Left orderability for surgeries on the $[1,1,2,2,2j]$ two-bridge knots

TL;DR

This paper establishes a cohomological criterion for identifying intervals of left-orderable Dehn surgeries on certain 3-manifolds and applies it to a family of two-bridge knots, providing evidence for a conjecture in the field.

Contribution

It introduces a new cohomological criterion for left-orderable Dehn surgeries and demonstrates its application to a specific family of two-bridge knots.

Findings

The family of two-bridge knots with continued fraction [1,1,2,2,2j] admits an interval of left-orderable Dehn surgeries.

The criterion successfully predicts left-orderability in these cases.

Provides positive evidence for a conjecture by Xinghua Gao.

Abstract

Let $M$ be a $\mathbb{Q}$-homology solid torus. In this paper, we give a cohomological criterion for the existence of an interval of left-orderable Dehn surgeries on $M$. We apply this criterion to prove that the two-bridge knot that corresponds to the continued fraction $[1,1,2,2,2j]$ for $j\geq 1$ admits an interval of left-orderable Dehn surgeries. This family of two-bridge knots gives some positive evidence for a question of Xinghua Gao.