Non-Left-Orderable Surgeries on 1-Bridge Braids

TL;DR

This paper investigates the relationship between L-spaces and left-orderability of fundamental groups for 3-manifolds obtained from surgeries on 1-bridge braids, confirming the Boyer-Gordon-Watson conjecture in specific cases.

Contribution

It computes knot groups for three families of 1-bridge braids and verifies the Boyer-Gordon-Watson conjecture using established criteria for these cases.

Findings

Confirmed the conjecture for certain 1-bridge braid surgeries.

Computed explicit knot groups and peripheral subgroups.

Validated the conjecture using criteria from previous works.

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.