Saddle point braids of braided fibrations and pseudo-fibrations

Let \(g_t\) be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of \(g_t\) are distinct for all t, they form a braid \(B_1\) on n strands. Likewise, if the critical points of \(g_t\) are distinct for all t, they form a braid \(B_2\) on \(n-1\) strands. In this paper we study the relationship between \(B_1\) and \(B_2\). Composing the polynomials \(g_t\) with the argument map defines a pseudo-fibration map on the complement of the closure of \(B_1\) in \({\mathbb {C}}\times S^1\), whose critical points lie on \(B_2\). We prove that for \(B_1\) a T-homogeneous braid and \(B_2\) the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links \(L_1\) and \(L_2\) there is a mixed polynomial \(f:{\mathbb {C}}^2\rightarrow {\mathbb {C}}\) in complex variables u, v and the complex conjugate \(\overline{v}\) such that both f and the derivative \(f_u\) have a weakly isolated singularity at the origin with \(L_1\) as the link of the singularity of f and \(L_2\) as a sublink of the link of the singularity of \(f_u\).