  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.
