L-space surgery and twisting operation

TL;DR

This paper investigates conditions under which twisting operations on knots produce infinite families of L-space knots, with applications to torus and Berge knots, revealing new examples with complex properties.

Contribution

It provides a sufficient condition for twist families to contain infinitely many L-space knots and constructs new examples with higher tunnel numbers.

Findings

Infinite L-space knot families can be generated via twisting under certain conditions.

Torus and Berge knots can produce infinite hyperbolic L-space knot families through twisting.

Existence of hyperbolic L-space knots with tunnel number greater than one within twist families.

Abstract

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. 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.