Galois covers of graphs and embedded topology of plane curves

TL;DR

This paper introduces the splitting graph, a generalization of the splitting number, to classify the embedded topology of specific plane curves, including Artal arrangements, enhancing understanding beyond fundamental group invariants.

Contribution

The paper generalizes the splitting number to the splitting graph and applies it to classify the embedded topology of Artal arrangements of plane curves.

Findings

Splitting graph effectively distinguishes embedded topologies.

Classification of Artal arrangements achieved using splitting graph.

Enhances topological analysis beyond fundamental group methods.

Abstract

The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the splitting graph. By using the splitting graph, we classify the embedded topology of plane curves consisting of one smooth curve and non-concurrent three lines, called Artal arrangements.