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The loop-linking number of line arrangements

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Abstract

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 29 combinatorial types which lead to similar ordered arithmetic Zariski pairs of 11 lines. To conclude this paper, we give a additivity 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.

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Notes

  1. We should also mention here the refinement obtained by Jiang and Yau [25] who proved that the homeomorphism type of the complement determines the intersection lattice.

  2. This name arises from the work of Zariski who pioneered the study of the topology of equisingular curves [40, 41]. In particular, he constructed the first example of two irreducible curves of degree 6 which have the same local singularities and non-isomorphic fundamental groups.

  3. In the general case of algebraic plane curves, a Zariski pair is said to be arithmetic if the equations of their arrangements are conjugated in a Galois field. Such pairs have been extensively studied by authors like Degtyarev, Oka, Shimada or Tokunaga. Note that the Zariski pairs of arrangements given in [4, 19] are arithmetic.

  4. The boundary manifold is the boundary of a regular tubular neighborhood of \(\bigcup _{L\in \mathcal {A}}L\). It is a graph 3-manifold determined by the combinatorics of \(\mathcal {A}\). This manifold has been studied in [13].

  5. See [34] for more details about complex reflection groups.

  6. The ordered (resp. oriented) topological type of \(\mathcal {A}\) is the class of homeomorphism of \(\mathbb {C}\mathbb {P}^2\) which fixes \(\bigcup _{L\in \mathcal {A}}L\) and respects a fixed complete order on \(\mathcal {A}\) (resp. the global orientation of \(\mathbb {C}\mathbb {P}^2\) and the local orientation of the meridians).

  7. If \(\mathcal {A}\) is a pencil then its topology is combinatorially determined.

  8. It is compact, orientable, irreducible, and has an embedded incompressible surface.

  9. Note that this isomorphism between the ordered combinatorics is unique.

  10. To lighten the notation a sub-arrangement \(\mathcal {A}_P=\{L_{i_1},\ldots ,L_{i_m}\}\) is denoted by \(\{i_1,\ldots ,i_m\}\).

  11. The braids \(B_{ {(P\rightarrow L)} }\) are given as tuple of integer \((i_1,\ldots ,i_k)\), where a positive integer i in the tuple indicates a positive crossing between the strands number i and \(|i+1|\); while a negative i indicates a negative crossing between strands numbers |i| and \(|i|+1\). The singular points \(P_{i_1,\ldots , i_m}\) are expressed as the tuple \([i_1,\dots ,i_m]\).

  12. A wiring diagram of \(\mathcal {M}_2\) (resp. \(\mathcal {M}_4\)) can be obtained from the one of \(\mathcal {M}_3\) (resp. \(\mathcal {M}_1\)) by the inversion of the sign of the virtual crossings.

  13. This can be viewed as a consequence of Theorem 5.5 presented below.

  14. We consider the orders given by the indices, i.e. \(L^k_i< L^k_j \Leftrightarrow i<j\).

  15. A more detailed list with the equations of the realizations and the automorphism groups is given in the first version of this paper available at arXiv:2004.03550.

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Acknowledgements

During this work the author have been supported, first by the postdoctoral grant #2017/15369-0 of the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), second by the Polish Academy of Sciences, and third by JSPS KAKENHI Grant Number JP17H06128. The author would like to thank J. Viu-Sos for all their rewarding discussions and all his remarks/comments which greatly contributed to improve the quality of this article. He is also grateful to E. Hollen for her efficient and careful proofreading of this manuscript.

