Left-orderable, non-L-space surgeries on knots

TL;DR

This paper explores the classification of knot surgeries on the 3-sphere, focusing on identifying conditions under which surgeries produce manifolds with left-orderable fundamental groups that are not L-spaces, supporting a conjecture about their categorization.

Contribution

The authors introduce a method to construct knots with infinitely many hyperbolic surgeries that are left-orderable and not L-spaces, advancing understanding of the conjecture by Boyer, Gordon, and Watson.

Findings

Constructed infinitely many hyperbolic knots with all nontrivial surgeries being left-orderable and non-L-spaces.

Provided evidence supporting the classification conjecture for knot surgeries.

Developed a new approach to identify non-L-space, left-orderable surgeries on knots.

Abstract

Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be classified into left-orderable surgeries or L-space surgeries. We introduce a way to provide knots with left-orderable, non-L-space surgeries. As an application we present infinitely many hyperbolic knots on each of which every nontrivial surgery is a hyperbolic, left-orderable, non-L-space surgery.