Non-left-orderability of cyclic branched covers of pretzel knots $P(3,-3,-2k-1)$

TL;DR

This paper proves that the fundamental groups of certain cyclic branched covers of specific pretzel knots are not left-orderable, contributing to the understanding of their algebraic and topological properties.

Contribution

It establishes the non-left-orderability of the fundamental groups of all cyclic branched covers of the pretzel knots P(3,-3,-2k-1), extending previous results to a broad class of knots.

Findings

Fundamental groups of these branched covers are non-left-orderable.

These manifolds are identified as L-spaces.

Results hold for all integers k and n ≥ 1.

Abstract

We prove the non-left-orderability of the fundamental group of the $n$-th fold cyclic branched cover of the pretzel knot $P(3,-3,-2k-1)$ for all integers $k$ and $n\ge 1$. These $3$-manifolds are $L$-spaces discovered by Issa and Turner.