Non-Left-Orderable Surgeries on 1-Bridge Braids

Abstract

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in $S^3$ can produce large families of L-spaces, it is natural to examine the conjecture on these 3-manifolds. Greene, Lewallen, and Vafaee have proved that all 1-bridge braids are L-space knots. In this paper, we consider three families of 1-bridge braids. First we calculate the knot groups and peripheral subgroups. We then verify the conjecture on the three cases by applying the criterion developed by Christianson, Goluboff, Hamann, and Varadaraj, when they verified the same conjecture for certain twisted torus knots and generalized the criteria of Clay and Watson and of Ichihara and Temma.