Maxwell's Theory of Solid Angle and the Construction of Knotted Fields

Abstract

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in ${\mathbb{R}}^3$. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighborhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.

arXiv comment20 pages, 9 figures

References in corpus (15)

• Reconfigurable knots and links in chiral nematic colloids
• To be or knot to be?
• Tying knots in light fields
• How superfluid vortex knots untie
• Nematics, Knots and Non-orientable Surfaces
• Knots in the Skyrme-Faddeev model
• Knotted Defects in Nematic Liquid Crystals
• Geometry of Calugareanu's theorem
• Knotted fields and explicit fibrations for lemniscate knots
• Untangling knots via reaction-diffusion dynamics of vortex strings
• Constructing a polynomial whose nodal set is the three-twist knot $5_2$
• The length of excitable knots
• When do knots in light stay knotted?
• Rings on strings in excitable media
• Harmonic field in knotted space

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• Many-defect solutions in planar nematics: interactions, spiral textures and boundary conditions
• Contact Topology and the Classification of Disclination Lines in Cholesteric Liquid Crystals
• Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids
• Multi-valued vortex solutions to the Schrödinger equation and radiation
• One-Shot Method for Computing Generalized Winding Numbers
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