A polynomial defined by the SL(2;C)-Reidemeister torsion for a homology 3-sphere obtained by a Dehn surgery along a (2p,q)-torus knot

TL;DR

This paper introduces a polynomial related to the SL(2;C)-Reidemeister torsion for 3-manifolds obtained via Dehn surgery on (2p,q)-torus knots, generalizing previous formulas using Chebyshev polynomials.

Contribution

The authors generalize Johnson's formula for the (2,3)-torus knot to arbitrary (2p,q)-torus knots using Chebyshev polynomials.

Findings

Derived a generalized polynomial formula for (2p,q)-torus knots

Connected Reidemeister torsion zeros to polynomial roots

Extended previous knot theory results

Abstract

Let K be a (2p,q)-torus knot and M_n is a 3-manifold obtained by 1/n-Dehn surgery along K. We consider a polynomial whose zeros are the inverses of the Reideimeister torsion of M_n for SL(2;C)-irreducible representations. Johnson gave a formula for the case of the (2,3)-torus knot under some modification and normalization. We generalize this formula by using Tchebychev polynomials.