On the twisted Alexander polynomial for metabelian representations into \(\mathrm{SL}_2(\mathbb C)\)
Let \(K\) be a knot in \(S^3\), and denote \(\pi\) for \(\pi_1(S^3-K)\). Assume \(\rho\in \mathrm{Hom}(\pi,\mathrm{SL}_2(\mathbb{C}))\) is irreducible. The group \(\pi\) is finitely presented as \(\langle g_1,\dots,g_k\;|\;r_1,\dots,r_{k-1}\rangle\) (Wirtinger presentation), and its Abelianization \(\pi/[\pi,\pi]\) is \(\langle t\rangle\cong \mathbb{Z}\); where \(t\) is a multiplicative generator. Define \(\Phi:\mathbb{Z}[\pi]\to \mathrm{End}(\mathbb{C}[t^{\pm t}]\times \mathbb{C}[t^{\pm t}] )\) by \(\sum a_i\gamma_i\mapsto \sum a_i\alpha(\gamma_i)\rho(\gamma_i)\) where \(\alpha:\pi\to \langle t\rangle\) is the Abelianization map. In these terms, \textit{M. Wada} [Topology 33, No. 2, 241--256 (1994; Zbl 0822.57006)] defined the \textit{twisted Alexander polynomial} as \(\det\left(\Phi(\partial r_i/\partial g_j)_{1\leq i,j\leq k-1}\right)/\det(\Phi(g_k-1));\) assuming \(\alpha(g_k)\not=1\) and where \(\partial r_i/\partial g_j\) are the Fox derivatives. It does not depend on the presentation, or the conjugacy class of \(\rho\). Similarly one can define the twisted Alexander polynomial for \(\mathrm{Ad}_\rho\). A representation is called \textit{metabelian} if \(\rho([\pi,\pi])\) is Abelian. An irreducible metabelian representation \(\rho\) is \textit{longitude-regular} if and only if there exist \(v\in \mathfrak{sl}_2(\mathbb{C})\) such that \(v\otimes \lambda\) is non-trivial in \(H_1(S^3-K;\mathrm{Ad}_\rho)\), where \(\lambda\) is a longitude in the boundary torus of \(S^3-K\). The main result in this paper gives an explicit factorization of the twisted Alexander polynomial of \(\mathrm{Ad}_\rho\) for irreducible metabelian longitude-regular representations \(\rho\).
scientific article; zbMATH DE number 6241361
scientific article