Twisted Alexander polynomials and ideal points giving Seifert surfaces

TL;DR

This paper demonstrates that the highest degree coefficient of twisted Alexander polynomials remains finite at certain ideal points associated with minimal genus Seifert surfaces, supporting a conjecture in knot theory.

Contribution

It proves that the highest degree coefficient function is finite at ideal points corresponding to minimal genus Seifert surfaces, advancing understanding of the relationship between Alexander polynomials and character varieties.

Findings

The highest degree coefficient function is finite at ideal points giving Seifert surfaces.

Supports a conjecture relating twisted Alexander polynomials to minimal genus Seifert surfaces.

Provides new insights into the structure of the $SL_2(C)$-character variety in knot theory.

Abstract

The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_2(\mathbb{C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler-Shalen theory. It implies a partial affirmative answer to a conjecture by Dunfield, Friedl and Jackson.