Configurations of few lines in 3-space. Isotopy, chirality and planar layouts (Q1206511)

The author considers line configurations where the lines are pairwise skew in \(\mathbb{R}^ 3\) or pairwise disjoint in \(\mathbb{R}\mathbb{P}^ 3\). A labelled configuration of \(n\) lines in projective 3-space can be represented by a point in the Cartesian product of \(n\) Grassmannians \(G_{2,4}\). The set of labelled configurations of \(n\) pairwise disjoint lines in \(\mathbb{R}\mathbb{P}^ 3\) forms an open submanifold of the space and is called the configuration space. The points of this space are called line configurations. Two isotopic line configurations can be transformed to each other by a continuous motion of the lines during which no pair becomes coplanar. It is sufficient to treat a configuration as a set of mutually skew lines in affine \(\mathbb{R}^ 3\). A vertical projection to the \(xy\)-plane can give a simple arrangement of lines, no two parallel and no three concurrent. A planar layout is such a projection with the additional information that one knows at each intersection which line is above the other. This simple arrangement is a line diagram and the problem of which line diagrams are realizable, i.e., arise from a planar layout of a configuration in 3-space, is open. Here the author shows the projective invariance of the realizability and shows that certain diagrams are non-realizable. He further demonstrates an infinite class of line diagrams -- alternating \(k\)-angles -- that are non-realizable, and a class that are realizable-shellable diagrams. Every planar layout of a configuration of at most five lines is shellable. He then considers the chirality of three skew lines and stacks, which are configurations whose lines are in horizontal planes. He proves that every pair of stacks with equal chiralities consisting of at most five lines is isotopic. Further, for up to five lines, two labelled configurations are isotopic if and only if they have the same chiralities. Unfortunately, much of the above does not apply to six lines or more.