Knot Dynamics

TL;DR

This paper uses computer experiments and knot theory to explore how certain complex knots and unknots behave under self-repulsion, revealing that some do not simplify to minimal energy states.

Contribution

It introduces the concept of hard unknots and complexified knots that resist reduction under self-repulsion, supported by computational experiments.

Findings

Certain complex unknots do not reduce to simple forms.

Complexified knots can resist simplification under self-repulsion.

Phenomena are consistent across different computational models.

Abstract

We examine computer experiments that can be performed to understand the dynamics of knots under self-repulsion. 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 self-repulsion, and we find complex versions of knots that will not reduce to simpler forms under the self-repulsion. 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 self-repulsion. 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 self-repulsion 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.