  In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the
embedded topology of a line arrangement which generalizes the
$\mathcal{I}$-invariant introduced by Artal, Florens and the author. This new
invariant is called the loop linking number in the present paper. We refine the
result of Cadegan-Schlieper by proving that the loop linking number is an
invariant of the homeomorphism type of the arrangement complement.
  We give two effective methods to compute this invariant, both are based on
the braid monodromy. As an application, we detect an arithmetic Zariski pair of
arrangements with 11 lines whose coefficients are in the 5th cyclotomic field.
Furthermore, we also prove that the fundamental groups of their complements are
not isomorphic; it is the Zariski pair with the fewest number of lines which
have this property. We also detect an arithmetic Zariski triple with 12 lines
whose complements have non-isomorphic fundamental groups. In the appendix, we
give 28 similar arithmetic Zariski pairs detected using the loop linking
number.
  To conclude this paper, we give a multiplicativity theorem for the union of
arrangements. This first allows us to prove that the complements of Rybnikov's
arrangements are not homeomorphic, and then leads us to a generalization of
Rybnikov's result. Lastly, we use it to prove the existence of
homotopy-equivalent lattice-isomorphic arrangements which have non-homeomorphic
complements.
