Chirality and the isotopy classification of skew lines in projective 3-space

Henry Crapo 1 R. Penne
1 ALGO - Algorithms
Inria Paris-Rocquencourt
Abstract : This article concerns isotopy invariants of finite configurations of skew lines in projective 3-space. We develop the theory of the chiral signature and of the Kauffman polynomial of a configuration. Invariance of the Kauffman polynomial under two types of diagram moves is shown by a direct combinatorial argument. The connection between a configuration and its plane projections is established in the context of oriented projective geometry, using oriented directrices and limit isotopies. Using a map to the Klein spherical model of projective space, we arrive at a linked-circle model of the configuration, and to a convenient ball-and-string model, the temari model. Linear graphs provide codes for chiral signatures, and permit the identification of those signatures which can be realized as simple stacked configurations of lines, which we call spindles. A catalogue of all unlabeled configurations of up to six lines, together with their Kauffman polynomials, is appended.
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Submitted on : Monday, May 29, 2006 - 11:37:58 AM
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Henry Crapo, R. Penne. Chirality and the isotopy classification of skew lines in projective 3-space. [Research Report] RR-1693, INRIA. 1992. ⟨inria-00076928⟩

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