A polynomial defined by the $\mathit{SL}(2;\mathbb{C})$-Reidemeister torsion for a homology 3-sphere obtained by Dehn-surgery along a torus knot

TL;DR

This paper introduces a polynomial related to the $ ext{SL}(2; ext{C})$-Reidemeister torsion for homology 3-spheres obtained via Dehn surgery on torus knots, providing explicit formulas and relations.

Contribution

It presents an explicit formula for the polynomial using Chebyshev polynomials and establishes 3-term relations, advancing understanding of Reidemeister torsion in this context.

Findings

Explicit polynomial formula using Chebyshev polynomials.

3-term relations for the defined polynomials.

Connection between polynomial zeros and Reidemeister torsion.

Abstract

Let $M_n$ be a homology 3-sphere obtained by $\frac1n$-Dehn surgery along a $(p,q)$-torus knot. We consider a polynomial $\sigma_{(p,q,n)}(t)$ whose zeros are the inverses of the Reideimeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations. We give an explicit formula of this polynomial by using Tchebychev polynomials of the first kind. Further we also give a 3-term relations of these polynomials.