Combinatorial obstructions to the lifting of weaving diagrams

This paper deals with various connections of oriented matroids [3] and weaving diagrams of lines in space [9], [16], [27]. We encode the litability problem of a particular weaving diagram D on n lines by the realizability problem of a partial oriented matroid χ _D with 2n elements in rank 4. We prove that the occurrence of a certain substructure in D implies that χD is noneuclidean in the sense of Edmonds, Fukuda, and Mandel [12], [14]. Using this criterion we construct an infinite class of minor-minimal noneuclidean oriented matroids in rank 4. Finally, we give an easy algebraic proof for the nonliftability of the alternating weaving diagram on a bipartite grid of 4×4 lines [16].