  The fundamental polarization singularities of monochromatic light are
normally associated with invariance under coordinated rotations: symmetry
operations that rotate the spatial dependence of an electromagnetic field by an
angle $\theta$ and its polarization by a multiple $\gamma\theta$ of that angle.
These symmetries are generated by mixed angular momenta of the form $J_\gamma =
L + \gamma S$ and they generally induce M\"obius-strip topologies, with the
coordination parameter $\gamma$ restricted to integer and half-integer values.
In this work we construct beams of light that are invariant under coordinated
rotations for arbitrary $\gamma$, by exploiting the higher internal symmetry of
'bicircular' superpositions of counter-rotating circularly polarized beams at
different frequencies. We show that these beams have the topology of a torus
knot, which reflects the subgroup generated by the torus-knot angular momentum
$J_\gamma$, and we characterize the resulting optical polarization singularity
using third-and higher-order field moment tensors, which we experimentally
observe using nonlinear polarization tomography.
