Lines in 3-space isotopy, chirality and weavings
Rudi Penne · 1992
We consider configurations of n pairwise disjoint lines in $\IR\IP\sp3$, or n mutually skew lines in $\IR\sp3$. Already for n = 3 we can distinguish two isotopy classes. The planar layout of a line configuration can be perfectly compared with knot diagrams in Knot Theory. In the same spirit as the Reidemeister moves, we find two diagram moves generating every isotopy. This leads to the concept of equivalence of pseudoline diagrams, a purely combinatorial treatment of line isotopy. Within this framework we deduce isotopy invariants as the Kauffman polynomial and the labeled braid trace. In fact, we can switch really to Knot Theory by regarding n mutually skew lines as a link of n unknotted components which are pairwise linked. This view point provides a topological level of reasoning about the isotopy problem. Now we arrive at the weaving problem. Which line diagrams can be lifted to 3-space? This is a relevant question in areas as pattern recognition and computer graphics. A degenerated version of the weaving problem asks to lift planar drawings of lines to 3-space such that incidences occur among certain prescribed pairs of lines. If a pair of lines is incident as soon as a certain set of incidences is satisfied then we say that the former incidence pair depends on the latter set, yielding a matroid structure. It is exactly these dependencies among the incidences which prevent certain weavings on the given drawing to be realizable. If we perform a polarity in the plane, mapping the lines of the given drawing to points, and if we interpret the given incidence pairs as bars between the corresponding points (joints), then the space of liftings satisfying the prescribed incidences is isomorphic to the space of infinitesimal motions of the resulting bar-and-join framework. The matroid structure has an easy characterization in the case that the joints are in generic positions, given by Laman's Theorem. The problem of lifting line incidences is completely settled if the given projection is sufficiently generic. We attack the question how to characterize those special positions that are described by an irreducible polynomial. (Abstract shortened with permission of author.)