2020/04/06 by Bode, Benjamin, Kamada, Seiichi
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2004.02468
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.