L-space surgery and twisting operation

A knot in the $3$ –sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, ie a rational homology $3$ –sphere with the smallest possible Heegaard Floer homology. Given a knot $K$ , take an unknotted circle $c$ and twist $K$ $n$ times along $c$ to obtain a twist family ${K_{n}}$ . We give a sufficient condition for ${K_{n}}$ to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot $K$ , we can take $c$ so that the twist family ${K_{n}}$ contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.

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Keywords

L-space surgery, L-space knot, twisting, seiferter, tunnel number

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Secondary: 57N10

References

Publication

Received: 28 April 2015

Revised: 16 August 2015

Accepted: 10 September 2015

Published: 1 July 2016

Authors

Kimihiko Motegi
Department of Mathematics Nihon University 3-25-40 Sakurajosui, Setagaya-ku Tokyo 156-8550 Japan
http://www.math.chs.nihon-u.ac.jp/~motegi/

Volume 16, issue 3 (2016)

Kimihiko Motegi

Abstract