Computing the Braid Monodromy of Completely Reducible $n$-gonal Curves

Abstract

Braid monodromy is an important tool for computing invariants of curves and surfaces. In this paper, the \emph{rectangular braid diagram (RBD)} method is proposed to compute the braid monodromy of a completely reducible $n$-gonal curve, i.e. the curves in the form $(y-y_1(x))...(y-y_n(x))=0$ where $n\in {\mathbb{Z}}^{+}$ and $y_i\in \mathbb{C}[x]$. Also, an algorithm is presented to compute the Alexander polynomial of these curve complements using Burau representations of braid groups. Examples for each computation are provided.