Inscribe a tetrahedron in a sphere. Make small disks in the surface of the sphere around the corners of the tetrahedron. Take the preimage of these disks under the
Hopf fibration. You get four 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!