A note on combinatorial type and splitting invariants of plane curves

TL;DR

This paper introduces the G-combinatorial type, a generalization of splitting invariants, to distinguish the embedded topology of plane curves using modified plumbing graphs, extending previous invariants.

Contribution

It defines the G-combinatorial type for plane curves and proves its invariance under certain homeomorphisms, enhancing tools for topological classification.

Findings

G-combinatorial type generalizes splitting invariants

Proven invariance under specific homeomorphisms

Distinguishes topology of quasi-triangular curves

Abstract

Splitting invariants describe how a plane curve "splits" by the pull-back under a Galois cover over the projective plane whose branch locus contains no component of the plane curve. They enable us to distinguish the embedded topology of several plane curves with the same fundamental group of the complements. In this note, we introduce a generalization of splitting invariants, called the G-combinatorial type, for plane curves by using the modified plumbing graph defined by Hironaka. We prove the invariance of the G-combinatorial type under certain homeomorphisms based on the arguments of graph manifolds by Waldhausen and plumbing graphs by Neumann. Furthermore, we distinguish the embedded topology of quasi-triangular curves by the G-combinatorial type, which are generalization of triangular curves studied by Artal, Cogolludo and Mart\'in.