Left-orderability for surgeries on twisted torus knots

TL;DR

This paper investigates the left-orderability of fundamental groups of 3-manifolds obtained by specific surgeries on twisted torus knots, establishing conditions under which these groups are or are not left-orderable.

Contribution

It provides new criteria for left-orderability of fundamental groups resulting from surgeries on twisted torus knots, expanding understanding in 3-manifold topology.

Findings

Groups are not left-orderable for certain surgery slopes

Groups are left-orderable when surgery slopes are close to zero

Results apply to a family of twisted torus knots

Abstract

We show that the fundamental group of the $3$-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if $\frac{p}{q}$ is sufficiently close to $0$.