Satellite operators with distinct iterates in smooth concordance

Abstract

Let P be a knot in an unknotted solid torus (i.e. a satellite operator or pattern), K a knot in S^3 and P(K) the satellite of K with pattern P. For any satellite operator P, this correspondence gives a function P : C -> C on the set of smooth concordance classes of knots. We give examples of winding number one satellite operators P and a class of knots K, such that the iterated satellites P^i(K) are distinct as smooth concordance classes, i.e. if i=/=j>0, P^i(K)=/=P^j(K), where each P^i is unknotted when considered as a knot in S^3. This implies that the operators P^i give distinct functions on C, providing further evidence for the fractal nature of C. There are several other applications of our result, as follows. By using topologically slice knots K, we obtain infinite families {P^i(K)} of topologically slice knots that are distinct in smooth concordance. We can also obtain infinite families of 2-component links (with unknotted components and linking number one) which are not smoothly concordant to the positive Hopf link. For a large class of L-space knots K (including the positive torus knots), we obtain infinitely many prime knots {P^i(K)} which have the same Alexander polynomial as K but are not themselves L-space knots.