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 $g:\mathbb{C}\times S^1\to\mathbb{C}$ 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 $\arg g:(\mathbb{C}\times S^1)\backslash B\to S^1$. We show that a certain satellite operation produces new P-fibered braids from known ones. We also prove that any braid $B$ with $n$ strands, $k_-$ negative and $k_+$ positive crossings can be turned into a P-fibered braid (and hence also into a braid whose closure is fibered) by adding at least $\tfrac{k_-+1}{n}$ negative or $\tfrac{k_+ +1}{n}$ positive full twists to it.