Knotted fields and explicit fibrations for lemniscate knots

TL;DR

This paper constructs explicit complex maps with nodal lines forming lemniscate knots, demonstrating their properties and potential applications in physics, including knotted fields and models like Skyrme-Faddeev.

Contribution

It provides an explicit construction of complex maps with lemniscate knot nodal lines and proves these maps are fibrations, extending to maps with isolated singularities.

Findings

Existence of explicit fibrations with lemniscate knot nodal lines

Application potential in physics for creating knotted fields

Extension to maps with weakly isolated singularities

Abstract

We give an explicit construction of complex maps whose nodal line 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 generalising 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 generalising to rational maps with application to the Skyrme-Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.