Octahedron Hopf preimage (corners)

Inscribe a octahedron in a sphere. Make small disks in the surface of the sphere around the corners of the octahedron. Take the preimage of these disks under the Hopf fibration. You get six linked rings. For each pair of rings originating from adjacent cube corners, add several lateral connections between the rings. Then stereographically project the whole thing onto 3D space and compress it radially using the atan function.
There, I already did that for you!