Low-multipole kernel of polarization potentials ($\ker ({ð}^2)=\{\ell =0,1\}$)

Twice-applied spin-raising or lowering annihilates the ordinary spherical-harmonic multipoles $\ell =0,1$. Polarization potentials therefore have a four-dimensional monopole/dipole ambiguity on the sphere. For an axisymmetric mode it is $A+B\cos \theta$. Setting these components to zero defines the observable $\ell \ge 2$ potentials without changing $Q$ or $U$.