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# Communications in Analysis and Geometry

## Volume 24 (2016)

### Number 3

### Some knots in $S^1 \times S^2$ with lens space surgeries

Pages: 431 – 470

DOI: http://dx.doi.org/10.4310/CAG.2016.v24.n3.a1

#### Authors

#### Abstract

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 $S^3$.