On left-orderable fundamental groups and Dehn surgeries on twist knots

TL;DR

This paper characterizes when the fundamental group of manifolds obtained by Dehn surgery on hyperbolic twist knots is left-orderable, based on the surgery slope and properties of the knot parameter m.

Contribution

It provides explicit conditions on the surgery slope for left-orderability of the fundamental group of manifolds from twist knots, extending understanding of their algebraic properties.

Findings

Left-orderability depends on the surgery slope and the parity of m.

Explicit slope intervals for left-orderability are given for even and odd m.

The results connect algebraic properties of the fundamental group with geometric surgery parameters.

Abstract

We show that the resulting manifold by $r$-surgery on the hyperbolic twist knot $K_m, \, m \ge 2$, has left-orderable fundamental group if the slope $r$ satisfies the condition $r \in (-4,2m)$ if $m$ is even, and $r \in [0,4] \cup (\frac{4m}{\omega}+4, 2m+4)$ if $m$ is odd, where $\omega>1$ is the unique real solution of the equation $x e^{x}=4m$.