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.
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Røgen, P. An index formula for the self-linking number of a space curve. Geom Dedicata 134, 197–202 (2008). https://doi.org/10.1007/s10711-008-9254-0
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DOI: https://doi.org/10.1007/s10711-008-9254-0
Keywords
- Self-linking number
- Writhe
- Winding number
- Rotation index
- Whitney degree
- Total torsion
- Total geodesic curvature