# Connected numbers and the embedded topology of plane curves

## TL;DR

This paper introduces a new invariant called the connected number for plane curves under Galois covers, which helps classify the embedded topology of specific arrangements like Artal arrangements.

## Contribution

It defines the connected number for plane curves and demonstrates its effectiveness in classifying the embedded topology of Artal arrangements.

## Key findings

- Connected number is similar to the splitting number.
- Classifies embedded topology of Artal arrangements of degree ≥ 4.
- Provides a new tool for topological classification of plane curves.

## Abstract

The splitting number of a plane irreducible curve for a Galois cover is effective to distinguish the embedded topologies of plane curves. In this paper, we define a connected number of any plane curve for a Galois cover whose branch divisor has no common components with the plane curve, which is similar to the splitting number. We classify the embedded topology of Artal arrangements of degree $b\geq 4$ by the connected number, where an Artal arrangement of degree $b$ is a plane curve consisting of one smooth curve of dgree $b$ and three total inflectional tangents.