The loop-linking number of line arrangements

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology a line arrangement which generalizes $$\mathcal {I}$$ -invariant introduced by Artal, Florens and author. This new is called loop-linking number in present paper. We refine result proving that homeomorphism type complement. give two effective methods to compute this invariant, both are based on braid monodromy. As application, we detect arithmetic Zariski pair arrangements with 11 lines whose coefficients 5th cyclotomic field. Furthermore, also prove fundamental groups their complements not isomorphic; it fewest have property. triple 12 non-isomorphic groups. appendix, 29 combinatorial types lead similar ordered pairs lines. To conclude paper, additivity theorem for union arrangements. first allows us Rybnikov’s homeomorphic, then leads generalization result. Lastly, use existence homotopy-equivalent lattice-isomorphic non-homeomorphic complements.