On Chebyshev polynomials and torus knots

Details On Chebyshev polynomials and torus knots On Chebyshev polynomials and torus knots

2009-12-23

arxiv.org/abs/0912.4674v2

Ukr. J. Phys. 55 (2010) 129-134

Physics

Mathematical Physics

6 pages (two-column UJP style)

Scientific paper

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.

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