Gauss-Bonnet's Formula and Closed Frenet Frames

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the tangent indicatrix of a space curve we obtain a new short proof of a formula for integrated torsion presented in an unpublished manuscript by C. Chicone and N.J. Kalton.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.