Topological Aspect of Knotted Vortex Filaments in Excitable Media

TL;DR

This paper provides a rigorous topological description of knotted vortex filaments in excitable media, revealing that the Hopf invariant equals the sum of linking and self-linking numbers, suggesting new topological constraints.

Contribution

It introduces a topological framework using $$-mapping theory to relate the Hopf invariant to linking and self-linking numbers of vortex filaments.

Findings

Hopf invariant equals sum of linking and self-linking numbers.

Topological current theory applied to vortex filament charge density.

Potential new topological constraints on knotted vortex filaments.

Abstract

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.