Stable and Unstable Vortex Knots in Excitable Media

TL;DR

This paper investigates the behavior of knotted vortices in excitable media modeled by FitzHugh-Nagumo, revealing both unstable dynamics and stable configurations for certain knots, with implications for understanding vortex evolution.

Contribution

It provides a systematic survey of knotted vortex dynamics, identifies stable vortex geometries, and characterizes their steady states within the FitzHugh-Nagumo model.

Findings

Unsteady, irregular vortex dynamics dominate most knots.

Stable vortex configurations exist for specific knots like Whitehead link and 6_2 knot.

FitzHugh-Nagumo dynamics can untangle complex geometries while preserving topology.

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 $6_2$ 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.