Twisted Alexander polynomials of tunnel number one Montesinos knots

TL;DR

This paper computes the twisted Alexander polynomials for all tunnel number one Montesinos knots using their $SL_2(C)$-representations, revealing new properties about their coefficients and degrees, and providing examples of knots with specific polynomial characteristics.

Contribution

It introduces a comprehensive calculation of twisted Alexander polynomials for this class of knots, highlighting differences from classical Alexander polynomials and identifying nonfibered knots with monic classical but non-monic twisted polynomials.

Findings

Computed twisted Alexander polynomials for all tunnel number one Montesinos knots.

Identified knots with monic classical Alexander polynomials but non-monic twisted polynomials.

Provided explicit leading coefficients and degrees for these polynomials.

Abstract

We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As a corollary, we get some interesting examples, that is, nonfibered knots with monic Alexander polynomials which have non-monic twisted Alexander polynomials.