Above we have a model for a link corresponding to the complete graph on 6 vertices. The "edges" are the 10 red loops (which together with the 5 green annuli acting as vertices already looks like a complete graph on 5 vertices), and the 5 blue loops (which correspond to the edges running from each of the 5 existing vertices out to the imaginary 6th vertex). The link, however, is slightly oversimplified. The actual link we are interested in is obtained from it by applying one full Dehn twist to each of the five green annuli. The local effect is modeled in the figure below.

If we focus on a sub-link consisting of two red link components that meet at a green annulus we note that the effect of the twist is that the sub-link goes from being a two component unlink to a Hopf link as shown below.