Given a finitely presented group \(G\) and a linear representation \(\rho\) of \(G\), one can define a rational function which is an invariant of \(G\), called the twisted Alexander polynomial associated with \(\rho\). These invariants generalise the classical Alexander polynomial for knots groups which is associated with the trivial \(1\)-dimensional representation. N. M. Dunfield et al. conjectured in [Exp. Math. 21, No. 4, 329–352 (2012; Zbl 1266.57008)] that the twisted Alexander polynomial of a hyperbolic knot group associated to a representation in \(SL(2,{\mathbb C})\) which is a lift of the holonomy detects the genus (equivalently the Thurston norm) and fibredness of the knot.
The authors consider the analogous question for oriented hyperbolic links and show that the natural generalisation of the above conjecture holds true for a family of \(2\)-bridge links, namely double twist links. They observe that the fibredness and genus of \(2\)-bridge links are already determined by the reduced Alexander polynomial, however they prove that there are double twist two-component links whose genus is not detected by the twisted Alexander polynomials associated to some parabolic representations, that is representations of the link group in \(SL(2,{\mathbb C})\) that send all the meridans of the link to matrices with trace equal to \(2\).