2023/09/10 by Bode, Benjamin, Hirasawa, Mikami
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2309.04972
Let gt be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt are distinct for all t, they form a braid B1 on n strands. Likewise, if the critical points of gt are distinct for all t, they form a braid B2 on n-1 strands. In this paper we study the relationship between B1 and B2. Composing the polynomials gt with the argument map defines a pseudo-fibration map on the complement of the closure of B1 in ℂ× S1, whose critical points lie on B2. We prove that for B1 a T-homogeneous braid and B2 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 L1 and L2 there is a mixed polynomial f:ℂ2→ℂ in complex variables u, v and the complex conjugate v such that both f and the derivative fu have a weakly isolated singularity at the origin with L1 as the link of the singularity of f and L2 as a sublink of the link of the singularity of fu.