Colored HOMFLY polynomials for the pretzel knots and links

TL;DR

This paper computes HOMFLY polynomials for a broad family of pretzel links using the evolution method, revealing a new decompositional structure involving elementary polynomials and Racah matrices.

Contribution

It introduces a novel decomposition of HOMFLY polynomials for pretzel links into elementary components derived from evolution eigenvalues and Racah matrices.

Findings

Explicit formulas for symmetric HOMFLY polynomials of pretzel links.

Identification of a new structural decomposition involving elementary polynomials.

Potential extension to non-symmetric representations remains open.

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.