Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants

TL;DR

This paper introduces a new class of generalized Chebyshev polynomials of hyperkind h and kind k, exploring their properties and connections to Alexander knot and link invariants.

Contribution

It extends standard Chebyshev polynomials to a two-parameter family and investigates their relationship with knot invariants.

Findings

Defined generalized equidistant Chebyshev polynomials T(k,h)

Established connections with Alexander knot and link invariants

Explored properties through horizontal and vertical generalizations

Abstract

We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k,h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind h and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invariants are investigated.