Algebraic properties of twisted Alexander polynomial and Reidemeister torsion of torus knots

TL;DR

This paper investigates algebraic properties of twisted Alexander polynomials and Reidemeister torsion for torus knots, revealing their algebraic integer nature and exploring their relation to TQFT.

Contribution

It proves coefficients of twisted Alexander polynomials are algebraic integers and extends results on Reidemeister torsion to Seifert fibered spaces.

Findings

Coefficients are algebraic integers over the character variety.

Reidemeister torsions are algebraic integers for many Seifert fibered spaces.

Power sums of Reidemeister torsions relate to TQFT.

Abstract

In this paper we prove that every coefficient of twisted Alexander polynomials of torus knots associated with irreducible $\mathrm{SL}_n(\Bbb C)$-representations is an $\Bbb A$-valued locally constant function on the $\mathrm{SL}_n(\Bbb C)$-character variety, where $\Bbb A$ is the ring of all algebraic integers over $\Bbb C$. Moreover, as a generalization of a recent result of Kitano and Nozaki, we show that $\mathrm{SL}_n(\Bbb C)$-Reidemeister torsions are algebraic integers for many Seifert fibered spaces. Also, we discuss the power sums of Reidemeister torsions of torus knots for low-dimensional irreducible representations that provide a mysterious relation to TQFT.