth concordance

Abstract

Each pattern P P in a solid torus gives a function P : C → C P:\mathcal {C} \rightarrow \mathcal {C} on the smooth knot concordance group, taking any knot K K to its satellite P ( K ) P(K) . We give examples of winding number one patterns P P and a class of knots K K , such that the iterated satellites P i ( K ) P^i(K) are distinct in concordance, i.e. if i ≠ j ≥ 0 i \neq j \geq 0 , P i ( K ) ≠ P j ( K ) P^i(K) \neq P^j(K) . This implies that the operators P i P^i give distinct functions on C \mathcal {C} , providing further evidence for the (conjectured) fractal nature of C \mathcal {C} . Our theorem also allows us to construct several sets of examples, such as infinite families of topologically slice knots that are distinct in smooth concordance, infinite families of 2–component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link, and infinitely many prime knots which have the same Alexander polynomial as an L L –space knot but are not themselves L L –space knots.