Gauss–Bonnet's Theorem and Closed Frenet Frames

Abstract

Abstract. Our main result is that integrated geodesic curvature of a (nonsimple) closed curve on the unit two-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the tangent indicatrix of a space curve we obtain a new short proof of a formula for integrated torsion presented in an unpublished manuscript by C. Chicone and N. J. Kalton.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 nonvanishing curvature bisects the unit two-sphere to nonsimple 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.