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 ${\mathrm {SL}}_2(\mathbb {R})$ -representations of knot groups, it was shown in [8, 16] 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 [6] 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 ${\mathrm {SL}}_2(\mathbb {R})$ -representations of the knot group.

如果通过r -surgery 沿K获得的 3-流形具有左可排序基本群，则斜率r称为结 $K \subset S^3$ 的左可排序斜率。考虑康威符号中的双捻结 $C(2m, \pm 2n)$ 和 $C(2m+1, -2n)$ ，其中 $m \ge 1$ 和 $n \ge 2$ 是整数。通过使用连续双曲线族 ${\mathrm {SL}}_2(\mathbb {R})$ - 节点组的表示，[8, 16] 表明 $(-4n, 4m)$ 中的 任何斜率 (resp. $ [0, \max \{4m, 4n\})$ ) 是左可排序斜率 $C(2m, 2n)$ (resp. $C(2m, - 2n)$ ) 并且在 [6] 中 $(-4n,0]$ 中的任何斜率都是 $C(2m+1 ,-2n)$ . 然而，这些结果的证明是不完整的，因为没有证明表示族的连续性。在本文中，我们完成了这些证明，此外，我们证明了 $(- 4n, 4m)$ 是由双曲线 ${\mathrm {SL}}_2(\mathbb {R})$ - 节点组表示检测到 的 $C(2m+1,-2n)$的左可排序 斜率 。