Left orderable surgeries of double twist knots

A rational number $r$ is called a left orderable slope of a knot $K \subset S^3$ifthe3-manifoldobtainedfrom$S^3$by$r$-surgeryalong$K$ has left orderable fundamental group. In this paper we consider the double twist knots $C(k,l)$ in the Conway notation. For any positive integers $m$ and $n$, we show that if $K$ is a double twist knot of the form $C(2m,-2n)$, $C(2m+1,2n)$ or $C(2m+1,-2n)$ then there is an explicit unbounded interval $I$ such that any rational number $r\in I$ is a left orderable slope of $K$.