Excitable and Magnetic Knots

Three-dimensional excitable media host vortex filaments that can be created with a range of knotted and linked topologies. The evolution of these excitable knots and links is both complex and fascinating, as shown by examples of knot untangling and the collision of knots and links. Even the simple threading of a circular filament by other filaments is shown to produce exotic behaviour. This is illustrated within a chemical excitable medium via numerical simulations that are validated by experimental results. Magnetic systems provide another medium that may host knotted filaments. In this case the filament is a curve of constant magnetization and there is a conserved integer-valued topological invariant, known as the Hopf charge. Computations for a frustrated magnet reveal that minimal energy configurations, called Hopfions, form knots and links for a sufficiently large Hopf charge.