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

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 $\left|q\right|\ge 5$. To this end, we completely list up all such lens spaces with $\left|q\right|<5$ and prove that they are obtained only by torus knots.