Evaluations of the twisted Alexander polynomials of 2-bridge knots at $\pm 1$

Abstract

Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2\ast Z/p$, $p$ being odd. Then there is an algebraic integer $s_0$ such that for any $K$ in $H(p)$, $G$ has a parabolic representation $\rho$ into $SL(2,Z[s_0])\subset SL(2,C)$. Let $\Delta (t)$ be the twisted Alexander polynomial associated to $\rho$. Then we prove that for any $K$ in $H(p)$, $\Delta (1)=-2s_0^{-1}$ and $\Delta (-1)=-2s_0^{-1}{\mu }^2$, where $s_0^{-1},\mu \in Z[s_0]$. The number $\mu$ can be recursively evaluated.