论文标题
循环链接行安排数
The loop linking number of line arrangements
论文作者
论文摘要
在他的博士学位论文，凯德根·史克利珀（Cadegan-Schlieper）构建了线条布置的嵌入式拓扑结构的不变性，该拓扑概括了由Artal，Florens和作者引入的$ \ Mathcal {I} $ - 不变的。这个新不变的称为本文中的循环链接号码。我们通过证明循环链接数是同态类型补充的循环链接数字来完善Cadegan-Schlieper的结果。 我们提供了两种有效的方法来计算这种不变的，这两者都是基于编织的单片。作为一种应用，我们检测到算术Zariski对的排列，其系数位于第5个环体场中。此外，我们还证明了他们的补充的基本群体不是同构的。这是具有最少数量的线条的Zariski对。我们还检测到算术Zariski Triple的12行，其补充具有非同构基本组。在附录中，我们使用循环链接编号给出了28个类似的算术Zariski对。 总而言之，我们为安排联合提供了多个定理。这首先使我们能够证明Rybnikov的安排的补充不是同构的，然后导致我们对Rybnikov的结果进行概括。最后，我们用它来证明存在具有非塑形补位的同型晶格均质排列的存在。
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.
