L-spaces, left-orderability and two-bridge knots

Abstract

We show that the 3-fold cyclic branched cover of any genus 2 two-bridge knot [Formula: see text] is an L-space and its fundamental group is not left-orderable. Therefore, the family of 3-fold cyclic branched cover of any genus 2 two-bridge knot [Formula: see text] verifies the [Formula: see text]-space conjecture. We also show that if [Formula: see text] is a two-bridge knot with [Formula: see text], [Formula: see text], then the fundamental group of the 5-fold cyclic branched cover of [Formula: see text] is not left-orderable, which will complete the proof that the fundamental group of the 5-fold cyclic branched cover of any genus 1 two-bridge knot is not left-orderable.