# Non left-orderable surgeries on L-space twisted torus knots

## TL;DR

This paper proves that for certain L-space twisted torus knots, the fundamental group of the 3-manifold obtained by specific surgeries is not left-orderable when the surgery coefficient exceeds a bound related to the knot's genus.

## Contribution

It establishes a new criterion linking surgery coefficients and left-orderability for a class of twisted torus knots, expanding understanding of their fundamental groups.

## Key findings

- Fundamental groups are not left-orderable for surgeries with coefficient ≥ 2g(K)-1.
- Applicable to L-space twisted torus knots with specified parameters.
- Provides a bound connecting surgery slope and knot genus.

## Abstract

We show that if $K$ is an L-space twisted torus knot $T^{l,m}_{p,pk \pm 1}$ with $p \ge 2$, $k \ge 1$, $m \ge 1$ and $1 \le l \le p-1$, then the fundamental group of the $3$-manifold obtained by $\frac{r}{s}$-surgery along $K$ is not left-orderable whenever $\frac{r}{s} \ge 2 g(K) -1$, where $g(K)$ is the genus of $K$.