On the twisted Alexander polynomial for representations into \(SL_{2}(\mathbb C)\) (Q2859566)

The twisted Alexander polynomial is an invariant of a knot together with (the conjugacy class of) a representation of the knot group. A representation is said to be non-abelian if its image is a non-abelian group.NEWLINENEWLINEThis paper studies two particular families of non-fibred 2-bridge knots. One family consists of twist knots. Each member of the other family is a `2-bridge knot assigned to a pair of relatively prime odd integers'. For these families, the author gives the number of non-abelian representations into \(SL(2,\mathbb{C})\) for which the degree of the twisted Alexander polynomial is strictly less than the maximal value of \(4g-2\) (where \(g\) is the genus of the knot), and the number for which the polynomial is monic. These numbers are established by direct calculation using Chebyshev polynomials.