Twisted Alexander polynomials of 2-bridge knots for parabolic representations

TL;DR

This paper demonstrates that twisted Alexander polynomials linked to parabolic representations can determine fiberedness and genus of many 2-bridge knots, confirming a conjecture for numerous hyperbolic knots.

Contribution

It establishes a connection between twisted Alexander polynomials and key knot invariants for a broad class of 2-bridge knots, confirming a conjecture for many hyperbolic cases.

Findings

Twisted Alexander polynomial determines fiberedness of 2-bridge knots.

It also determines the genus of these knots.

The results confirm a conjecture for infinitely many hyperbolic knots.

Abstract

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.