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Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

Mikami Hirasawa

http://orcid.org/0000-0002-8219-1263

Department of Mathematics, Nagoya Institute of Technology, Nagoya, Aichi 466-8555, Japan

E-mail Address: hirasawa.mikami@nitech.ac.jp

Corresponding author.

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and

Kunio Murasugi

Department of Mathematics, University of Toronto, Toronto, ON M5S2E4, Canada

E-mail Address: murasugi@math.toronto.edu

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https://doi.org/10.1142/S0218216518500153Cited by:1 (Source: Crossref)

Abstract

Let $M$ $M$ be a non-abelian semi-direct product of a cyclic group $ℤ / n$ and an elementary abelian $p$ -group $A = \oplus k (ℤ / p)$ of order $p^{k}$ , $p$ being a prime and $gcd (n, p) = 1$ . Suppose that the knot group $G (K)$ of a knot $K$ in the $3$ -sphere is represented on $M$ . Then we conjectured (and later proved) that the twisted Alexander polynomial $Δ_{γ, K} (t)$ associated to $γ : G (K) \to M \to G L (p^{k}, ℤ)$ is of the form: $\frac{Δ_{K} (t)}{1 - t} F (t^{n})$ , where $Δ_{K} (t)$ is the Alexander polynomial of $K$ and $F (t^{n})$ is an integer polynomial in $t^{n}$ . In this paper, we present a proof of the following. For a $2$ -bridge knot $K (r)$ in $H (p)$ , if $n = 2$ and $k = 1$ , then $F (t^{2})$ is written as $f (t) f (t^{- 1})$ , where $H (p)$ is the set of $2$ -bridge knots whose knot groups map on that of $K (1 / p)$ with $p$ odd.

Keywords:

AMSC: 57M25, 57M27

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