Knot Dynamics
Abstract
We examine computer experiments that can be performed to understand the dynamics of knots under selfrepulsion. In the course of specific computer exploration we use the knot theory of rational knots and rational tangles to produce classes of unknots with complex initial configurations that we call hard unknots, and corresponding complex configurations that are topologically equivalent to simpler knots. We shall see that these hard unknots and complexified knots give examples that do not reduce in the experimental space of the computer program. That is, we find unknotted configurations that will not reduce to simple circular forms under selfrepulsion, and we find complex versions of knots that will not reduce to simpler forms under the selfrepulsion. It is clear to us that the phenomena that we have discovered depend very little on the details of the computer program as long as it conforms to a general description of selfrepulsion. Thus, we suggest on the basis of our experiments that sufficiently complex examples of hard unknots and sufficiently complex examples of complexified knots will not reduce to global minimal energy states in selfrepulsion environments. In the course of the paper we make the character of these examples precise. It is a challenge to other program environments to verify or disprove these assertions.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.12538
 Bibcode:
 2021arXiv210912538K
 Keywords:

 Mathematics  Geometric Topology;
 57K10
 EPrint:
 LaTeX document, 28 pages, 33 figures