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Length of excitable knots.

Fabian Maucher1,2, Paul Sutcliffe1

  • 1Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom.

Physical Review. E
|January 20, 2018
PubMed
Summary

Numerical simulations show that FitzHugh-Nagumo evolution preserves knot topology in vortex strings. This method reveals a minimal string length for each knot, comparable to ideal knot ropelength.

Area of Science:

  • Complex systems
  • Knot theory
  • Fluid dynamics

Background:

  • Excitable media exhibit complex dynamic behaviors.
  • Knot theory is crucial for understanding topological structures.
  • Vortex strings in physical systems can form complex knots.

Purpose of the Study:

  • To investigate the long-term dynamics of knotted vortex strings in an excitable medium.
  • To explore the topological stability of knots under FitzHugh-Nagumo evolution.
  • To determine the minimal length of knotted vortex strings and compare it to ideal knot ropelength.

Main Methods:

  • Extensive numerical simulations of an excitable medium.
  • Applying the FitzHugh-Nagumo model to simulate vortex string evolution.
  • Analysis of knot topology preservation and minimal length determination for torus knots up to crossing number 11.

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Main Results:

  • FitzHugh-Nagumo evolution consistently preserves the knot topology of vortex strings.
  • A well-defined minimal length for each knot is achieved, comparable to ideal knot ropelength.
  • The medium boundary plays a significant role in stabilizing knot length.

Conclusions:

  • The FitzHugh-Nagumo model offers a viable field theory approach for studying knot dynamics.
  • Knot topology is stable under simulated excitable medium evolution.
  • There is no single attractor for a given knot topology, indicating diverse dynamic possibilities.