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

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 $\frac{m}{l}< 2 \lfloor \frac{2n+4}{3} \rfloor$.