The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers

2009/12/16 by Yuanan Diao, Claus Ernst, Diao, Yuanan +5 · 2 citations
Computer Science · Mathematics · #57M25 #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.0912.3282

openalex publication_date 2009/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln⁵ (Cr(K)) for any knot K. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number.

Cited by

• Milnor invariants and thickness of spherical links
• Gordian split links in the Gehring ropelength problem

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• Braid Index Bounds Ropelength From Below