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

In this paper, we provide an explicit construction of continuous paths of ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$-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 $n≠4$, is left-orderable if $\frac{m}{l}<2\lfloor \frac{2n+4}{3}\rfloor$.