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Self-Linking Numbers of a Space Polygons

by Thomas F. Banchoff

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.