Knotted fields and explicit fibrations for lemniscate knots. (30th June 2017)

• Record Type: Journal Article
• Authors: Bode, B.; Dennis, M. R.; Foster, D.; King, R. P.
• Is Part Of: Proceedings. Volume 473:Number 2202(2017)
• Publication Date: 2017-06-30
• Subjects: knot -- singularity -- braid -- applied topology; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2016.0829
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ℓ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme–Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.