Combinatorial obstructions to the lifting of weaving diagrams

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Abstract

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 χD 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]. © 1993 Springer-Verlag New York Inc.

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Richter-Gebert, J. (1993). Combinatorial obstructions to the lifting of weaving diagrams. Discrete & Computational Geometry, 10(1), 287–312. https://doi.org/10.1007/BF02573982

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