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L-space surgery and twisting operation

Kimihiko Motegi

Algebraic & Geometric Topology 16 (2016) 1727–1772

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 {Kn}. We give a sufficient condition for {Kn} 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 {Kn} 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.

L-space surgery, L-space knot, twisting, seiferter, tunnel number
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 57N10
Received: 28 April 2015
Revised: 16 August 2015
Accepted: 10 September 2015
Published: 1 July 2016
Kimihiko Motegi
Department of Mathematics
Nihon University
3-25-40 Sakurajosui, Setagaya-ku
Tokyo 156-8550