Sweeps, arrangements and signotopes

AbstractSweeping is an important algorithmic tool in geometry. In the first part of this paper we define sweeps of arrangements and use the “Sweeping Lemma” to show that Euclidean arrangements of pseudolines can be represented by wiring diagrams and zonotopal tilings. In the second part we introduce a further representation for Euclidean arrangements of pseudolines. This representation records an “orientation” for each triple of lines. It turns out that a “triple orientation” corresponds to an arrangement exactly if it obeys a generalized transitivity law. Moreover, the “triple orientations” carry a natural order relation which induces an order relation on arrangements. A closer look on the combinatorics behind this leads to a series of signotope orders closely related to higher Bruhat orders. We investigate the structure of higher Bruhat orders and give new purely combinatorial proofs for the main structural properties. Finally, we reconnect the combinatorics of the second part to geometry. In particular, we show that the maximum chains in the higher Bruhat orders correspond to sweeps