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 $f:{\mathbb{R}}^5\to {\mathbb{R}}^2$ 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 $\mathbb{C}\times S^1\times S^1$ without branch points can be constructed as the vanishing set of a holomorphic polynomial $f:{\mathbb{C}}^3\to \mathbb{C}$ on $\mathbb{C}\times S^1\times S^1\subset {\mathbb{C}}^3$. Both constructions allow us to give upper bounds on the degree of the polynomials.

arXiv comment32 pages, 4 figures

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