Left orderable surgeries of double twist knots

摘要

A rational number r is called a left orderable slope of a knot K ∈ S~3 if the 3-manifold obtained from S~3 by r-surgery along 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 ∈ I is a left orderable slope of K.