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-Watson. As an application of this sufficient condition, we show that for any $(p,q)$ 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 D{\c a}bkowski, Przytycki and Togha.