Constructing links of isolated singularities of polynomials ℝ4 → ℝ2

Abstract

We show that if a braid [Formula: see text] can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328 ) can be extended to a construction of a polynomial [Formula: see text] with the closure of [Formula: see text] as the link of an isolated singularity of [Formula: see text], showing that the closure of [Formula: see text] is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of [Formula: see text] over [Formula: see text].