Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants

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.

7 pages, two-column UJP style, talk at the 3rd Walter Thirring International School on Fundumentals of Astroparticle and Quantum Physics (17-23 September, 2017, BITP, Kyiv, Ukraine)