Gauss-Bonnet's Formula and Closed Frenet Frames

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    Our main result is that integrated geodesic curvature of a (non-simple) closed curve on the unit 2-sphere equals a half integer weighted sum of the areas of the connected components of the complement of the curve. These weights that gives a spherical analogy to the winding number of closed plane curves are found using Gauss-Bonnet's theorem after cutting the curve into simple closed sub-curves. If the spherical curve is the tangent indicatrix of a space curve we obtain a new short proof of a formula for integrated torsion presented in an unpublished manuscript by C. Chicone and N.J. Kalton. Applying our result to the principal normal indicatrix we generalize a theorem by Jacobi stating that:a simple closed principal normal indicatrix of a closed space curve with non-vanishing curvature bisects the unit 2-sphere to non-simple principal normal indicatrices. Some errors in the literature are corrected.We show that a factorization of a knot diagram into simple closed sub-curves defines an immersed disc with the knot as boundary and use this to give an upper bound on the unknotting number of knots.
    Original languageEnglish
    JournalGeometriae Dedicata
    Issue number3
    Pages (from-to)295-315
    Publication statusPublished - 1998

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