Our maint result is that integrated geodesic curvature of a
non-single 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
analog to the winding number of closed plane curves are found
using Gauss-Bonnet's Theorem after cutting the curve into simple
closed sub-curves. At this point an error in the litterature is
corrected. If the spherecal curve is the tangent indicatrix of a
space-curve we obtain a new short proof of a formula for
integrated torsion presented in a well ciculated unpubliched
manuscript by C. Chicone and N.J. Kalton and point out the
connection to Kroneckers Drehziffer. Applying our result to the
principal normal indicatrix we genneralice a theorem by Jacobi
stating:A simple closed principal normal indicatrix of a closed
space-curve with non-vaniching curvature bisects the unit 2-sphere
to non-simple principal normal indicatrices.We point out how
factorization of knot diagrams into simple closed sub-surves
define immersed discs with the knot as boundary and use this to
give a upper bound on the unknotting number of
knots.Keywords:integral formulas, total curvature, total torsion,
index of spherical curves, unknotting number

Number of pages | 20 |
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Publication status | Published - 1997 |
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