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

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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≥ 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.