Left-orderability and cyclic branched coverings

Abstract

We provide an alternative proof of a sufficient condition for the fundamental group of the $n^{th}$ cyclic branched cover of $S^3$ along a prime knot $K$ to be left-orderable, which is originally due to Boyer, Gordon and Watson. As an application of this sufficient condition, we show that for any $\left(p,q\right)$ two-bridge knot, with $p\equiv 3\text{mod }4$, there are only finitely many cyclic branched covers whose fundamental groups are not left-orderable. This answers a question posed by Da̧bkowski, Przytycki and Togha.