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Knotted fields and explicit fibrations for lemniscate knots. (English) Zbl 1402.57006

Summary: 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 (\(1,\ell\)) 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.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
14P25 Topology of real algebraic varieties

Software:

KnotInfo; Knot Atlas
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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