Colored HOMFLY polynomials for the pretzel knots and links

Abstract

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 SUq (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.