Left-orderable surgeries on the knot $6_2$ via hyperbolic $\widetilde{PSL}(2,\mathbb{R})$-representations

TL;DR

This paper presents a new technique for detecting left-orderability of fundamental groups after Dehn surgery on knots, especially 2-bridge knots, using hyperbolic $ ilde{PSL}(2,R)$-representations, demonstrated on the knot $6_2$.

Contribution

It introduces a novel method leveraging hyperbolic $ ilde{PSL}(2,R)$-representations to determine left-orderability of groups resulting from surgeries, specifically applied to the knot $6_2$.

Findings

All surgeries on $6_2$ with slopes in $(-4,8)$ have left-orderable fundamental groups.

The method effectively detects left-orderability for a range of slopes.

The approach is particularly useful for 2-bridge knots.

Abstract

We introduce a new method of detecting when the fundamental group of a Dehn surgery on a knot admits a left-ordering, a method which is particularly useful for 2-bridge knots. As an illustration of this method, we show that all Dehn surgeries on the knot $6_2$ with slope in the interval $(-4, 8)\cap\mathbb{Q}$ have left-orderable fundamental groups by exhibiting a family of hyperbolic $\widetilde{PSL}(2,\mathbb{R})$-representations of the knot complement group.