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Knotted fields and explicit fibrations for lemniscate knots.

B Bode¹, M R Dennis¹, D Foster¹

¹H H Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences

|July 11, 2017

Summary

Researchers constructed complex maps with nodal lines forming lemniscate knots. These maps, useful in physics and quantum mechanics, are proven to be fibrations, extending to singularities.

Keywords:

applied topology braid knot singularity

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Area of Science:

Complex analysis
Knot theory
Mathematical physics

Background:

Lemniscate knots are spiral knots generalizing torus knots.
They are defined as closures of braids with specific Lissajous figure strand movements.
Understanding complex maps with specific nodal line structures is crucial for various physics applications.

Purpose of the Study:

To explicitly construct complex maps with lemniscate knot nodal lines.
To investigate the properties and existence of such maps as fibrations.
To explore potential applications in physics, including knotted fields and quantum mechanics.

Main Methods:

Explicit construction of complex maps.
Review of lemniscate knot properties (braid closures, Lissajous figures).
Proof of existence and fibration properties with parameter selection.

Main Results:

Successful explicit construction of complex maps with lemniscate nodal lines.
Demonstration that these maps are fibrations under specific conditions.
Extension of the construction to maps with weakly isolated singularities.

Conclusions:

Complex maps with lemniscate knot nodal lines can be explicitly constructed.
These maps exhibit fibration properties, offering a new mathematical framework.
The construction has potential applications in creating knotted fields in physics and quantum mechanics.