Geometriae Dedicata

, Volume 134, Issue 1, pp 197–202

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

Original Paper

DOI: 10.1007/s10711-008-9254-0

Cite this article as:
Røgen, P. Geom Dedicata (2008) 134: 197. doi:10.1007/s10711-008-9254-0


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.


Self-linking numberWritheWinding numberRotation indexWhitney degreeTotal torsionTotal geodesic curvature

Mathematics Subject Classification (2000)


Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of DenmarkKongens LyngbyDenmark