Constructing a polynomial whose nodal set is any prescribed knot or link

TL;DR

This paper presents an algorithm to explicitly construct polynomials in complex variables whose zero sets on the 3-sphere form any prescribed knot or link, linking braid data to polynomial degree bounds.

Contribution

It introduces a method to construct polynomials with prescribed knot or link nodal sets from braid representations, providing degree bounds based on braid data.

Findings

Constructs polynomials with prescribed knot/link nodal sets

Provides bounds on polynomial degree in terms of braid data

Establishes a link between braid representations and polynomial properties

Abstract

We describe an algorithm that for every given braid $B$ explicitly constructs a function $f:\mathbb{C}^{2}\rightarrow\mathbb{C}$ such that $f$ is a polynomial in $u$, $v$ and $\overline{v}$ and the zero level set of $f$ on the unit three-sphere is the closure of $B$. The nature of this construction allows us to prove certain properties of the constructed polynomials. In particular, we provide bounds on the degree of $f$ in terms of braid data.