Saddle point braids of braided fibrations and pseudo-fibrations

Abstract

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 visualisation of this fact. Our work implies that for every pair of links $L_1$ and $L_2$ there is a mixed polynomial $f:\mathbb{C}^2\to\mathbb{C}$ 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 $f_u$.