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

TL;DR

This paper presents a method to explicitly construct complex polynomials in three dimensions whose zero sets form the three-twist knot $5_2$, with potential applications in physical systems involving knotted fields.

Contribution

The authors develop a general procedure to generate explicit polynomials with prescribed knotted nodal lines, extending previous work on simpler knots and exploring mathematical properties of these maps.

Findings

Constructed a polynomial with nodal lines forming the three-twist knot $5_2$

Linked phase critical points with Morse-Novikov number, showing non-fibred nature

Extended the method to other six-crossing knots

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.