Colored knot polynomials for arbitrary pretzel knots and links

D Galakhov,D Melnikov,A Mironov,A Morozov,A Sleptsov

doi:10.1016/j.physletb.2015.02.029

Abstract

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of pretzel knots and links. The answer for the Jones and HOMFLY is fully and explicitly expressed through the Racah matrix of Uq(SUN), and looks related to a modular transformation of toric conformal block.

Highlights

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g + 1)parametric family of Pretzel knots and links
Orientation of lines does not matter, when one considers the Jones polynomials. These polynomials are defined only for the symmetric representations [r] and, do not change under arbitrary permutations of parameters ni
The answer for the colored Jones polynomials for this entire family can be written in full generality, and is wonderfully simple r g r

Summary

Introduction

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g + 1)parametric family of Pretzel knots and links. Orientation of lines does not matter, when one considers the Jones polynomials (not HOMFLY!). These polynomials are defined only for the symmetric representations [r] and, do not change under arbitrary permutations of parameters ni (though the knot/link itself has at best the cyclic symmetry ni −→ ni+1, and even this is true only for particular orientations).

Results

Conclusion