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We establish a number of results about smooth and topological concordance of knots inS  1   ×   S  2    . The  winding number  of a knot inS  1   ×   S  2is defined to be its class inH  1    (   S  1   ×   S  2   ;  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 inS  1   ×   S  2is slice (respectively, topologically slice) if it bounds a smooth (respectively, locally flat) disk inD  2   ×   S  2    . 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.

Concordance of knots in S 1 × S 2