Two curious strongly invertible L-space knots

TL;DR

This paper introduces two strongly invertible L-space knots with unique properties, providing counterexamples to a conjecture and exploring their exceptional algebraic and topological features.

Contribution

The paper presents explicit examples of strongly invertible L-space knots that challenge existing conjectures and exhibit unusual algebraic properties.

Findings

The knots' surgeries are not double branched covers of Khovanov thin links.

These knots have formal semigroups that are actual semigroups.

They serve as counterexamples to Watson's conjectural characterization.

Abstract

We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.