Knotted and linked phase singularities in monochromatic waves

Exact solutions of the Helmholtz equation are constructed, possessing wavef ront dislocation lines (phase singularities) in the form of knots or links where the wave function vanishes ('knotted nothings'). The construction pro ceeds by making a nongeneric structure with a strength n dislocation loop t hreaded by a strength m dislocation line, and then perturbing this. In the resulting unfolded (stable) structure, the dislocation loop becomes an (in, it) torus knot if in and n are coprime, and N linked rings or knots if rn and n have a common factor N: the loop or rings are threaded by an m-strand ed helix. In our explicit implementation, the wave is a superposition of Be ssel beams,, accessible to experiment. Paraxially, the construction fails.