Excitable knots


There are a range of chemical, physical and biological excitable media that support spiral wave vortices. Examples include the Belousov-Zhabotinsky redox reaction, the chemotaxis of slime mould and action potentials in cardiac tissue. Usually spatio-temporal three-dimensional interacting vortex strings, e.g. superfluid vortices [Kleckner16], are prone to reconnection and untying. However, it appears that there are types of reaction-diffusion equations - for example the so-called FitzHugh-Nagumo equations which constitute the simplest mathematical model of cardiac tissue as an excitable medium - that may give rise to strong short-ranged repulsive interactions between vortex strings. The latter may lead to a preservation of topology of vortex strings [Winfree84,Sutcliffe03]. We use a Biot-Savart construction to initialize a given knot as a vortex string in the FitzHugh-Nagumo equations. We show, that the evolution can be used to untangle knotted vortex strings [Maucher16]. Furthermore, the evolution yields a well-defined minimal length for a range of knots that is comparable to the ropelength of ideal knots [Maucher17]. We highlight the role of the medium boundary in stabilizing the length of the knot and discuss the implications beyond torus knots. By applying Moffatt’s test we show that there is not a unique attractor within a given knot topology.

[Kleckner16] D. Kleckner, L. H. Kauffman, W. T. M. Irvine, Nature Physics 12, 650 (2016)

[Winfree84] A. T. Winfree and S. H. Strogatz, Nature 311, 611 (1984)

[Sutcliffe03] P. Sutcliffe, A.T. Winfree Phys. Rev. E 68, 016218 (2003)

[Maucher16] F. Maucher, P. Sutcliffe, Phys. Rev. Lett. 116, 178101 (2016)

[Maucher17] F. Maucher, P. Sutcliffe, (submitted) (2017)