  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.
