Cabling procedure for the colored HOMFLY polynomials

Abstract

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and $\mathcal{R}$ -matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and $\mathcal{R}$ -matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental $\mathcal{R}$ -matrices and clarifying some conjectures formulated in previous papers.