Saddle point braids of braided fibrations and pseudo-fibrations

Let gₜ gt be a loop in the space of monic complex polynomials in one variable of fixed degree n . If the roots of gₜ gt are distinct for all t , they form a braid B₁ B1 on n strands. Likewise, if the critical points of gₜ gt are distinct for all t , they form a braid B₂ B2 on $$n-1$$ n-1 strands. In this paper we study the relationship between B₁ B1 and B₂ B2 . Composing the polynomials gₜ gt with the argument map defines a pseudo-fibration map on the complement of the closure of B₁ B1 in C× S¹ C×S1 , whose critical points lie on B₂ B2 . We prove that for B₁ B1 a T-homogeneous braid and B₂ B2 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 L₁ L1 and L₂ L2 there is a mixed polynomial f:C²→ C f:C2→C in complex variables u , v and the complex conjugate v̄ v¯ such that both f and the derivative fᵤ fu have a weakly isolated singularity at the origin with L₁ L1 as the link of the singularity of f and L₂ L2 as a sublink of the link of the singularity of fᵤ fu .