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arXiv · 1412.8432

Colored HOMFLY polynomials for the pretzel knots and links

A. Mironov·A. Morozov·A. Sleptsov
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Abstract

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g+1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU_q(N) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R.

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A. Mironov, A. Morozov, A. Sleptsov. 2015-04-02. Colored HOMFLY polynomials for the pretzel knots and links. https://doi.org/10.1007/jhep07(2015)069

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