Berge–Gabai knots and L–space satellite operations

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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Keywords

L–space, Berge–Gabai knot, satellite knot, Dehn surgery

Mathematical Subject Classification 2010

Primary: 57M25, 57M27, 57R58

References

Publication

Received: 26 June 2014

Accepted: 8 August 2014

Published: 15 January 2015

Authors

Jennifer Hom
Department of Mathematics Columbia University New York, NY 10027 USA

Tye Lidman
Department of Mathematics University of Texas at Austin Austin, TX 78712 USA

Faramarz Vafaee
Mathematics Department California Institute of Technology Pasadena, CA 91125 USA

Volume 14, issue 6 (2014)

Jennifer Hom, Tye Lidman and Faramarz Vafaee

Abstract