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

Abstract

The explicit formula, which expresses the Alexander polynomials ∆n,3(t) of torus knots T(n, 3) as a sum of the Alexander polynomials ∆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 ∆n,3(t) through Chebyshev polynomials. The latter result is extended to general torus knots T(n, l) with n and l coprime.