Knotted polarizations and spin in three-dimensional polychromatic waves

We consider complex three-dimensional polarizations in the interference of several vector wave fields with different commensurable frequencies and polarizations. We show that the resulting polarizations can form knots, and interfering three waves is sufficient to generate a variety of Lissajous, torus, and other knot types. We describe the spin angular momentum, generalized Stokes parameters, and degree of polarization for such knotted polarizations, which can be regarded as partially polarized. Our results are generic for any vector wave fields, including, e.g., optical and acoustic waves. As a directly observable example, we consider knotted trajectories of water particles in the interference of surface water (gravity) waves with three different frequencies.