Left orderable surgeries of double twist knots II

A slope r is called a left orderable slope of a knot K subset of S-3 if the 3-manifold obtained by r-surgery along K has left orderable fundamental group. Consider double twist knots C(2m, +/- 2n) and C(2m + 1, -2n) in the Conway notation, where m >= 1 and n >= 2 are integers. By using continuous families of hyperbolic SL2(R)-representations of knot groups, it was shown in [8, 16] that any slope in (-4n, 4m) (resp. [0, max{4m, 4n})) is a leY 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 SL2(R)-representations of the knot group.