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.
ASJC Scopus subject areas
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty