$\mathrm{SL}_2(\mathbb R)$-representations and left-orderable surgeries of $(-2, 3, 2n+1)$-pretzel knots

TL;DR

This paper constructs explicit paths of $ ext{SL}_2( ext{R})$-representations for certain pretzel knot groups and uses these to determine conditions under which the resulting 3-manifold groups are left-orderable.

Contribution

It provides an explicit construction of continuous $ ext{SL}_2( ext{R})$-representation paths for $(-2,3,2n+1)$-pretzel knots and relates these to left-orderability of surgery manifolds.

Findings

Fundamental groups of certain surgeries are left-orderable under specified conditions.

Explicit $ ext{SL}_2( ext{R})$-representation paths are constructed for the knot groups.

Left-orderability is established for surgeries with slope less than a specific bound.

Abstract

In this paper, we provide an explicit construction of continuous paths of $\mathrm{SL}_2(\mathbb R)$-representations of the knot groups of $(-2,3,2n+1)$-pretzel knots. As an application, we show that the fundamental group of the $3$-manifold obtained from the $3$-sphere by $\frac{m}{l}$-surgery along the $(-2,3,2n+1)$-pretzel knot, where $n \ge 3$ is an integer and $n \not= 4$, is left-orderable if $\frac{m}{l}< 2 \lfloor \frac{2n+4}{3} \rfloor$.