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 is obtained by Berge’s surgery on a nontrivial nontorus doubly primitive knot in the 3–sphere, then . To this end, we completely list up all such lens spaces with and prove that they are obtained only by torus knots.
"Knots in lens spaces with the $3$–sphere surgery." Algebr. Geom. Topol. 8 (1) 53 - 79, 2008. https://doi.org/10.2140/agt.2008.8.53