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

TL;DR

This paper investigates the values of twisted Alexander polynomials at ±1 for certain 2-bridge knots, revealing algebraic relations involving a specific algebraic integer and providing a recursive method to evaluate associated parameters.

Contribution

It establishes explicit formulas for the twisted Alexander polynomial at ±1 for knots in the set H(p), linking these values to an algebraic integer and a recursively computable parameter.

Findings

elta(1) = -2s_0^{-1}

elta(-1) = -2s_0^{-1}^2

 recursive evaluation method for nd 

Abstract

Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2 * Z/p$, $p$ 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.