The Trefoil Soliton

The Kiepert trefoil is an algebraic curve with remarkable geometric and number theoretic properties. Ludwig Kiepert, generalizing ideas due to Serret and Liouville, determined that it could be parametrized by arc length in terms of elliptic functions. In this note, we observe some other properties o...

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Bibliographic Details
Published in:Mathematics (Basel) 2022-05, Vol.10 (9), p.1512
Main Author: Singer, David A.
Format: Article
Language:English
Subjects:
Conflicts of interest
Elliptic functions
Evolution
filament equation
Geometry
Lagrange multiplier
Solitary waves
soliton
Three dimensional flow
trefoil
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Summary:The Kiepert trefoil is an algebraic curve with remarkable geometric and number theoretic properties. Ludwig Kiepert, generalizing ideas due to Serret and Liouville, determined that it could be parametrized by arc length in terms of elliptic functions. In this note, we observe some other properties of the curve. In particular, the curve is a special example of a buckled ring, and thus a solitary wave solution to the planar filament equation, evolving by rotation. It is also a solitary wave solution to a flow in the (three-dimensional) filament hierarchy, evolving by translation.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10091512