#### Volume 16, issue 3 (2016)

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L-space surgery and twisting operation

### Kimihiko Motegi

Algebraic & Geometric Topology 16 (2016) 1727–1772
##### Abstract

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 $\left\{{K}_{n}\right\}$. We give a sufficient condition for $\left\{{K}_{n}\right\}$ 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 $\left\{{K}_{n}\right\}$ 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.

##### Keywords
L-space surgery, L-space knot, twisting, seiferter, tunnel number
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 57N10
##### 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/