On the minimum ropelength of knots and links

2001/03/31 by Jason Cantarella, Rob Kusner, John M. Sullivan +1 · 11 citations
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.DG #math.GT #msc:49Q10 #msc:53A04 #msc:57M25

paper · pdf · doi:10.1007/s00222-002-0234-y

published as Invent. Math. 150 (2002) 257-286 · 29 pages, 14 figures; New version has minor additions and corrections; new section on asymptotic growth of ropelength; several new references

arxiv created 2002/03/01 · openalex publication_date 2002/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01 · arxiv updated 2026/08/03

Abstract

The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C^1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp.

Cited by

• Geometric Constraints in Link Isotopy
• Profiles of Critical Flat Ribbon Knots
• The second hull of a knotted curve
• Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links
• Geometric complexity of embeddings in ℝ^d R d
• Errata and Comments for "Energy of knots and conformal geometry"
• Criticality for the Gehring link problem
• Quadrisecants give new lower bounds for the ropelength of a knot
• Curves of Finite Total Curvature
• Borromean rings [wikipedia]
• Hopf link [wikipedia]