Twisted Alexander polynomials and character varieties of 2-bridge knot groups (Q2888820)

The twisted Alexander polynomial for knots in the 3-sphere is a generalization of the classical Alexander polynomial and classical results about the Alexander polynomial of fibered knots extend to the twisted case. For instance, it is known that the twisted Alexander polynomials associated with nonabelian \(SL(2,\mathbb{C})\)-representations of a fibered knot are all monic and that if a knot of genus \(g\) is fibered then any of these polynomials is of degree \(4g-2\).NEWLINENEWLINEIn this paper, the authors consider the fibering problem for knots from the viewpoint of the \(SL(2,\mathbb{C})\)-character variety. They prove that for a nonfibered 2-bridge knot, there exists an irreducible curve component in the character variety of nonabelian \(SL(2,\mathbb{C})\)-representations of the knot which contains only a finite number of monic characters. In addition, they prove that for a (possibly nonfibered) 2-bridge knot of genus \(g\), in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree \(4g-2\).