The fundamental interplay between topology and dynamics is revealed when knots and links form spontaneously during evolution of a system. This phenomenon is often manifested in material filaments such as agitated strings1 and excited molecules2. Examples of topologically complex three-dimensional fields, from across the sciences, also manifest knot dynamics. In contrast to material filaments, the whole of space of a knotted field must be filled in a way consistent with any localized knot structure. The idea of Lord Kelvin to describe atoms as vortex knots in ether3 has found its continuation in nonlinear field theories in the form of knotted, or, more generally, topological solitons4. Knotted solitons5,6 were introduced as energy minimizers of a three-dimensional nonlinear Lagrangian for a vector order parameter, the latter is similar to pseudo-spin in Bose-Einstein condensates (BECs)7,8 and multicomponent superconductors9,10, as well as the director field in liquid crystals11,12 and electric and magnetic field lines in null-solutions to Maxwell equations13,14. Other systems display persistent, tangled defect line structures, such as scroll wave sources in chemical and biological reaction-diffusion systems15, localized excitations in bistable metamaterials16, knotted disclination lines around colloids in nematic liquid crystals17, and quantized vortex filaments in turbulent superfluids18,19 and trapped matter waves20–22.