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

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

2008-08-22

arxiv.org/abs/0808.3058v1

Mathematics

Geometric Topology

Scientific paper

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 $s_0^{-1}, \mu \in Z[s_0]$. The number $\mu$ can be recursively evaluated.

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