Knots in lens spaces with the 3–sphere surgery

Saito, Toshio

doi:10.2140/agt.2008.8.53

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

Toshio Saito

Algebraic & Geometric Topology 8 (2008) 53–79

DOI: 10.2140/agt.2008.8.53

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 $(p, q)$ is obtained by Berge’s surgery on a nontrivial nontorus doubly primitive knot in the 3–sphere, then $| q | \geq 5$ . 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

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57M25

References

Publication

Received: 22 February 2006

Accepted: 7 November 2007

Published: 8 February 2008

Authors

Toshio Saito
Graduate School of Humanities and Sciences Nara Women’s University Kitauoya Nishi-machi Nara 630–8506 Japan

Volume 8, issue 1 (2008)