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This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot.

This paper details experiments in simulated physical movement of knots under self-repelling forces. We use the program KnotPlot of Robert Scharein [

The reader should note that there have been many years of computer experiments involving ropelength and knot energy. This paper makes no pretense at being original except in our observations of what can be done with the specific apparatus of the KnotPlot program. Furthermore, one can and should be skeptical about whether the results that we see using KnotPlot are special to that program. In particular, we show how certain phenomena appear in starting from certain specific configurations. It would be very interesting to take these same configurations and run them through other programs and to look at the whole situation with a more theoretical eye. Also there is no attempt here to review theoretical work by other researchers. This paper is about the specific explorations that are available from KnotPlot and similar programs. It will be of value to persons who wish to perform similar experiments of their own. In order to give the reader access to some of this work we have particularly listed references [

Here are some more caveats about the experiments that we present. There’s a whole science of numerical mathematics. The first thing to know about steepest descent methods is that straight gradient flow doesn’t work well in high dimensions. It almost always gets stuck in narrow valleys, slowing down with tiny oscillations. Here it’s essential to use a “conjugate gradient” method. (This is standard in Brakke’s “evolver”, which has been used for many of the serious knot energy simulations, but is not available in KnotPlot. Thus the experiments shown here can only be a beginning. The standard Ohara energy (with power 2—the mathematically interesting case where length need not be fixed) is in fact Moebius-invariant (as was proved in the paper of Freedman/He/Wang [

We encourage the reader to do experiments with KnotPlot as a first approximation and to ask many questions and search out better programs and read the literature. This paper is intended as a practical introduction to this subject.

Recall that a knot is an embedding of a circle in three dimensional spaces. The topological type of the knot is its equivalence class under ambient isotopy where two knots are said to be ambient isotopic if one can place them in a continuously varying family of knots starting with one, and ending with the other. We say that a knot is unknotted if its ambient isotopy class contains a simple planar circle, the unknot.

In general it is a difficult problem to tell whether a knot is knotted or unknotted. There are combinatorial algorithms that can determine knottedness from a graphical representation of a knot. On the other hand, there are physically motivated algorithms that apparently can unknot a knot. One of these is motivated by the following idea: Coat the knot with electrical charge and then let it self-repel without changing its length. If it is unknotted, one hopes that the self-repulsion will push it apart from itself into an obviously unknotted form.

This paper recounts experiments that the author has performedon self-repelling knots using the computer program KnotPlot [^{d}

In the computer model, the knot is represented by a string of vectors (_{1}, _{2}, ..., _{n}_{1} is connected to _{2}, _{2} to _{3} and so on with _{n}_{1}. Thus the knot itself contains the points corresponding to the vectors _{k}_{k}_{1}

On can also calculate the Simon energy ^{2} force (_{i} − _{j}

Part of the circumstances related to the behaviour of the energy functional is to do with an additional feature of the program KnotPlot. Along with implementing a repelling force, the program also models the connections between the points _{k}_{k+}_{1} as springs with restoring force behaving according to Hooke's law (force proportional to the extension of the length of the distance beyond a base distance corresponding to the length of the contracted spring). One can operate the program so the springs are either damped or undamped. With damped springs the program operates as though the springs are fully contracted, and no energy is exchanged with the springs. In undamped mode, the knot exchanges energy with its springs, moves around, oscillates and the energy functional will be seen to go up and down. Putting a knot in this process in the undamped mode is a way to subject it to perturbation that may allow it more freedom of movement than it can have in the damped mode. We will see how this plays out in the experiments.

There are two versions of the trefoil knot when you regard it as a torus knot. You view it as a (2,3) torus knot, winding twice around the torus in the longitudinal direction and three times around in the meridianal direction, as depicted in

The (2,3) torus knot with

In

The other version of the trefoil knot is the (3,2) torus knot as shown in

The (3,2) torus knot at

The end result for the (3,2) torus knot,

In

Note that this final result in

For the (4,2) torus link, we find a very similar phenomenon. Starting in the (4,2) configuration the link descends to the form shown in

(4,2) torus link with

And then on perturbation through the undamped repelling process, this relatively stable form goes into oscillation and descends to the form shown in

Full Descent of the (4,2) Torus Link,

We now turn to a way to geometrically measure configurations of knots in three-dimensional space that is related to the concept of

Here we point out that since there is the ropelength RL(^{−d}

ELR(T,6) ~32.68, electrical ropelength for the trefoil knot.

In practice

Trefoil with

It is also clear that the electrical ropelength will not in general be close to the minimal ropelength for a knot. When we work with a complex knot it is often the case that the force/energy minimum for the knot has some parts of the knot considerably closer to each other than other parts. This creates an inequity that results in the electrical ropelength being greater than the minimal ropelength.

Here is a remarkable phenomenon. When we start with a very symmetrical version of the (3,4) torus knot and use an undamped force evolution, the KnotPlot program retains the symmetry and stabilizes with the image shown in

(3,4) Torus knot minimized via damped, evolution.

(3,4) Torus knot minimized via undamped evolution,

With this example, we end our survey of experiments with energy and ropelength using the KnotPlot program. These experiments leave many open questions. We would like to know how to better explore the landscape of configurations of a knot or link and to determine the pattern of relative and absolute minima for the Simon energy. We would like to know if, in principle, pursuing an unknot down its self-repelling evolution will unknot it (with sufficient beads of course). We would like to better understand the geometric configurations of the minima and their relation to their ropelengths. We encourage the reader to carry out experiments of the sort described in this paper.