Summary: A slope \(r\) is called a left orderable slope of a knot \(K \subset S^3\) if the 3-manifold obtained by \(r\)-surgery along \(K\) has left orderable fundamental group. Consider double twist knots \(C(2m, \pm 2n)\) and \(C(2m+1, -2n)\) in the Conway notation, where \(m \ge 1\) and \(n \ge 2\) are integers. By using continuous families of hyperbolic \(\text{SL}_2(\mathbb{R})\)-representations of knot groups, it was shown in [R. Hakamata and M. Teragaito, Can. Math. Bull. 57, No. 2, 310–317 (2014; Zbl 1305.57010), A. T. Tran, J. Math. Soc. Japan 67, No. 1, 319–338 (2015; Zbl 1419.57028)] that any slope in \((-4n, 4m)\) (resp. \([0, \max\{4m, 4n\})\)) is a left orderable slope of \(C(2m, 2n)\) (resp. \(C(2m, -2n)\)) and in [X. Gao, “Slope of orderable Dehn filling of two-bridge knots”, Preprint, arXiv:1912.07468] that any slope in \((-4n,0]\) is a left orderable slope of \(C(2m+1,-2n)\). However, the proofs of these results are incomplete, since the continuity of the families of representations was not proved. In this paper, we complete these proofs, and, moreover, we show that any slope in \((-4n, 4m)\) is a left orderable slope of \(C(2m+1,-2n)\) detected by hyperbolic \(\text{SL}_2(\mathbb{R})\)-representations of the knot group.
For part I, see [A. T. Tran, J. Math. Soc. Japan 73, No. 3, 753–765 (2021; Zbl 1479.57052)].