The fundamental polarization singularities of light are generally symmetric under coordinated rotations: that is, transformations which rotate the spatial dependence of the fields by an angle $\theta$ and the field polarization by a fraction of that angle, as generated by 'mixed' angular momenta of the form $L + S$. Generically, the coordination parameter has been thought to be restricted to integer or half-integer values. In this work we show that this constraint is an artifact of the restriction to monochromatic elds, and that a wider variety of optical singularities is available when more than one frequency is involved. We show that these new optical singularities present novel field topologies, isomorphic to torus knots, and we show how they can be characterized both analytically and experimentally. Moreover, the generator for the symmetry group of these singularities, whose algebraic structure is deeply related to the torus-knot topology of the beams, is conserved in nonlinear optical interactions.