Communications in Analysis and Geometry
Volume 24 (2016)
Some knots in $S^1 \times S^2$ with lens space surgeries
Pages: 431 – 470
We propose a classification of knots in $S^1 \times S^2$ that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knot in $S^1 \times S^2$ may be obtained from a Berge–Gabai knot in a Heegaard solid torus of $S^1 \times S^2$, 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 $S^1 \times S^2$. Thus we bring the classification of lens space surgeries on knots in $S^1 \times S^2$ in line with the Berge Conjecture about lens space surgeries on knots in $S^3$.