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 $K_{[-2q,2s,-2t,2l]}$ 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 $K_{[-2q,2s,-2t,2l]}$ verifies the $L$-space conjecture. We also show that if $K_{[2k,-2l]}$ is a 2-bridge knot with $k\geq 2$, $l>0$, then the fundamental group of the 5-fold cyclic branched cover of $K_{[2k,-2l]}$ is not left-orderable, which will complete the proof that the fundamental group of the 5-fold cyclic branched cover of any genus one two-bridge knot is not left-orderable.