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1976 (Publication #24)

Self-Linking Numbers of Space Polygons

Indiana University Mathematics Journal 25 (1976), 1171-1188.

View a review of this article at MathSciNet

ABSTRACT

If a smooth closed space curve, X, whose curvature is nowhere vanishing is moved along a small distance, e, along its principal normal vector to form a new curve, X(e), then the linking number of X with X(e) is independent of e for sufficiently small e. This number is defined to be the self-linking number of X.

This paper's purpose is to develop the theory and all the ramifications of self-linking numbers for polygons in 3-space. The proofs rely upon a projection theorem which allows one to calculate self-linking numbers easily by projecting the curve to a plane adn then summing the number of crossing and signed pairs of inflection points.

Section 1 proves this basic theorem using deformation methods. The seconds section uses these techniques to prove a finite form of the Gauss integral. Section 3 derives integral formulas for self-linking numbers of polygons. The fourth section demonstrates the connection between these formulas and the definitions of Calugareanu and Pohl. The fifth section examines the behavior of self-linking numbers under deformations, and the last section shows how these ideas can be generalized to higher dimensions.alculate self-linking numbers easily by projecting the curve toa plane adn then summing the number of crossing and signed pairs of inflection points.