ABSTRACT
It's easy to see that the three solutions in Figure 1 have this toroidal property. But it's not at all obvious there aren't other, non-toroidal solutions (or, for that matter, other toroidal solutions). We'll present a somewhat unsatisfactory proof below (unsatisfactory in that we'd like to encourage people to look for a simpler, more conceptual proof-ours is partly a brute-force process of elimination). But first, we need to in troduce another, non-geometric symmetry that's hidden in the problem, which we'll call existential [Sar56] symmetry.
