Some knots in S¹ × S² with lens space surgeries

We propose a classification of knots in S¹ × S² that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knot in S¹ × S² may be obtained from a Berge–Gabai knot in a Heegaard solid torus of S¹ × 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. Assuming results of Cebanu, we are able to further conclude that these three families constitute all the doubly primitive knots in S¹ × S². Thus we bring the classification of lens space surgeries on knots in S¹ × S² in line with the Berge Conjecture about lens space surgeries on knots in S³.