2004/08/31 by Elizabeth Denne, Yuanan Diao, John M. Sullivan +1 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Agronomy #Combinatorics #Geometric and Algebraic Topology #Knot (papermaking) #Knot invariant #Knot theory #Mathematical analysis #Mathematics #Pure mathematics #Trefoil #Trefoil knot #Tricolorability #Upper and lower bounds #math.DG #math.GT #msc:49Q10 #msc:53A04 #msc:57M25 #semigroups and automata theory
paper · pdf · doi:10.2140/gt.2006.10.1
published as Geom. Topol. 10 (2006) 1-26 · v3 is the version published by Geometry & Topology on 25 February 2006
openalex publication_date 2006/02/25 · arxiv created 2009/02/09 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Using the existence of a special quadrisecant line, we show the ropelength of any nontrivial knot is at least 15.66. This improves the previously known lower bound of 12. Numerical experiments have found a trefoil with ropelength less than 16.372, so our new bounds are quite sharp. 57M25; 49Q10, 53A04 More technically, the thickness .K/ of a space curve K is defined by Gonzalez and Maddocks [11] to be twice the infimal radius r .a; b; c/ of circles through any three distinct points of K . It is known from the work of Cantarella, Kusner and Sullivan [4] that .K/ D 0 unless K is C 1;1 , meaning that its tangent direction is a Lipschitz function of arclength. When K is C 1 , we can define normal tubes around K , and then indeed .K/ is the supremal diameter of such a tube that remains embedded. We note that in the existing literature thickness is sometimes defined to be the radius rather than diameter of this thick tube.