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    Benoît Guerville-Ballé

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Appendices

Appendix A: Values for the arrangements \(\mathcal {M}_i\)

1.1 Generator of \({{\,\mathrm{TLG}\,}}(C,\mathbb {Z}/5\mathbb {Z})\)

The generator \(\Lambda _0\) of \({{\,\mathrm{TLG}\,}}(C,\mathbb {Z}/5\mathbb {Z})\) is given by the following characters \(\lambda _{ {(P\rightarrow L)} }\).

figure c

1.2 The upper-linking numbers of \(\mathcal {M}_1\)

The values of the upper-linking numbers of the braid \(B_{ {(P\rightarrow L)} }\) are given below. The values denoted with a \(=^*\) symbol have been simplified using the property \(\sum _{L\in \mathcal {A}} m_L = 0\).

figure d

Appendix B: List of arithmetic Zariski pairs with 11 lines

Each one of following combinatoricsFootnote 15 admits four realizations Galois conjugated in the 5th cyclotomic fields and its tensor linking groups with coefficients in \(\mathbb {Z}/5\mathbb {Z}\) is isomorphic to \(\mathbb {Z}/5\mathbb {Z}\). Furthermore, their realizations can be distinguished using the loop-linking number. The following list has been obtained using a computer-aid checking of all the lines combinatorics with 11 lines. First, we listed the combinatorics which admits a non-trivial tensor linking group with coefficients in \(\mathbb {Z}/p\mathbb {Z}\) for \(p=5,7,9,11\). Then, we determined the moduli-space of each of these combinatorics, and for the one which have several connected-components, we computed their loop-linking numbers. To our knowledge and our computations, the following list together with the examples obtained from the ordered Zariski pair of Sect. 4.1 and the one of [19], is the complete list of the combinatorics which leads to an arithmetic Zariski pair with 11 lines that can be detected by the loop-linking number with coefficient in \(\mathbb {Z}/p\mathbb {Z}\) for \(p=5,7,9,11\).

