Constructing a polynomial whose nodal set is the three-twist knot $5_2$

Abstract

We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot $5_2$. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain complex polynomial which depends on an additional parameter. For sufficiently small values of this parameter, the nodal lines form the three-twist knot. Further mathematical properties of this map are explored, including the relationship of the phase critical points with the Morse-Novikov number, which is nonzero as this knot is not fibred. We also find analogous functions for other knots with six crossings. The particular function we find, and the general procedure, should be useful for designing knotted fields of particular knot types in various physical systems.

19 pages, 6 figures

References in corpus (7)

• Knots in the Skyrme-Faddeev model
• Weaving knotted vector fields with tunable helicity
• Global Defect Topology in Nematic Liquid Crystals
• Vortex knots in tangled quantum eigenfunctions
• The Topology of Dislocations in Smectic Liquid Crystals
• Untangling knots via reaction-diffusion dynamics of vortex strings
• Cabling in the Skyrme-Faddeev model

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• Aspects of Defect Topology in Smectic Liquid Crystals
• Maxwell's Theory of Solid Angle and the Construction of Knotted Fields
• Anisotropic non-linear optical response of nodal loop materials
• Nonlinear dynamics of a charged particle in a strong non-null knot wave background
• On real algebraic links in the 3-sphere associated with mixed polynomials