To study the topology of polarization fields in homogeneous anisotropic media, we formulate the classic Christoffel equation in the polarization variables and solve it for the slowness vectors of plane waves corresponding to a given polarization. This task might emerge in passive seismology when neither ray nor wavefront-normal direction of a body wave recorded by a single three-component seismometer is available, so that velocities or slownesses of plausible wave arrivals have to be inferred from the recorded direction of particle motion. Our analysis shows that, unless the Christoffel equation degenerates and yields an infinite number of different slowness vectors, the finite nonzero number of its real-valued solutions varies from one to four. Unexpectedly, we find a subset of triclinic solids in which the polarization field contains holes — there exist finite-size solid angles of polarization directions unattainable to any plane wave.

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