To any given pair consisting in the fundamental group \(G\) of the exterior of a knot \(K\) and a linear representation \(\rho\) of \(G\), one can associate an invariant called the twisted Alexander polynomial of \(K\) associated to \(\rho\). These generalise the classical Alexander polynomial which corresponds to the case of the trivial, one-dimensional representation.
The author computes the twisted Alexander polynomials associated to representations into \(\mathrm{SL}_2({\mathbb C})\) for the family of hyperbolic pretzel knots of the form \((-2,3,2n+1)\). The representations considered contain the holonomy representations, that is lifts of the discrete and faithful representaion of the knot group into \(\mathrm{PSL}_2({\mathbb C})\) defining the hyperbolic structure of the knot complement.
The computations show that a conjecture of N. M. Dunfield, S. Friedl and N. Jackson [Exp. Math. 21, No. 4, 329–352 (2012; Zbl 1266.57008)] holds for the class of knots considered. The conjecture states that the twisted Alexander polynomials associated to the holonomy representations detect the genus and the fibredness of the knots, in the following sense: the polynomials are of degree \(4g-2\), where \(g\) is the genus of the knot, and are monic if and only if the knot is fibred. For instance, in the case of the family considered, the knots are hyperbolic provided that \(n\neq 0,1,2\) and fibred for all \(|n|>1\); for \(n\neq 0,1,2\), their twisted Alexander polynomials have degrees \(4g-2\), where \(g=|n+1|+1\), and are monic if and only if \(|n|>1\).