Some knots in S¹ x S² with lens space surgeries

We propose a classification of knots in S¹ x S² that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S¹ x S² may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S¹ x S², as observed by Rasmussen. We show that there are yet two other families of knots: those that lie on the fiber of a genus one fibered knot and the `sporadic' knots. All these knots in S¹ x S² are both doubly primitive and spherical braids. This classification arose from generalizing Berge's list of doubly primitive knots in S³, though we also examine how one might develop it using Lisca's embeddings of the intersection lattices of rational homology balls bounded by lens spaces as a guide. We conjecture that our knots constitute a complete list of doubly primitive knots in S¹ x S² and reduce this conjecture to classifying the homology classes of knots in lens spaces admitting a longitudinal S¹ x S² surgery.