Topological Aspect of Knotted Vortex Filaments in Excitable Media

Details Topological Aspect of Knotted Vortex Filaments in Excitable Media Topological Aspect of Knotted Vortex Filaments in Excitable Media

2007-12-27

arxiv.org/abs/0712.4196v1

Chin. Phys. Lett.25, 353-356(2008)

Nonlinear Sciences

Pattern Formation and Solitons

4 pages, no figures, Accepted by Chin. Phys. Lett

Scientific paper

10.1088/0256-307X/25/2/001

Scroll waves exist ubiquitously in three-dimensional excitable media. It's rotation center can be regarded as a topological object called vortex filament. In three-dimensional space, the vortex filaments usually form closed loops, and even linked and knotted. In this letter, we give a rigorous topological description of knotted vortex filaments. By using the $\phi$-mapping topological current theory, we rewrite the topological current form of the charge density of vortex filaments and use 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.

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