Twisted Alexander polynomials on curves in character varieties of knot groups

TL;DR

This paper investigates the relationship between twisted Alexander polynomials and knot properties, proposing new insights into how character varieties can detect fiberedness and genus of knots, especially for nonfibered knots.

Contribution

It explores the existence of specific curve components in character varieties related to monic twisted Alexander polynomials and their role in detecting knot fiberedness and genus.

Findings

Identifies a curve component related to the conjecture for nonmonic Alexander polynomials.

Discusses the potential of character varieties to detect knot fiberedness.

Provides theoretical insights into the relationship between twisted Alexander polynomials and knot invariants.

Abstract

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