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Classical pretzel knots and left orderability

Arafat Khan

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

E-mail Address: arafat@utdallas.edu

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and

Anh T. Tran

https://orcid.org/0000-0003-1179-3088

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

E-mail Address: att140830@utdallas.edu

Corresponding author.

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https://doi.org/10.1142/S0129167X21500804Cited by:1 (Source: Crossref)

Abstract

We consider the classical pretzel knots $P (a_{1}, a_{2}, a_{3})$ , where $a_{1}, a_{2}, a_{3}$ are positive odd integers. By using continuous paths of elliptic ${SL}_{2} (ℝ)$ -representations, we show that (i) the 3-manifold obtained by $\frac{m}{l}$ -surgery on $P (a_{1}, a_{2}, a_{3})$ has left orderable fundamental group if $\frac{m}{l} < 1$ and (ii) the $n th$ -cyclic branched cover of $P (a_{1}, a_{2}, a_{3})$ has left orderable fundamental group if $n > 2 π / arccos (1 - 2 / (1 + a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{1}))$ .

Communicated by Yasuyuki Kawahigashi

Keywords:

AMSC: 57K10, 57M12

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