Left orderable surgeries of double twist knots II

Khoi, Vu The ; Teragaito, Masakazu ; Tran, Anh T.

Canadian mathematical bulletin, Tome 64 (2021) no. 3, pp. 624-637

Résumé

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 double twist 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 continuous families of hyperbolic ${\mathrm {SL}}_2(\mathbb {R})$-representations of knot groups, it was shown in [8, 16] 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 [6] 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 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.

Détail

DOI : 10.4153/S0008439520000703

Mots-clés : Dehn surgery, left orderable, L-space, Riley polynomial, two-bridge knot