Left-orderability, branched covers and double twist knots

For some families of two-bridge knots, including double twist knots with genus at least four, we determine precisely the set integers n > 1 n>1 such that fundamental group alttext="n"> encoding="application/x-tex">n -fold cyclic branched cover 3-sphere along these is left-orderable. There are figure-eight knot, for which this empty. We give first class hyperbolic not type, can be completely determined.