Concordance of knots in S-1 x S-2

Abstract

We establish a number of results about smooth and topological concordance of knots in S1xS2. The winding number of a knot in S1xS2 is defined to be its class in H1(S1xS2;Z)Z. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in S1xS2 is slice (respectively, topologically slice) if it bounds a smooth (respectively, locally flat) disk in D2xS2. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than +/- 1, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally, we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are pairwise topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and are pairwise topologically, but not smoothly, concordant.