#### Volume 8, issue 1 (2008)

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Knots in lens spaces with the $3$–sphere surgery

### Toshio Saito

Algebraic & Geometric Topology 8 (2008) 53–79
##### Abstract

In this paper, we improve a necessary condition which we gave in [Topology Appl. 154 (2007) 1502–1515], for 1–bridge braids in lens spaces to admit an integral surgery yielding the 3–sphere. As an application, we prove that if the lens space of type $\left(p,q\right)$ is obtained by Berge’s surgery on a nontrivial nontorus doubly primitive knot in the 3–sphere, then $|q|\ge 5$. To this end, we completely list up all such lens spaces with $|q|<5$ and prove that they are obtained only by torus knots.

##### Keywords
Dehn surgery, lens space, doubly primitive knot, dual knot
Primary: 57M27
Secondary: 57M25