Nœud de trèfle | EPFL Graph Search

In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. The trefoil knot is named after the three-leaf clover (or trefoil) plant. The trefoil knot can be defined as the curve obtained from the following parametric equations: The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus : Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3 (a cuspidal cubic). If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot. The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic.) Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve. The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it.

Computation and visualization of ideal knot shapes

Mathias Carlen

We investigate numerical simulations and visualizations of the problem of tying a knot in a piece of rope. The goal is to use the least possible rope of a fixed, prescribed radius to tie a particular knot, e.g. a trefoil, a figure eight, and so on. The ropelength of the knot, the ratio to be minimized, is its length divided by its radius. An overview of existing algorithms to minimize the ropelength is given. They are based on different discretizations. Our work builds on the biarc discretization, for which we have developed an entire C++ library "libbiarc". The library contains a variety of tools to manipulate curves, knots or links. The biarc discretization is particularly well suited to evaluation of thickness. To compute ideal knot shapes we use simulated annealing software, which is also included in "libbiarc", on a biarc discretization. Simulated annealing is a stochastic optimization algorithm that randomly changes the point or tangent data. In the quest to find appropriate moves for this process we arrived upon a Fourier representation for knots, which allows global changes to the curve in the annealing process. Moreover, with the Fourier representation we can enforce symmetries that a given knot might have. To identify these symmetries we use visualization of simulations where symmetry was not enforced. Visualization of knot shapes and their properties is another important aspect in this work. It ranges from simple graphs of the curvature of a knot, through 2-dimensional plots of certain distance, circle or sphere functions, to 3-dimensional images of contact properties. Specially designed color gradients have been developed to emphasize crucial regions of the plots. We show that the contact set of ideal torus knots is a curve that is ambient isotopic to the knot itself, which is a result first suggested by visualization. A combination of numerics and visualization made us aware of a closed trajectory within the trefoil knot, a 9-billiard. Consequently the symmetries and the billiard make it possible to represent the trefoil with only two curve sub segments. We also anneal and visualize knot shapes in the unit 3-sphere or S3. In particular we present the contact set of a candidate for optimality, whose curved contact chords form Villarceau circles, which in turn span a Clifford torus embedded in the unit 3-sphere. Finally some knots and contact surfaces are constructed as physical 3D models using 3D printers.

EPFL2010

Computation and visualization of ideal knot shapes

Mathias Carlen

EPFL2010