$$\begin{aligned} C_1=\big \{&\{1, 2\}, \{1, 3, 4, 5\}, \{1, 6, 8\}, \{1, 7, 11\}, \{1, 9, 10\}, \{2, 3, 6, 7\}, \{2, 4, 11\}, \{2, 5, 10\}\\&\{2, 8, 9\}, \{3, 8, 11\}, \{3, 9\}, \{3, 10\}, \{4, 6, 9\}, \{4, 7\}, \{4, 8\}, \{4, 10\}, \{5, 6\}, \{5, 7\}\\&\{5, 8\}, \{5, 9, 11\}, \{6, 10\}, \{6, 11\}, \{7, 8, 10\}, \{7, 9\}, \{10, 11\} \big \} \\&\\ C_2=\big \{&\{1, 2, 3\}, \{1, 4\}, \{1, 5, 7, 8\}, \{1, 6, 9\}, \{1, 10, 11\}, \{2, 4, 5, 6\}, \{2, 7, 11\}, \{2, 8\}, \{2, 9, 10\}\\&\{3, 4, 9\}, \{3, 5\}, \{3, 6, 8\}, \{3, 7\}, \{3, 10\}, \{3, 11\}, \{4, 7, 10\}, \{4, 8, 11\}, \{5, 9, 11\}\\&\{5, 10\}, \{6, 7\}, \{6, 10\}, \{6, 11\}, \{7, 9\}, \{8, 9\}, \{8, 10\} \big \} \\&\\ C_3=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 6, 10\}, \{2, 5, 8\}, \{2, 7\}, \{2, 9, 11\}\\&\{3, 4, 7\}, \{3, 5\}, \{3, 6, 11\}, \{3, 8\}, \{3, 9\}, \{3, 10\}, \{4, 8, 11\}, \{4, 9\}, \{5, 6, 9\}, \{5, 7\}\\&\{5, 10\}, \{5, 11\}, \{6, 8\}, \{7, 8, 10\}, \{7, 9\}, \{7, 11\}, \{9, 10\} \big \} \\&\\ C_4=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 6, 10\}, \{2, 5, 8\}, \{2, 7, 9\}\\&\{2, 11\}, \{3, 4, 7\}, \{3, 5\}, \{3, 6, 11\}, \{3, 8\}, \{3, 9\}, \{3, 10\}, \{4, 8, 11\}, \{4, 9\}, \{5, 6, 9\}\\&\{5, 7\}, \{5, 10\}, \{5, 11\}, \{6, 8\}, \{7, 8, 10\}, \{7, 11\}, \{9, 10\}, \{9, 11\} \big \} \\&\\ C_5=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8\}, \{2, 5, 6\}, \{2, 7, 9, 10\}\\&\{2, 11\}, \{3, 4, 10\}, \{3, 5\}, \{3, 6, 9\}, \{3, 7\}, \{3, 8, 11\}, \{4, 6, 11\}, \{4, 7\}, \{4, 9\}, \{5, 7, 8\}\\&\{5, 9\}, \{5, 10\}, \{5, 11\}, \{6, 8\}, \{6, 10\}, \{7, 11\}, \{8, 10\}, \{9, 11\} \big \} \\&\\ C_6=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8, 10\}, \{2, 5\}, \{2, 6, 9\}, \{2, 7, 11\}\\&\{3, 4, 6\}, \{3, 5\}, \{3, 7\}, \{3, 8, 11\}, \{3, 9\}, \{3, 10\}, \{4, 7, 9\}, \{4, 11\}, \{5, 6, 11\}, \{5, 7\}\\&\{5, 8\}, \{5, 9, 10\}, \{6, 8\}, \{6, 10\}, \{7, 8\}, \{7, 10\}, \{9, 11\} \big \} \\&\\ C_7=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8, 10\}, \{2, 5\}, \{2, 6, 11\}, \{2, 7, 9\}\\&\{3, 4, 7\}, \{3, 5\}, \{3, 6, 8\}, \{3, 9\}, \{3, 10\}, \{3, 11\}, \{4, 6\}, \{4, 9, 11\}, \{5, 6, 9\}, \{5, 7, 10\}\\&\{5, 8\}, \{5, 11\}, \{6, 10\}, \{7, 8\}, \{7, 11\}, \{8, 11\}, \{9, 10\} \big \} \\&\\ C_8=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8, 10\}, \{2, 5\}, \{2, 6, 11\}, \{2, 7, 9\}\\&\{3, 4, 9\}, \{3, 5, 6\}, \{3, 7\}, \{3, 8\}, \{3, 10\}, \{3, 11\}, \{4, 6\}, \{4, 7, 11\}, \{5, 7\}, \{5, 8, 11\}\\&\{5, 9\}, \{5, 10\}, \{6, 8\}, \{6, 9, 10\}, \{7, 