Left orderable surgeries of double twist knots

Abstract

A rational number 𝑟 is called a left orderable slope of a knot 𝐾 ⊂ 𝑆³ if the 3-manifold obtained from 𝑆³ 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.