Chirality and the isotopy classification of skew lines in projective 3- space
The following question is stated: Given two indexed families \(L = \{L_ 1, \dots, L_ n\}\), and \(M = \{M_ 1, \dots, M_ n\}\), each consisting of \(n\) mutually nonintersecting lines in real affine or projective three- space, can the lines of the family \(L\) be moved in a continuous manner and in such a way that at no time do two of the lines intersect, until they arrive at the positions of the corresponding lines of the set \(M\), for \(i = 1, \dots, n\)? If so, the families \(L\), \(M\) are called affinely or projectively isotopic. How many distinct configurations of \(n\) lines, up to affine or projective isotopy, are there? The paper is divided in 16 sections. The affine problem is trivial and so the paper studies the projective case in a constructive and algebro- geometric way (with Grassmann-Plücker coordinates). The authors define the chiral signature of a configuration of \(n\) unoriented lines (an arithmetic invariant) and compute all abstract chiral signatures for up to eight lines. Other properties of line configurations are stated in terms of oriented lines and the linking numbers of pairs of such lines. In the context of oriented projective geometry, there are the following possibilities: 1. to consider the configuration in perspective, from a certain point of view \(z\), and 2. to scan the line configuration with respect to a line through \(z\). The perspective view of line configuration gives rise to the combinatorial structure of a link chirotope, while the scan of the configuration yields a geometric braid (sections 8 and 12, respectively). Concepts which are connected with plane projections are studied in sections 5-7, singularities are studied in section 9. In section 12, an easy way to construct an associated planar link diagram is given, starting from a planar layout of a given configuration. In section 13 the question of flexible isotopies versus rigid isotopies is studied: Are there distinct isotopy classes of line configurations which are topologically indistinguishable? Section 14 gives a direct proof that a Kauffman polynomial computed on planar layouts in the projective plane is an invariant under diagram moves. In sections 15 and 16 some special classes of configurations (called spindles) are studied.
scientific article; zbMATH DE number 549719