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Left orderability and cyclic branched coverings of rational knots $C (2 p, 2 m, 2 n + 1)$

Bradley Meyer

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

E-mail Address: bradley.meyer@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/S0218216523500621Cited by:0 (Source: Crossref)

Abstract

We consider cyclic branched coverings of a 3-parameter family of rational knots in $S^{3}$ and study the left orderability of their fundamental groups. We first compute the nonabelian ${SL}_{2} (ℂ)$ -character varieties of the rational knots $C (2 p, 2 m, 2 n + 1)$ in the Conway notation, where $p, m, n$ are integers. We then study real points on these varieties and finally use them to determine the left orderability of the fundamental groups of cyclic branched coverings of $C (2 p, 2 m, 2 n + 1)$ .

Keywords:

AMSC: 57K31, 57K10, 57M12

References

1. S. Boyer, C. Gordon and L. Watson , On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013) 1213–1245. Web of Science, Google Scholar
2. G. Burde and H. Zieschang , Knots, De Gruyter Studies in Mathematics, Vol. 5 (De Gruyter, Berlin, 2003). Google Scholar
3. M. Dabkowski, J. Przytycki and A. Togha , Non-left-orderable 3-manifold groups, Can. Math. Bull. 48 (2005) 32–40. Web of Science, Google Scholar
4. C. Gordon , Riley’s conjecture on $SL (2, ℝ)$ representations of $2$ -bridge knots, J. Knot Theory Ramif. 26 (2017) 1740003. Link, Web of Science, Google Scholar
5. Y. Hu , The left orderability and cycle branched coverings, Algebraic Geom. Topol. 15 (2015) 399–413. Web of Science, Google Scholar
6. V. Khoi , A cut-and-paste method for computing the Seifert volumes, Math. Ann. 326 (2003) 759–801. Web of Science, Google Scholar
7. B. Meyer and A. Tran , Left orderability of cyclic branched covers of rational knots $C (2 n + 1, 2 m, 2)$ , Topol. Appl. 310 (2022) 108018. Web of Science, Google Scholar
8. P. Ozsváth and Z. Szabó , On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300. Web of Science, Google Scholar
9. R. Riley , Nonabelian representations of 2-bridge knot groups, Q. J. Math. Oxford Ser. (2) 35 (1984) 191–208. Web of Science, Google Scholar
10. A. Tran , Nonabelian representations and signatures of double twist knots, J. Knot Theory Ramif. 25 (2016) 1640013. Link, Web of Science, Google Scholar
11. A. Tran , On left-orderability and cyclic branched coverings, J. Math. Soc. Jpn. 67(3) (2015) 1169–1178. Web of Science, Google Scholar
12. H. Turner , Left-oderability, branched coverings and double twist knots, Proc. Am. Math. Soc. 149 (2021) 1343–1358. Web of Science, Google Scholar