Chirality and the isotopy classification of skew lines in projective 3- space
type: Item
instance of: scholarly article
title: Chirality and the isotopy classification of skew lines in projective 3- space
zbMATH Open document ID: 0802.51024
author: Henry Crapo
author: Rudi Penne
published in: Advances in Mathematics
publication date: 1994-12-12T00:00:00Z
full work available at URL: RR-1693.pdf
review text: 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 to…
Mathematics Subject Classification ID: 51M30
Mathematics Subject Classification ID: 51N15
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 549719
zbMATH Keywords: moving lines
zbMATH Keywords: projectively isotopic
zbMATH Keywords: Grassmann-Plücker coordinates
zbMATH Keywords: chiral signature
zbMATH Keywords: line configurations
zbMATH Keywords: oriented projective geometry
zbMATH Keywords: plane projections
reviewed by: Hans-Dietrich Hecker
MaRDI profile type: Publication
OpenAlex ID: W2091064956
DOI: 10.1006/AIMA.1994.1001
description: scientific article; zbMATH DE number 549719