Cabling procedure for the colored HOMFLY polynomials

TL;DR

This paper details a systematic cabling procedure for computing colored HOMFLY polynomials, including the derivation of projectors and R-matrices, with theoretical insights into the group structure involved.

Contribution

It provides explicit matrix forms for projectors and R-matrices, enabling the calculation of HOMFLY polynomials in any representation and knot, and offers a group theory explanation of the cabling method.

Findings

Explicit matrix forms for projectors and R-matrices are constructed.

The cabling procedure can, in principle, compute HOMFLY polynomials in any representation.

Group theory explains the structure of fundamental R-matrices and supports existing conjectures.

Abstract

In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and $\mathcal{R}$-matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{R}$-matrices allow one in principle (neglecting the computational difficulties) to find the HOMFLY polynomial in any representation for any knot. We also discuss the group theory explanation of the cabling procedure. This leads to the explanations of the form of the fundamental $\mathcal{R}$-matrices and illuminates several conjectures proposed in previous papers.