Curves of Finite Total Curvature

2008/01/01 by John M. Sullivan · 1 citation
Mathematics · Engineering · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Advanced Numerical Analysis Techniques #Mathematics #Curvature #Knot (papermaking) #Class (philosophy) #Differential geometry of curves #Total curvature #Differential geometry #Willmore energy #Differential (mechanical device) #Geometry #Pure mathematics #Mathematical analysis #Mean curvature #Mean curvature flow #Differential equation #Computer science #Physics #Ordinary differential equation

paper · doi:10.1007/978-3-7643-8621-4_7

openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz.

Citations

• Using the Borsuk–Ulam Theorem
• Alternating Quadrisecants of Knots
• Filling Riemannian manifolds
• Criticality for the Gehring link problem
• Intégrale, Longueur, Aire
• Allgemeine Untersuchungen über Rectification der Curven
• The distortion of a knotted curve
• The second hull of a knotted curve
• Quadrisecants give new lower bounds for the ropelength of a knot
• On the minimum ropelength of knots and links
• Weakly Differentiable Functions
• On the Total Curvature of Knots

Cited by

• Fáry–Milnor theorem [wikipedia]

Related

• Curves of Finite Total Curvature
• A Survey of the Differential Geometry of Discrete Curves
• Total Curvature of Graphs after Milnor and Euler
• Discrete curve theory in space forms: planar elastic and area-constrained elastic curves
• Total curvature and isotopy of graphs in R³