Some knots in S¹ x S² with lens space surgeries

Baker, K. L., Buck, D. and Lecuona, A. G. (2016) Some knots in S¹ x S² with lens space surgeries. Communications in Analysis and Geometry, 24(3), pp. 431-470. (doi: 10.4310/CAG.2016.v24.n3.a1)

Abstract

We propose a classification of knots in S¹ x S² that admit a longitudinal surgery to a lens space. Any lens space obtainable by longitudinal surgery on some knot in S¹ x S² may be obtained from a Berge–Gabai knot in a Heegaard solid torus of S¹ x 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 x S². Thus we bring the classification of lens space surgeries on knots in S¹ x S² in line with the Berge Conjecture about lens space surgeries on knots in S³.

• Item Type: Articles
• Additional Information: This work is partially supported by grant #209184 to Kenneth L. Baker from the Simons Foundation, by the Spanish GEOR MTM2011-22435 to Ana G. Lecuona, and Leverhulme Trust grant RP2013-K-017 and EPSRC grants G039585/1 and H031367/1 to Dorothy Buck.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Garcia Lecuona, Professor Ana
• Authors: Baker, K. L., Buck, D., and Lecuona, A. G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Communications in Analysis and Geometry
• Publisher: International Press
• ISSN: 1019-8385
• ISSN (Online): 1944-9992
• Published Online: 22 June 2016
• Copyright Holders: Copyright © 2016 International Press of Boston, Inc.
• First Published: First published in Communications in Analysis and Geometry 24(3): 431-470
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

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