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Minimality of rational knots $C (2 n + 1, 2 m, 2)$

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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Anna Pham

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

E-mail Address: tpham22@math.wisc.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/S0218216524500019Cited by:0 (Source: Crossref)

Abstract

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots $C (2 n + 1, 2 m, 2)$ in the Conway notation, where $m \neq 0$ and $n \neq 0, - 1$ are integers. When $| m | \geq 2$ , we show that the nonabelian ${SL}_{2} (ℂ)$ -character variety of $C (2 n + 1, 2 m, 2)$ is irreducible and therefore $C (2 n + 1, 2 m, 2)$ is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.

Keywords:

AMSC: 57K10, 57K31

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