Classical pretzel knots and left orderability

TL;DR

This paper investigates the left orderability of fundamental groups of 3-manifolds derived from classical pretzel knots, using elliptic SL(2,R)-representations to establish conditions based on surgeries and branched covers.

Contribution

It introduces a method using elliptic SL(2,R)-representations to determine left orderability for manifolds from pretzel knots, providing explicit criteria for surgeries and branched covers.

Findings

Left orderability for surgeries with slope less than 1.

Left orderability for cyclic branched covers beyond a specific order.

Application of elliptic SL(2,R)-representations to knot group analysis.

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 $n > 2\pi / \arccos(1-2/(1+a_1 a_2 + a_2 a_3 + a_3 a_1))$.