Narrow Results

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 ${\mathrm{\text{SL}}}_2(ℝ)$-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 $n\mathrm{\text{th}}$-cyclic branched cover of $P(a_1,a_2,a_3)$ has left orderable fundamental group if $n>2\pi /arccos(1-2/(1+a_1{a}_2+a_2{a}_3+a_3{a}_1))$.