Non-homotopicity of the linking set of algebraic plane curves

TL;DR

This paper demonstrates that the linking set invariant of algebraic plane curves can distinguish between curves with the same fundamental group, showing it is not solely determined by the fundamental group.

Contribution

It proves that the linking set invariant is not determined by the fundamental group, highlighting its independent discriminative power for algebraic plane curves.

Findings

Linking set distinguishes certain Zariski pairs.

Linking set is not determined by the fundamental group.

Provides new insights into curve invariants.

Abstract

The linking set is an invariant of algebraic plane curves introduced by Meilhan and the first author. It has been successfully used to detect several examples of Zariski pairs, i.e. curves with the same combinatorics and different embedding in $\mathds{CP}^2$. Differentiating Shimada's $\pi_1$-equivalent Zariski pair by the linking set, we prove, in the present paper, that this invariant is not determined by the fundamental group of the curve.