Geometriae Dedicata

, Volume 134, Issue 1, pp 197–202 | Cite as

An index formula for the self-linking number of a space curve

Original Paper

Abstract

Given an embedded closed space curve with non-vanishing curvature, its self-linking number is defined as the linking number between the original curve and a curve pushed slightly off in the direction of its principal normals. We present an index formula for the self-linking number in terms of the writhe of a knot diagram of the curve and either (1) an index associated with the tangent indicatrix and its antipodal curve, (2) two indices associated with a stereographic projection of the tangent indicatrix, or (3) the rotation index (Whitney degree) of a stereographic projection of the tangent indicatrix minus the rotation index of the knot diagram.

Keywords

Self-linking number Writhe Winding number Rotation index Whitney degree Total torsion Total geodesic curvature 

Mathematics Subject Classification (2000)

57M25 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark

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