Vortex filament equation and its solutions and energies
Epperson, Kelly Scott
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First, we will give the derivation of the Vortex Filament Equation (VFE) and discuss the concept of conserved quantities. Then we will discuss Fourier knots and their relationship to the VFE. We will also give some results pertaining to Fourier knots evolving under the VFE, and make some conjectures on future avenues of research. Finally, we will discuss knot energies, focusing on the Mobius or O‚ÄôHara energy and the Brylinski Beta function of a knot, again presenting results and offering conjectures. We will show the results of numerical experiments at appropriate times in the text.