Realization of intermediate links of line arrangements

TL;DR

This paper explores the topological and combinatorial properties of line arrangements by analyzing the link configurations obtained through intersections with spheres and bands, highlighting differences between real and complex arrangements.

Contribution

It introduces new insights into how line arrangements relate to link configurations, especially contrasting real and complex cases and their realizability via different intersections.

Findings

Certain link configurations are realizable by complex but not real line arrangements.

Different intersection methods (sphere vs. band) produce distinct link configurations.

Relations between link configurations from bands and spheres are established.

Abstract

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangements; (b) if we intersect the arrangements with a vertical band instead of a sphere, what link configurations can be obtained? (c) relations between link configurations obtained by bands and spheres.