Twisting and satellite operations on P-fibered braids

Abstract

A geometric braid $B$ can be interpreted as a loop in the space of monic complex polynomials with distinct roots. This loop defines a function $g : \mathbb{C} \times S^1 \to C$ that vanishes on $B$. We define the set of P‑fibered braids as those braids that can be represented by loops of polynomials such that the corresponding function g induces a fibration arg $g : (\mathbb{C} \times S^1) \setminus B \to S^1$. We show that a certain satellite operation produces new P‑fibered braids from known ones. We also use P‑fibered braids to prove that any braid $B$ with $n$ strands, $k_{-}$ negative and $k_{+}$ positive crossings can be turned into a braid whose closure is fibered by adding at least $\frac{k_{-} +1}{n}$ negative or $\frac{k_{+} +1}{n}$ positive full twists to it. Using earlier constructions of P‑fibered braids we prove that every link is a sublink of a real algebraic link, i.e., a link of an isolated singularity of a polynomial map $\mathbb{R}^4 \to \mathbb{R}^2$.