Computation and visualization of ideal knot shapes

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.

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