### Abstract

We propose a classification of knots in S

^{1}x S^{2}that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knots in S^{1}x S^{2}may be obtained from a Berge-Gabai knot in a Heegaard solid torus of S^{1}x 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. All these knots in S^{1}x S^{2}are both doubly primitive and spherical braids. This classification arose from generalizing Berge's list of doubly primitive knots in S^{3}, 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 S^{1}x S^{2}and reduce this conjecture to classifying the homology classes of knots in lens spaces admitting a longitudinal S^{1}x S^{2}surgery.Original language | English |
---|---|

Number of pages | 35 |

Journal | ArXiv e-prints |

Publication status | Published - 27 Feb 2013 |

### Keywords

- math.GT
- 57M27

## Fingerprint Dive into the research topics of 'Some knots in S<sup>1</sup> x S<sup>2</sup> with lens space surgeries'. Together they form a unique fingerprint.

## Cite this

Baker, K. L., Buck, D., & Lecuona, A. G. (2013). Some knots in S

^{1}x S^{2}with lens space surgeries.*ArXiv e-prints*.