Stable and unstable vortex knots in excitable media

Abstract

We study the dynamics of knotted vortices in a bulk excitable medium using the FitzHugh-Nagumo model. From a systematic survey of all knots of at most eight crossings we establish that the generic behaviour is of unsteady, irregular dynamics, with prolonged periods of expansion of parts of the vortex. The mechanism for the length expansion is a long-range 'wave slapping' interaction, analogous to that responsible for the annihilation of small vortex rings by larger ones. We also show that there are stable vortex geometries for certain knots; in addition to the unknot, trefoil and figure eight knots reported previously, we have found stable examples of the Whitehead link and 62 knot. We give a thorough characterisation of their geometry and steady state motion. For the unknot, trefoil and figure eight knots we greatly expand previous evidence that FitzHugh-Nagumo dynamics untangles initially complex geometries while preserving topology. arXiv:1809.04567v2 [nlin.PS]

Shaping the longitudinal electric field component of light

“…Hence, it permits to consider the opposite case to the previous section, where we can retain the evanescent part of the beam, however we have to spectrally filter parts of the beam around the origin k 2 ⊥ = 0. Vortex knots have been studied in a range of different fields during the last two decades including water waves, 19 as soliton solutions in the Skyrme-Faddev model, 20 in frustrated magnets, 21 in excitable media [22][23][24][25] and in optics. 3, 26-28 A proposal for imprinting optical knots onto BECs has been made in.…”

Colonies of threaded rings in excitable media

“…Filament dynamics has a complicated dependence on local properties, such as curvature and twist [7,8], but even more elusive is the influence of global topological properties on filament motion. Numerical simulations reveal elaborate evolutions for filaments that are knotted or linked [9][10][11][12][13][14], including knot untangling without untying. These results make it apparent that an important mechanism that drives filament dynamics is the location of wave-front collision interfaces, but these evolve in a subtle manner due to small differences in the frequencies of wave emissions from differing parts of a filament.…”

“…It represents a quasi-two-dimensional arena in which to study filament motion induced by wave slapping due to differences in wave frequencies. This is a crucial mechanism that drives the more complicated motion of filament knots and links [9][10][11][12][13][14] in a fully three-dimensional context. The work presented here provides a clear motivation to extend current experimental studies on thrings [19], to advance capabilities that will provide experimental results for comparison to other systems displaying complex swimming dynamics and clustering, such as colloids, bacteria, and nematic liquid shells [24], where chemical signaling and topology also play a prominent role.…”

Threaded Rings that Swim in Excitable Media

“…To date, there are no experimental examples of knotted or linked filaments in any excitable medium. However, numerical simulations [10][11][12][13][14][15][16][17] provide predictions for their behaviour that are waiting to be confirmed by experiments, if ingenious methods can be engineered and implemented to create the required initial conditions and image the result.…”