Topological Aspect of Knotted Vortex Filaments in Excitable Media

Scroll waves exist ubiquitously in three-dimensional excitable media. The rotation centre can be regarded as a topological object called the vortex filament. In three-dimensional space, the vortex filaments usually form closed
loops, and can be even linked and knotted. We give a rigorous topological description of knotted vortex filaments. By using the Ф-mapping topological current theory, we rewrite the topological current form of the charge density of vortex filaments, and using this topological current we reveal that the Hopf invariant of vortex filaments is just the sum of the linking and self-linking numbers of the knotted vortex filaments. We think that the precise expression of the Hopf invariant may imply a new topological constraint on knotted vortex filaments.