Knotted fields and explicit fibrations for lemniscate knots

Benjamin Bode, Mark Dennis, David Foster, Robert P King

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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, ) 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.
Original languageEnglish
Article number20160829
Number of pages22
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume473
Issue number2202
Early online date7 Jun 2017
DOIs
Publication statusPublished - 7 Jun 2017

Structured keywords

  • SPOCK

Keywords

  • applied topology
  • singularity
  • braid
  • knot theory

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