Non-homotopicity of the linking set of algebraic plane curves

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 $\mathbb{CP}^2$. 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.