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On Chebyshev Polynomials and Torus Knots
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 wit...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Відділення фізики і астрономії НАН України
2010
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| Subjects: | |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/13295 |
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| Summary: | 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 to prove their direct connection with the HOMFLY polynomials and the skein relation of the latter. |
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