The loop linking number of line arrangements

Abstract

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology of a line arrangement which generalizes the 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 homotopyequivalent lattice-isomorphic arrangements which have non-homeomorphic complements.

On the non-connectivity of moduli spaces of arrangements: the splitting-polygon structure

“…This construction will produce line arrangements of 11 lines whose moduli space have four connected components. Arithmetically, these arrangements will be between the Rybnikov arrangements which are not Galois conjugated in their field of definition, and the arrangements obtain by the author in [12,13] which are arithmetic quadruples in the 5th cyclotomic field. Indeed, they will be arithmetic quadruples, but with the Klein group as Galois group of their field of definition.…”

“…By applying the Alexander Invariant test of level 2 (described in [4,5]), we obtain the following theorem. We don't give here more details about the proof, since the strategy is the exactly the same as in [4,5,13], and since the author used the same program to perform the computation. We refer to these articles for more details (the construction of the Alexander Invariant test is done in [4] while a Sagemath code is given in [5]).…”

“…Due to their particular arithmetic property, it would be interesting to see if the invariants developed by Bannai, Shirane and Tokunaga [6,22,25] could distinguish their topology. Furthermore, neither the linking-invariants [7,13] nor the torsion order of the first lower central series quotients of their fundamental groups [23,8] can distinguish it.…”

“…As mentioned in the abstract, the questions related to the combinatorial nature of some properties of a hyperplane arrangement are numerous in the literature. If some of them have been solved by the affirmative, as for the number of chambers of a real arrangement [27], the cohomology ring of the complement [19], the rank of the lower central series quotients of the fundamental group of its complement [10] or the deletion and addition-deletion theorems of free arrangements [2,1]; some others obtained a negative answer, as for the embedded topology of a complex arrangements or the fundamental group of its complements, see [21,12,5,13], (also negative for the smaller class of real complexified arrangements [3,14]), the torsion of the lower central series quotients [8] or the existence of unexpected curves [14]. Naturally, the number of problems which are still open (or conjectural) is larger; like the famous Terao's conjecture [24,20], the combinatorial nature of the characteristic varieties [15] or of the homology of the Milnor fiber [11,Problem 4.5], to name some but a few.…”