Twisted Alexander polynomials on Mangum-Shanahan curves (Q6621557)

Let \(M_n\) be a once-punctured torus bundle with tunnel number one. The manifold \(M_n\) is a finite volume hyperbolic \(3\)-manifold with exactly one cusp when \(|n|> 2.\) So, up to homeomorphism, the \(M_n\)'s form an infinite family of \(3\)-manifolds, \(\{M_n\}_{n \in \mathbb{Z}}\). Let \(\rho_{\tau}:\pi_1(M_n)\rightarrow SL_3(\mathbb{C})\) be a Mangum and Shanahan representation. The authors provide an explicit formula for the twisted Alexander polynomial of \(M_n\) associated with \(\rho_{\tau}:\pi_1(M_n)\rightarrow SL_3(\mathbb{C})\) by using the presentation \(\pi_1(M_n)=\langle\alpha,\beta \mid \beta^n=\omega\rangle,\) where \(\omega\) is a simple word in the generators \(\alpha, \beta\) independent of \(n\). The authors also show that the Reidemeister torsion tor\((M_{-3}, \rho_{\tau})\) of the complement of the figure-eight knot \(M_{-3}\) is constant on Mangum-Shanahan curves \(\rho_{\tau}\) \((\tau \in \mathbb{C}\setminus \{0\})\).