Left orderable surgeries of double twist knots II

Abstract

A slope $r$ is called a left orderable slope of a knot $K\subset S^3$ if the 3-manifold obtained by $r$-surgery along $K$ has left orderable fundamental group. Consider two-bridge knots $C(2m,\pm 2n)$ and $C(2m+1,-2n)$ in the Conway notation, where $m\ge 1$ and $n\ge 2$ are integers. By using \textit{continuous} families of hyperbolic ${\mathrm{SL}}_2(\mathbb{R})$-representations of knot groups, it was shown in \cite{HT-genus1, Tr} that any slope in $(-4n,4m)$ (resp. $[0,\max \{4m,4n\})$) is a left orderable slope of $C(2m,2n)$ (resp. $C(2m,-2n)$) and in \cite{Ga} 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 \textit{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 ${\mathrm{SL}}_2(\mathbb{R})$-representations of the knot group.

14 pages, 1 figure

References in corpus (2)

• Orderability of Homology Spheres Obtained by Dehn Filling
• Left orderable surgeries of double twist knots

Cited by in corpus (1)

• A unified Casson-Lin invariant for the real forms of SL(2)