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

Abstract

In this paper we prove that every coefficient of twisted Alexander polynomials of torus knots associated with irreducible ${\mathrm{SL}}_n(\mathbb{C})$-representations is an $\mathbb{A}$-valued locally constant function on the ${\mathrm{SL}}_n(\mathbb{C})$-character variety, where $\mathbb{A}$ is the ring of all algebraic integers over $\mathbb{C}$. Moreover, as a generalization of a recent result of Kitano and Nozaki, we show that ${\mathrm{SL}}_n(\mathbb{C})$-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.

Accepted for publication in Topology and its Applications. Some results in Section 6 overlap with the recent preprint arXiv:2605.19460 of Terashima and Yamaguchi