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

Understand

Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2 * Z/p$, $p$ 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 $s_0^{-1}, \mu \in Z[s_0]$.