L-space surgery and twisting operation

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.