8\}, \{7, 10\}, \{9, 11\} \big \} \\&\\ C_9=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8, 10\}, \{2, 5, 6\}, \{2, 7\}, \{2, 9, 11\}\\&\{3, 4, 7\}, \{3, 5\}, \{3, 6, 9\}, \{3, 8\}, \{3, 10\}, \{3, 11\}, \{4, 6, 11\}, \{4, 9\}, \{5, 7\}, \{5, 8\}\\&\{5, 9, 10\}, \{5, 11\}, \{6, 8\}, \{6, 10\}, \{7, 8, 11\}, \{7, 9\}, \{7, 10\} \big \} \\&\\ C_{10}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8, 10\}, \{2, 5, 6\}, \{2, 7\}, \{2, 9, 11\}\\&\{3, 4, 7\}, \{3, 5\}, \{3, 6, 9\}, \{3, 8\}, \{3, 10\}, \{3, 11\}, \{4, 6, 11\}, \{4, 9\}, \{5, 7, 8\}, \{5, 9, 10\}\\&\{5, 11\}, \{6, 8\}, \{6, 10\}, \{7, 9\}, \{7, 10\}, \{7, 11\}, \{8, 11\} \big \} \\&\\ C_{11}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 8, 10\}, \{2, 5, 6\}, \{2, 7, 9\}\\&\{2, 11\}, \{3, 4, 7\}, \{3, 5\}, \{3, 6, 11\}, \{3, 8\}, \{3, 9\}, \{3, 10\}, \{4, 6\}, \{4, 9, 11\}, \{5, 7, 8\}\\&\{5, 9\}, \{5, 10\}, \{5, 11\}, \{6, 8\}, \{6, 9, 10\}, \{7, 10\}, \{7, 11\}, \{8, 11\} \big \} \\&\\ C_{12}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 10\}, \{2, 5, 6, 8\}, \{2, 7\}, \{2, 9, 11\}\\&\{3, 4, 6, 11\}, \{3, 5\}, \{3, 7, 9\}, \{3, 8\}, \{3, 10\}, \{4, 7, 8\}, \{4, 9\}, \{5, 7\}, \{5, 9, 10\}, \{5, 11\}\\&\{6, 9\}, \{6, 10\}, \{7, 10\}, \{7, 11\}, \{8, 10\}, \{8, 11\} \big \} \\&\\ C_{13}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10, 11\}, \{2, 4, 10\}, \{2, 5, 9, 11\}, \{2, 6\}, \{2, 7, 8\}\\&\{3, 4, 7\}, \{3, 5, 8\}, \{3, 6, 11\}, \{3, 9\}, \{3, 10\}, \{4, 6, 8\}, \{4, 9\}, \{4, 11\}, \{5, 6\}, \{5, 7\}\\&\{5, 10\}, \{6, 9, 10\}, \{7, 9\}, \{7, 10\}, \{7, 11\}, \{8, 10\}, \{8, 11\} \big \} \\&\\ C_{14}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 4, 6, 11\}, \{2, 5, 9\}, \{2, 7, 10\}\\&\{2, 8\}, \{3, 4, 7\}, \{3, 5\}, \{3, 6\}, \{3, 8\}, \{3, 9, 11\}, \{3, 10\}, \{4, 8, 9\}, \{4, 10\}, \{5, 6, 10\}\\&\{5, 7\}, \{5, 8\}, \{5, 11\}, \{6, 9\}, \{7, 9\}, \{7, 11\}, \{8, 10, 11\} \big \} \\&\\ C_{15}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 8, 10\}, \{1, 7, 9\}, \{1, 11\}, \{2, 4, 6\}, \{2, 5, 9, 11\}, \{2, 7, 8\}\\&\{2, 10\}, \{3, 4, 7\}, \{3, 5, 10\}, \{3, 6, 11\}, \{3, 8\}, \{3, 9\}, \{4, 8, 9\}, \{4, 10\}, \{4, 11\}, \{5, 6\}\\&\{5, 7\}, \{5, 8\}, \{6, 7\}, \{6, 9\}, \{7, 10, 11\}, \{8, 11\}, \{9, 10\} \big \} \\&\\ C_{16}=\big \{&\{1, 2, 3\}, \{1, 4, 5\}, \{1, 6, 10\}, \{1, 7, 11\}, \{1, 8, 9\}, \{2, 4, 6\}, \{2, 5, 7\}, \{2, 8, 10\}, \{2, 9, 11\}\\&\{3, 4\}, \{3, 5, 9\}, \{3, 6\}, \{3, 7, 8\}, \{3, 10\}, \{3, 11\}, \{4, 7, 10\}, \{4, 8, 11\}, \{4, 