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

Abstract

We describe an algorithm that for every given braid [Formula: see text] explicitly constructs a function [Formula: see text] such that [Formula: see text] is a polynomial in [Formula: see text], [Formula: see text] and [Formula: see text] and the zero level set of [Formula: see text] on the unit three-sphere is the closure of [Formula: see text]. The nature of this construction allows us to prove certain properties of the constructed polynomials. In particular, we provide bounds on the degree of [Formula: see text] in terms of braid data.