I'm trying to understand how "coloring" the component of a link changes the link. I'm looking at the picture for the section on the colored Jones polynomial on the link provided, and was wondering if somebody could tell me what the contents of the rectangle with the label [n cable]. Am i correct in assuming that the larger rectangle on the left just has n copies of the link L running parallel to each other? Thanks!!
https://en.wikipedia.org/wiki/Jones_polynomial#Colored_Jones_polynomial
The rectangle with contents "$N$ cable" does not represent a block of the link diagram. It labels the number of duplicates of the knot in the cabling, in particular, there are $N$ copies of the knot in the cabling.
The poorly described diagram seems to be representing a knot as an unspecified braid, represented by the large empty rectangle on the left, that has been closed, represented by the four strands that meet the large rectangle along its top and bottom edges and pass around to the right of the diagram, that has an $N$-cabling, represented by the label that you ask about. (For the description as a closed braid, compare/contrast Figure 4 at https://arxiv.org/pdf/1804.07910.pdf .) For Jones polynomial calculations, representing the knot as a closed braid is a substantial (conceptual) aid to computation.
A cabling is a specific kind of satellite knot. Start with a closed regular neighborhood, $n(K)$, of the knot $K$. (This neighborhood is homeomorphic to a solid torus, but rather than imagine a "doughnut", unless you really are thinking of the colored Jones polynomial of the unknot, the solid torus is knotted.) On the surface of $n(K)$, draw a torus knot. This torus knot is a cabling of $K$.
For the purpose of the colored Jones polynomial, it matters how many longitudinal copies of the knot are cut by a meridional disk of the solid torus (minimized under isotopy), but it doesn't really matter how they revolve around the core of the solid torus as they proceed along the knot as long as each strand closes itself off. That is, as long as the number of $2\pi/N$ revolutions is a multiple of $N$. (In general, if there are $m$ of $2\pi/N$ revolutions, there are $\gcd(m,N)$ parallel closed curves on the surface, each of which is an $N/\gcd(m,N)$ cabling of the original knot.)