9\}, \{5, 6, 11\}\\&\{5, 8\}, \{5, 10\}, \{6, 7\}, \{6, 8\}, \{6, 9\}, \{7, 9\}, \{9, 10\}, \{10, 11\} \big \} \\&\\ C_{17}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 6\}, \{1, 7\}, \{1, 8, 10\}, \{1, 9, 11\}, \{2, 4, 7\}, \{2, 5, 8, 9\}, \{2, 6, 11\}\\&\{2, 10\}, \{3, 4, 9\}, \{3, 5\}, \{3, 6, 10\}, \{3, 7\}, \{3, 8\}, \{3, 11\}, \{4, 8\}, \{4, 10, 11\}, \{5, 7, 10\}\\&\{5, 11\}, \{6, 7\}, \{6, 8\}, \{6, 9\}, \{7, 8, 11\}, \{7, 9\}, \{9, 10\} \big \} \\&\\ C_{18}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 6\}, \{1, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 4, 7, 9\}, \{2, 5, 8\}, \{2, 6\}, \{2, 10, 11\}\\&\{3, 4, 11\}, \{3, 5\}, \{3, 6\}, \{3, 7\}, \{3, 8, 10\}, \{3, 9\}, \{4, 8\}, \{4, 10\}, \{5, 7\}, \{5, 9, 11\}\\&\{5, 10\}, \{6, 7, 10\}, \{6, 8, 11\}, \{6, 9\}, \{7, 11\}, \{8, 9\} \big \} \\&\\ C_{19}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 6\}, \{1, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 4, 7, 9\}, \{2, 5, 11\}, \{2, 6, 8\}\\&\{2, 10\}, \{3, 4\}, \{3, 5, 10\}, \{3, 6\}, \{3, 7, 11\}, \{3, 8\}, \{3, 9\}, \{4, 8\}, \{4, 10, 11\}, \{5, 7\}\\&\{5, 8, 9\}, \{6, 7, 10\}, \{6, 9\}, \{6, 11\}, \{8, 10\}, \{8, 11\}, \{9, 11\} \big \} \\&\\ C_{20}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 6\}, \{1, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 4, 9, 11\}, \{2, 5, 10\}, \{2, 6, 8\}\\&\{2, 7\}, \{3, 4, 7\}, \{3, 5\}, \{3, 6\}, \{3, 8, 9\}, \{3, 10, 11\}, \{4, 8\}, \{4, 10\}, \{5, 7, 11\}, \{5, 8\}\\&\{5, 9\}, \{6, 7, 10\}, \{6, 9\}, \{6, 11\}, \{7, 9\}, \{8, 10\}, \{8, 11\} \big \} \\&\\ C_{21}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 6\}, \{1, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 4, 11\}, \{2, 5, 7, 9\}, \{2, 6\}, \{2, 8, 10\}\\&\{3, 4, 10\}, \{3, 5, 11\}, \{3, 6, 8\}, \{3, 7\}, \{3, 9\}, \{4, 7\}, \{4, 8, 9\}, \{5, 8\}, \{5, 10\}, \{6, 7, 11\}\\&\{6, 9\}, \{6, 10\}, \{7, 10\}, \{8, 11\}, \{9, 11\}, \{10, 11\} \big \} \\&\\ C_{22}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 7\}, \{1, 6, 9\}, \{1, 8, 11\}, \{1, 10\}, \{2, 4, 6, 8\}, \{2, 5, 10\}, \{2, 7, 11\}\\&\{2, 9\}, \{3, 4\}, \{3, 5\}, \{3, 6\}, \{3, 7, 10\}, \{3, 8, 9\}, \{3, 11\}, \{4, 9, 10\}, \{4, 11\}, \{5, 6\}\\&\{5, 8\}, \{5, 9, 11\}, \{6, 7\}, \{6, 10, 11\}, \{7, 8\}, \{7, 9\}, \{8, 10\} \big \} \\&\\ C_{23}=\big \{&\{1, 2, 3\}, \{1, 4, 5, 7\}, \{1, 6, 10\}, \{1, 8\}, \{1, 9, 11\}, \{2, 4, 6, 8\}, \{2, 5, 11\}, \{2, 7\}, \{2, 9, 10\}\\&\{3, 4\}, \{3, 5\}, \{3, 6\}, \{3, 7, 11\}, \{3, 8, 