A linking invariant for algebraic curves

TL;DR

This paper introduces a new topological invariant for algebraic plane curves, generalizing previous invariants and capable of distinguishing curves with identical combinatorics but different topologies.

Contribution

The authors develop a novel linking invariant for algebraic curves, extending the I-invariant, with practical computation methods and applications to Zariski pair classification.

Findings

Invariant distinguishes Zariski pairs with same combinatorics

Provides computational tools via braid monodromy and connected numbers

Successfully differentiates known topologically distinct curves

Abstract

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computing this invariant, using a modification of the usual braid monodromy or using the connected numbers introduced by Shirane. As an application, we show that this invariant distinguishes several Zariski pairs, i.e. pairs of curves having same combinatorics, yet different topologies. The former is the well known Zariski pair found by Artal, composed of a smooth cubic with 3 tangent lines at its inflexion points. The latter is formed by a smooth quartic and 3 bitangents.