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The invariant $\mathcal{I}(\mathcal{A},\xi,\gamma)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnikov arrangements form an ordered Zariski pairs (i.e. two arrangements with the same combinatorial information and different ordered topologies). Finally, we extend this method to a particular family of arrangements and thus we obtain a method to construct new examples of Zariski pairs.

Multiplicativity of the $\mathcal{I}$-invariant and topology of glued arrangements