I will be doing a few presentations at an undergraduate research conference later, the topics of which are somewhat tentative at the moment, but one of the things I wanted to go over were some of the introductory bits (say, first 10ish pages) in the paper On Tutte's Chromatic Invariant, and with an advisor I found an excellent idea for a visual aid demonstration on how the trace form on the partition algebra can be seen as a monomial in terms of the join operation by using a spherical, baseball-esque covering.
These are what the partition diagrams I am interested in should look like (the middle one):
Notice that in the middle one, there is "vertex-splitting." For non-planar partitions, different shaded regions may overlap, and we have to choose (arbitrarily) which one will go over which. What I want to do, conceptually, is put two partition diagrams over top each other and glue their boundaries, then blow it up so that it is a sphere: one side of the sphere will show one partition diagram, and the other side will show the other partition diagram. The resulting figure will illustrate the "trace form" $\rm tr(p_\pi p_\sigma^{t})$ of the two partitions $\pi$ and $\sigma$ (this is detailed in the linked paper, though somewhat tersely.)
Ideally, then, the sphere and the shaded regions will look very pretty and I will be able to rotate it at my whim, and I will be able to animate say a dozen and a half of these figures and put them onto a slide.
What sorts of techniques or functions or whatever would Mathematica veterans recommend a newbie look into to accomplish this task?