Classical pretzel knots and left orderability

Abstract

We consider the classical pretzel knots $P(a_1,a_2,a_3)$, where $a_1,a_2,a_3$ are positive odd integers. By using continuous paths of elliptic ${\mathrm{SL}}_2(\mathbb{R})$-representations, we show that (i) the 3-manifold obtained by $\frac{m}{l}$-surgery on $P(a_1,a_2,a_3)$ has left orderable fundamental group if $\frac{m}{l}<1$, and (ii) the $n^{\mathrm{th}}$-cyclic branched cover of $P(a_1,a_2,a_3)$ has left orderable fundamental group if $n>2\pi /\arccos (1-2/(1+a_1{a}_2+a_2{a}_3+a_3{a}_1))$.

arXiv comment12 pages, 1 figure