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Left orderable surgeries of double twist knots II

Part of: Ordered structures Low-dimensional topology

Published online by Cambridge University Press:  27 August 2020

Vu The Khoi ,
Masakazu Teragaito  and
Anh T. Tran
Vu The Khoi
Affiliation:
Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi10307, Vietnam e-mail: vtkhoi@math.ac.vn
Masakazu Teragaito
Affiliation:
Department of Mathematics and Mathematics Education, Hiroshima University, 1-1-1 Kagamiyama, Higashi-Hiroshima, Hiroshima7398524, Japan e-mail: teragai@hiroshima-u.ac.jp
Anh T. Tran*
Affiliation:
Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, 75080-3021, USA
*
e-mail: att140830@utdallas.edu
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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 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.

Keywords

Dehn surgeryleft orderableL-spaceRiley polynomialtwo-bridge knot

MSC classification

Secondary: 06F15: Ordered groups 57M27: Invariants of knots and 3-manifolds 57M25: Knots and links in $S^3$
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 624 - 637
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The first author has been partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2015.20. The second author has been partially supported by JSPS KAKENHI Grant Number JP20K03587. The third author has been partially supported by a grant from the Simons Foundation (354595).

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