Knotted surfaces as vanishing sets of polynomials

Abstract

We present an algorithm that takes as input any element 𝐵 of the loop braid group and constructs a polynomial 𝑓 : ℝ⁵ → ℝ² such that the intersection of the vanishing set of 𝑓 and the unit 4-sphere contains the closure of 𝐵. The polynomials can be used to create real analytic time-dependent vector fields with zero divergence and closed flow lines that move as prescribed by 𝐵. We also show how a family of surface braids in ℂ ×𝑆¹ ×𝑆¹ without branch points can be constructed as the vanishing set of a holomorphic polynomial 𝑓 : ℂ³ → ℂ on ℂ ×𝑆¹ ×𝑆¹ ⊂ ℂ³. Both constructions allow us to give upper bounds on the degree of the polynomials.