Narrow Results

Let $M$ be a non-abelian semi-direct product of a cyclic group $\mathbb{Z}/n$ and an elementary abelian $p$-group $A=\oplus k(\mathbb{Z}/p)$ 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 ${\mathrm{\Delta }}_{\gamma ,K}(t)$ associated to $\gamma :G(K)\to M\to GL(p^k,\mathbb{Z})$ is of the form: $\frac{{\mathrm{\Delta }}_K(t)}{1-t}F(t^n)$, where ${\mathrm{\Delta }}_K(t)$ 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.