Twisted Alexander polynomials and incompressible surfaces given by ideal points

TL;DR

This paper explores the relationship between twisted Alexander polynomials and incompressible surfaces in 3-manifolds, showing that certain algebraic functions associated with these polynomials remain finite at specific ideal points related to Thurston norm minimizing surfaces.

Contribution

It establishes a connection between ideal points of character varieties and incompressible surfaces via twisted Alexander polynomials, extending Culler-Shalen theory.

Findings

Regular functions from twisted Alexander polynomials are finite at ideal points.

Ideal points corresponding to Thurston norm minimizing surfaces relate to finite algebraic values.

The work links algebraic invariants to geometric structures in 3-manifolds.

Abstract

We study incompressible surfaces constructed by Culler-Shalen theory in the context of twisted Alexander polynomials. For a $1$st cohomology class of a $3$-manifold the coefficients of twisted Alexander polynomials induce regular functions on the $SL_2(\mathbb{C})$-character variety. We prove that if an ideal point gives a Thurston norm minimizing non-separating surface dual to the cohomology class, then the regular function of the highest degree has a finite value at the ideal point.