Abstract
The average shape of the trefoil knot tied on a floppy, hard rope subject to thermal fluctuations has been determined. The fluctuations of the shape of knots were performed by random bending. As a result of the changing shape procedure large sets of deformed conformations of the initial knot were obtained. Afterwards, these sets were subject to the shape-fitting procedure. It has been found that the conformation is different from the ideal conformation of the knot.
Highlights
Suppose a closed knot has been tied on a piece of a rope subject to thermal fluctuations
An essential question arises: What is the average shape of the fluctuating knot? The answer to this question depends both on the physical properties of the rope in which the knot is tied and on what we mean by “the average shape”
As Saka and Takano have found, the trefoil knot tied on the elastic rope subject to thermal fluctuations becomes asymmetrical: the threefold symmetry characteristic of its tight conformation [4,5] vanishes —one of its three foils becomes larger than the two other foils
Summary
Suppose a closed knot has been tied on a piece of a rope subject to thermal fluctuations. The question concerning the average shape of a fluctuating knot, for knots tied on an elastic rope, has been recently asked and answered by Saka and Takano [2] The answer that they provide depends on the details of the interaction potentials defining the elastic properties of the model rope they used, and on the temperature of the thermal bath within which the simulated knot was submerged. As Saka and Takano have found, the trefoil knot tied on the elastic rope subject to thermal fluctuations becomes asymmetrical: the threefold symmetry characteristic of its tight conformation [4,5] vanishes —one of its three foils becomes larger than the two other foils. This has allowed us to precisely measure their lengths and determine how the lengths change with the increasing length of the rope in which the knot had been tied
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