Knotted fields and explicit fibrations for lemniscate knots

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.

• Publication year

  2016

• Venue

  Proceedings of the Royal Society A

• Publication date

  2016-11-08

• Fields of study

  Mathematics, Physics, Medicine

• Identifiers
  DOI 10.1098/rspa.2016.0829arXiv 1611.02563PMID 28690405PMCID PMC5493943
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar, PubMed

• No claims are published for this paper.

• No concepts are published for this paper.

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