Here are a few pictures of another high-genus fullerene I made last month. This structure contains 12 necks, each of them is made of 5 octagons again.
To make this molecule, I have used approximately six long threads with around 4 meters each.
It is clear that we have to weave the inner part of a HG-fullerene first, in this case, 12 necks connected to each other.
Note also that this structure is, I believe, one of the two simplest high-genus fullerenes with icosahedral symmetry one can create. Another icosahedral HG-fullerene has 10 heptagons in each of 12 necks.