On Chebyshev polynomials and torus knots

TL;DR

This paper explores the use of q-numbers and q,p-numbers, related to Chebyshev polynomials, to derive polynomial invariants for torus knots, connecting them with Alexander and HOMFLY polynomials through skein relations.

Contribution

It introduces a novel approach using q,p-numbers to derive and connect polynomial invariants of torus knots with classical knot polynomials.

Findings

q-numbers relate to Alexander polynomials for T(s,2) knots

q,p-numbers lead to generalized two-variable Alexander polynomials

established connection between these polynomials and HOMFLY polynomials

Abstract

In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q,p-numbers, the generalized two-variable Alexander polynomials, and prove their direct connection with the HOMFLY polynomials and the skein relation of the latter.