Stable and Unstable Vortex Knots in Excitable Media

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 behavior 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 characterization 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.