On the minimum ropelength of knots and links

By Jason H. Cantarella, Robert B. Kusner, John M. Sullivan Jr.. 2001

242 citations recorded by the source corpus.

Abstract.The ropelength of a knot is the quotient of its length by 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 C1,1 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.

Subfields: Applied Mathematics, Computational Mechanics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2526577f28123279b38eb49c09eac6c9e7f8b996

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In Applied Mathematics

In Computational Mechanics

In Geometry and Topology

Citations in the corpus

EXISTENCE OF IDEAL KNOTS (2003, methodology): “presented in [ 11 ] show that the ideal configurations of some linked (but unknotted) curves are also not C2. Thus the C1,1 regularity established in”
Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm (2002, methodology): “1 on the meridian of Li and 0 on the meridian of every other component of L. In Example 8.3, we use Corollary 10.5 to estimate ropelength for a specific link from [ CKS ].”
Tangent-point self-avoidance energies for curves (2010, background): “[15], [7], [21]–[23], [12], [13], and has been studied numerically; see e.”
What are the Longest Ropes on the Unit Sphere? (2011, background): “Related variational problems in the context of nonlinearly elastic curves and rods with the same notion of thickness, in particular the existence of ideal knots, were treated in [13]; for ideal knots and links see also [3,11].”
Characterization of ideal knots (2004, background): “In [7] such conditions are studied under the hypothetical smoothness assumption that ideal knots were C(2)-curves, which we believe cannot be expected in general, since there are examples of ideal links, exhibited in [3], which are only C(1,1), and there seems to be strong numerical evidence that on”
Criticality for the Gehring link problem (2004, methodology): “telymanycomponents. (This was originally observed by Milnor [25].) Thus in the work to come, we need only to consider tame links. 7 Examples of critical links 7.1 The known length-minimizing links In [4], we showed that if one component of a link is linked to k others then its length is at least a c”
On rectifiable curves with Lp-bounds on global curvature: self-avoidance, regularity, and minimizing knots (2007, background): “In that context the global radius of curvature introduced by Gonzalez and Maddocks [20] turned out to be both a mathematically precise and analytically tractable notion to tackle energy minimization problems in nonlinear elasticity and knot theory for curves and rods with a given thickness [22], [8]”
Curves of Finite Total Curvature (2006, background): “istance between its endpoints. Chern, in §5 of his beautiful essay [Che89] in his MAA book, gives a proof for C2 curves and remarks (without proof) that it also applies to piecewise smooth curves. In [CKS02] we noted that Chern’s proof actually applies to C 1, curves, that is to curves with a Lipsch”
Regularizing and self-avoidance effects of integral Menger curvature (2010, unknown): “This result was the starting point for a number of variational applications involving curves or elastic rods with positive thickness and under topological constraints [15, 5, 13, 11, 12].”
The Thurston norm, fibered manifolds and twisted Alexander polynomials (2005, methodology): “In [ CKS02 ] the authors consider the link in Figure 6. Denote the component represented by a circle by L1, and the other com-”