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For a fibered knot in the 3-sphere the twisted Alexander polynomialassociated to an SL(2,C)-character is known to be monic. It is conjectured thatfor a nonfibered knot there is a curve component of the SL(2,C)-charactervariety containing only finitely many characters whose twisted Alexanderpolynomials are monic, i.e. finiteness of such characters detects fiberednessof knots. In this paper we discuss the existence of a certain curve componentwhich relates to the conjecture when knots have nonmonic Alexander polynomials.We also discuss the similar problem of detecting the knot genus.

Twisted Alexander polynomials on curves in character varieties of knot groups