  Let P be a knot in a solid torus, K a knot in 3-space and P(K) the satellite
knot of K with pattern P. This defines an operator on the set of knot types and
induces a satellite operator P:C--> C on the set of smooth concordance classes
of knots. There has been considerable interest in whether certain such
functions are injective. For example, it is a famous open problem whether the
Whitehead double operator is weakly injective (an operator is called weakly
injective if P(K)=P(0) implies K=0 where 0 is the class of the trivial knot).
We prove that, modulo the smooth 4-dimensional Poincare Conjecture, any strong
winding number one satellite operator is injective on C. More precisely, if P
has strong winding number one and P(K)=P(J), then K is smoothly concordant to J
in S^3 x [0,1] equipped with a possibly exotic smooth structure. We also prove
that any strong winding number one operator is injective on the topological
knot concordance group. If P(0) is unknotted then strong winding number one is
the same as (ordinary) winding number one. More generally we show that any
satellite operator with non-zero winding number n induces an injective function
on the set of Z[1/n]-concordance classes of knots. We extend some of our
results to links.
