Alexander Polynomial Invariants of Torus Knots T(n,3) and Chebyshev Polynomials

TL;DR

This paper derives explicit formulas linking Alexander polynomials of torus knots T(n,3) to Chebyshev polynomials, extending to general torus knots T(n,l) with coprime n and l, revealing new algebraic relationships.

Contribution

The paper provides explicit formulas expressing Alexander polynomials of T(n,3) in terms of T(k,2) and Chebyshev polynomials, extending these results to T(n,l) knots.

Findings

Explicit formula for elta_{n,3}(t) in terms of elta_{k,2}(t)

Expression of Alexander polynomials via Chebyshev polynomials

Extension to general torus knots T(n,l) with coprime n and l

Abstract

The explicit formula, which expresses the Alexander polynomials \Delta_{n,3}(t) of torus knots T(n,3) as a sum of the Alexander polynomials \Delta_{k,2}(t) of torus knots T(k,2), is found. Using this result and those from our previous papers, we express the Alexander polynomials \Delta_{n,3}(t) through Chebyshev polynomials. The latter result is extended to general torus knots T(n,l) with n and l coprime.