Harmonics on the factored three-sphere and the Hopf map

Laplacian eigenmodes on homogeneous Clifford–Klein factors of the three–sphere are obtained as pullbacks of harmonics on the orbifolded two–sphere using the Hopf map. A method of obtaining these polyhedral, or crystal, harmonics using binary invariants is presented which has computational advantages over those based on projection techniques, or those using invariants constructed in terms of Cartesian coordinates. In addition, modes transforming according to the irreps of the deck group are found in easy fashion using the covariants already conveniently calculated by Desmier and Sharp and by Bellon. Agreement is found with existing results.