On T(n,4) torus knots and Chebyshev polynomials

TL;DR

This paper explores expressing Alexander polynomials of certain torus knots as sums of Chebyshev polynomials, introducing new numerical invariants for links and knots based on these expansions.

Contribution

It presents a novel representation of Alexander polynomials for T(n,4) torus knots using Chebyshev polynomials, enabling new invariants for knot characterization.

Findings

Alexander polynomials expressed as sums of Chebyshev polynomials

Introduction of numerical invariants for links and knots

Potential for new classification tools based on polynomial expansions

Abstract

The Alexander polynomials \Delta_{n,3}(t) and \Delta_{n,4}(t) are presented as a sum of the Alexander polynomials \Delta_{k,2}(t). These polynomials are also expressed in the form of a sum of Chebyshev polynomials of the second kind. These expansions allow one to introduce the "coordinates" in corresponding bases, which are proposed to be the numerical invariants characterizing links and knots.