关于Chebyshev多项式和环面纽结
On Chebyshev polynomials and torus knots
摘要：
在这项工作中，我们证明了q-数和它们的两个参数的推广，q，p-数，可以用来获得一些多项式不变量的环面结和链接。首先，我们证明了与切比雪夫多项式密切相关的q-数也可以与环面纽结类T（s，2）的Alexander多项式相关，s为奇数，并用于寻找相应的绞结关系。然后，我们发展了这一过程，借助于q，p-数，得到了广义二元Alexander多项式，并证明了它们与HOMFLY多项式的直接联系和HOMFLY多项式的skein关系。
英文摘要：
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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