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

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    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.
    Original languageEnglish
    JournalGeometriae Dedicata
    Volume134
    Issue number1
    Pages (from-to)197-202
    ISSN0046-5755
    DOIs
    Publication statusPublished - 2008

    Keywords

    • writhe
    • total geodesic curvature
    • rotation index
    • winding number
    • self-linking number
    • Whitney degree
    • total torsion

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