Abstract
We propose a classification of knots in S1 × S2 that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knot in S1 × S2 may be obtained from a Berge–Gabai knot in a Heegaard solid torus of S1 × S2, 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. Assuming results of Cebanu, we are able to further conclude that these three families constitute all the doubly primitive knots in S1 × S2. Thus we bring the classification of lens space surgeries on knots in S1 × S2 in line with the Berge Conjecture about lens space surgeries on knots in S3.
Original language | English |
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Pages (from-to) | 431-470 |
Journal | Communications in Analysis & Geometry |
Volume | 24 |
Issue number | 3 |
DOIs | |
Publication status | Published - 1 Jan 2016 |