In the paper Configurations of points and topology of real line arrangements, Math. Ann. 374 (2019), no. 1-2, 1–35, doi: 10.1007/s00208-018-1673-0, B. Guerville and J. Viu give families of line arrangements which define Zariski pairs distinguished by a link invariant. This paper can be found also in arXiv:1702.00922. Moreover, a simple way to compute this linking invariant is given.
In a subsequent paper, Fundamental groups of real arrangements and torsion in the lower central series quotients, Exp. Math. 29 (2020), no. 1, 28–35,
doi: 10.1080/10586458.2018.1428131, these authors together with E. Artal compute the fundamental groups for one of these pairs and prove they are not isomorphic. This paper can also be found in arXiv:1704.04152.
We provide two notebooks for the involved computations, which can be either downloaded or executed online in 
The first notebook studies carefully the arrangements while the second one can be used for other arrangements.