10\}, \{3, 9\}, \{4, 9\}, \{4, 10, 11\}, \{5, 6\}, \{5, 8, 9\}\\&\{5, 10\}, \{6, 7, 9\}, \{6, 11\}, \{7, 8\}, \{7, 10\}, \{8, 11\} \big \} \\&\\ C_{24}=\big \{&\{1, 2, 3, 4\}, \{1, 5, 6\}, \{1, 7, 8\}, \{1, 9\}, \{1, 10, 11\}, \{2, 5\}, \{2, 6, 10\}, \{2, 7, 9\}, \{2, 8, 11\}\\&\{3, 5, 7\}, \{3, 6\}, \{3, 8\}, \{3, 9, 10\}, \{3, 11\}, \{4, 5\}, \{4, 6, 7\}, \{4, 8, 9\}, \{4, 10\}, \{4, 11\}\\&\{5, 8, 10\}, \{5, 9, 11\}, \{6, 8\}, \{6, 9\}, \{6, 11\}, \{7, 10\}, \{7, 11\} \big \} \\&\\ C_{25}=\big \{&\{1, 2, 3, 4\}, \{1, 5, 6\}, \{1, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 5, 7\}, \{2, 6, 9\}, \{2, 8, 11\}, \{2, 10\}\\&\{3, 5\}, \{3, 6, 8\}, \{3, 7, 9\}, \{3, 10\}, \{3, 11\}, \{4, 5\}, \{4, 6, 10\}, \{4, 7, 11\}, \{4, 8\}, \{4, 9\}\\&\{5, 8, 10\}, \{5, 9, 11\}, \{6, 7\}, \{6, 11\}, \{7, 10\}, \{8, 9\}, \{10, 11\} \big \} \\&\\ C_{26}=\big \{&\{1, 2, 3, 4\}, \{1, 5, 6\}, \{1, 7, 8\}, \{1, 9, 10\}, \{1, 11\}, \{2, 5, 9\}, \{2, 6, 7\}, \{2, 8, 11\}, \{2, 10\}\\&\{3, 5, 7\}, \{3, 6\}, \{3, 8, 9\}, \{3, 10\}, \{3, 11\}, \{4, 5\}, \{4, 6, 9\}, \{4, 7, 11\}, \{4, 8\}, \{4, 10\}\\&\{5, 8, 10\}, \{5, 11\}, \{6, 8\}, \{6, 10, 11\}, \{7, 9\}, \{7, 10\}, \{9, 11\} \big \} \\&\\ C_{27}=\big \{&\{1, 2, 3, 4\}, \{1, 5, 6\}, \{1, 7, 10\}, \{1, 8, 9\}, \{1, 11\}, \{2, 5, 7\}, \{2, 6, 8\}, \{2, 9, 10\}, \{2, 11\}\\&\{3, 5, 9\}, \{3, 6, 11\}, \{3, 7\}, \{3, 8\}, \{3, 10\}, \{4, 5\}, \{4, 6, 7\}, \{4, 8, 11\}, \{4, 9\}, \{4, 10\}\\&\{5, 8, 10\}, \{5, 11\}, \{6, 9\}, \{6, 10\}, \{7, 8\}, \{7, 9, 11\}, \{10, 11\} \big \} \\&\\ C_{28}=\big \{&\{1, 2, 4\}, \{1, 3, 5\}, \{1, 6, 7\}, \{1, 8, 9\}, \{1, 10\}, \{1, 11\}, \{2, 3\}, \{2, 5, 6\}, \{2, 7, 8\}, \{2, 9, 10\}\\&\{2, 11\}, \{3, 4, 7\}, \{3, 6, 8\}, \{3, 9, 11\}, \{3, 10\}, \{4, 5, 9\}, \{4, 6, 10\}, \{4, 8\}, \{4, 11\}\\&\{5, 7, 11\}, \{5, 8\}, \{5, 10\}, \{6, 9\}, \{6, 11\}, \{7, 9\}, \{7, 10\}, \{8, 10, 11\} \big \} \\&\\ C_{29}=\big \{&\{1, 2, 5\}, \{1, 3, 7\}, \{1, 4, 8\}, \{1, 6, 9\}, \{1, 10, 11\}, \{2, 3, 9\}, \{2, 4, 10\}, \{2, 6, 8\}, \{2, 7, 11\}\\&\{3, 4, 6\}, \{3, 5, 10\}, \{3, 8, 11\}, \{4, 5, 7\}, \{4, 9, 11\}, \{5, 6\}, \{5, 8\}, \{5, 9\}, \{5, 11\}, \{6, 7\}\\&\{6, 10\}, \{6, 11\}, \{7, 8\}, \{7, 9\}, \{7, 10\}, \{8, 9\}, \{8, 10\}, \{9, 10\} \big \} \end{aligned}$$

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Guerville-Ballé, B. The loop-linking number of line arrangements. Math. Z. 301, 1821–1850 (2022). https://doi.org/10.1007/s00209-021-02953-x

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