Knotted surfaces as vanishing sets of polynomials

We present an algorithm that takes as input any element B of the loop braid group and constructs a polynomial f : R-5 -> 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 timedependent 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 x S-1 x S-1 without branch points can be constructed as the vanishing set of a holomorphic polynomial f : C-3 -> C on C x S-1 x S-1 subset of C-3. Both constructions allow us to give upper bounds on the degree of the polynomials.