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Berge–Gabai knots and L–space satellite operations

Hom, Jennifer and Lidman, Tye and Vafaee, Faramarz (2014) Berge–Gabai knots and L–space satellite operations. Algebraic and Geometric Topology, 14 (6). pp. 3745-3763. ISSN 1472-2747. doi:10.2140/agt.2014.14.3745.

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Let P(K) be a satellite knot where the pattern P is a Berge–Gabai knot (ie a knot in the solid torus with a nontrivial solid torus Dehn surgery) and the companion K is a nontrivial knot in S^3. We prove that P(K) is an L–space knot if and only if K is an L–space knot and P is sufficiently positively twisted relative to the genus of K. This generalizes the result for cables due to Hedden [Int. Math. Res. Not. 2009 (2009) 2248–2274] and Hom [Algebr. Geom. Topol. 11 (2011) 219–223].

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Additional Information:© 2014 Mathematical Sciences Publishers. Received: 26 June 2014. Accepted: 8 August 2014. Published: 15 January 2015. We would like to thank Matthew Hedden for helpful discussions and his interest in our work. We are also grateful to Josh Greene for pointing out Remark 1.2, to Allison Moore, David Shea Vela-Vick and Rachel Roberts for help with the proof of Lemma 1.4, to Ko Honda for a helpful discussion and to Liam Watson for comments on an earlier version of this paper. Hom was partially supported by NSF grant DMS-1307879. Lidman was partially supported by NSF grant DMS-0636643.
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Subject Keywords:L–space, Berge–Gabai knot, satellite knot, Dehn surgery
Issue or Number:6
Classification Code:MSC 2010: Primary: 57M25, 57M27, 57R58
Record Number:CaltechAUTHORS:20150327-060832382
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:56152
Deposited By: Ruth Sustaita
Deposited On:27 Mar 2015 19:30
Last Modified:10 Nov 2021 20:55

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