Left-orderablity for surgeries on $(-2,3,2s+1)$-pretzel knots

TL;DR

This paper investigates the left-orderability of fundamental groups resulting from Dehn surgeries on certain pretzel knots, establishing conditions under which these groups are or are not left orderable based on the surgery slope.

Contribution

It provides new criteria for left-orderability of fundamental groups after surgery on $(-2,3,2s+1)$-pretzel knots, linking the slope to the group's orderability.

Findings

Groups are not left orderable if slope $rac{p}{q} \\ge 2s+3$.

Groups are left orderable near slope zero.

Results depend on the parameter $s$ of the pretzel knot.

Abstract

In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a $(-2,3,2s+1)$-pretzel knot ($s\ge 3$) with slope $\frac{p}{q}$ is not left orderable if $\frac{p}{q}\ge 2s+3$, and that it is left orderable if $\frac{p}{q}$ is in a neighborhood of zero depending on $s$.