Gauss-Bonnet's Theorem and Closed Frenet Frames

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