This model simulates the shape of a tight clasp, that is, a ropelength-minimizing configuration of two linked arcs with endpoints fixed in parallel planes. Here, ropelength means the total length of the curves divided by their thickness, the radius of the largest embedded normal tube around them. We use the characterization from [1] of thickness as the infimal radius of circles through three points on the curves.

It is known that thickness equals either the minimum radius of
curvature of the curve or the minimum critical self-distance,
whichever is less. In particular, a curve of positive thickness is
**C ^{1,1}**.

Ropelength minimization is intended as a mathematical model of
tying knots tight in real rope. If the radius of the rope is **1**,
then we assume that the physical rope is the unit normal tube
around some core curve **K**. If this tube is embedded but
no larger tube is, then the thickness of **K** equals **1**,
and its ropelength is just its length.
To minimize ropelength means to use as short a length of such rope
as possible.

For two linked loops that remain distance **2** from each other,
the length-minimizing configuration is two round circles in perpendicular
planes; this was the solution to the Gehring link problem [2]. In [3],
we established the existence of **C ^{1,1}** minimizers
for any link type, and gave more examples of tight links.
For example, in the tight configuration of a simple linked chain,
each component (except at the ends) is a stadium curve, consisting of
two semicircles and two straight segments.

The clasp problem asks to minimize the ropelength of a pair of simply linked arcs with free boundary (endpoints) in parallel planes. It seemed natural to expect that in the minimizer the two curves would lie in perpendicular planes, and each would again consist of straight segments and a semicircle. But in work currently underway [4], we establish a balancing criterion for critical points of ropelength; this criterion is not satisfied by that obvious configuration. Indeed it is not hard to exhibit a ropelength-decreasing perturbation. We expect that we will eventually be able to use our balance criterion to get equations describing this tight clasp exactly.

This model presents the results of Evolver [5] simulations of the
tight clasp. As described in [6], we fix length and maximize not
thickness but a discretization of a smooth energy functional.
If we let **k(x,y)** denote the curvature of the circle
through **x** and tangent to the curve at **y**, then
the reciprocal of thickness is the supremum of **k(x,y)**.
Instead of minimizing this supremum, we minimize the **L ^{p}**
norm of

The simulation results lead us to conjecture that
the minimizing clasp, though necessarily **C ^{1,1}**,
by results of [2], has curvature discontinuities not only at the ends
of the straight segments, where it jumps from

The first image is an orthogonal projection of the clasp, where one strand appears in its exact shape and the other is collapsed to a line.

The other three images show graphs of curvature as a function of
arclength along one of the arcs of the clasp. We show the graph for the
numerical minimizer for three different values of **p** and **n**
(the number of edges on a polygonal approximation to each arc):
**p=4096, n=256** and **p=8192, n=256** and **p=4096, n=512**.
The close similarity of all three supports the conjecture that this
model is a good approximation to the ropelength minimizer or
tight clasp.

The dotted horizontal lines show curvature levels **0** and
**1/2** and **1** (assuming the clasp has been scaled to
have thickness **1**). Near the base of each component,
there is evidently an arc of a circle of radius **2** around
the tip of the other component, while near the tip there is
a small arc of a unit circle; this kink is constrained by its own
curvature, not by touching the other component.

Limited experimentation suggests that this is indeed the minimizing configuration of the clasp. If we add a twisting perturbation that might allow the two components to have helical segments, this is removed by the gradient descent flow, and the clasp returns to the configuration shown here.

Model produced with: Evolver 2.16

Keywords | thick knots; minimum ropelength | |

MSC-2000 Classification | 57M25 (49Q10, 53A04) | |

Zentralblatt No. | 05264884 |

- Oscar Gonzalez and John H. Maddocks:
*Global Curvature, Thickness, and the Ideal Shapes of Knots*, Proc. Nat. Acad. Sci. (USA)**96**(1999), 4769-4773, . - Edelstein, Michael and Schwarz, Binyamin:
*On the length of linked curves*, Israel J. Math.**23**(1976), 94-95, . - Jason Cantarella and Robert B. Kusner and John M. Sullivan:
*On the Minimum Ropelength of Knots and Links*, Invent. Math.**150**, 2 (2002), 257-286, http://www.arXiv.org/abs/math.GT/0103224. - Jason Cantarella and Joseph Fu and Rob Kusner and John M. Sullivan and Nancy Wrinkle:
*Criticality for Ropelength*(2003), in preparation. - Ken Brakke:
*The Surface Evolver*, Exper. Math.**1**, 2 (1992), 141-165, http://www.susqu.edu/facstaff/b/brakke/evolver/. - John M. Sullivan:
*Approximating Ropelength by Energy Functions*, in Physical Knots, AMS Contemp. Math. (2002), 181-186, http://www.arXiv.org/abs/math.GT/0203205.

- Master File: clasp_Master.jvx
- Applet File: clasp_Master.jvx
- Preview: clasp_Preview.gif
- Image: clasp.gif
- Image: clasp256p4096.gif
- Image: clasp256p8192.gif
- Image: clasp512p4096.gif
- Original File (private file format): clasp_Original.fe

The original file is a Surface Evolver .fe file, but requires a not-yet-released version of Evolver that includes the new knot thickness energies. The first image is an orthogonal projection of the clasp; the other three are graphs of curvature as a function of arclength along the strand of the clasp.

Submitted: Thu Nov 15 7:39:57 CET 2001.

Revised: Tue Feb 11 15:00:29 CET 2003.

Accepted: Sun May 25 18:34:39 CET 2003.

University of Illinois, Department of Mathematics

1409 W Green St

Urbana, IL, USA 61801-2975

jms@uiuc.edu

http://torus.math.uiuc.edu/jms/