Chapter 18: Energy of a Knot

There is something about tying a knot perfectly, it can be such a satisfying experience. The knot can seem to take on a beautiful, symmetric shape almost on its own. It can seem to be a sort of little miracle, because the string or rope isn’t carrying the symmetry before the tying — it is just running through your hands or perhaps lying in a disorganized pile on the table or floor or deck. But in tying the knot, and perhaps pulling it tight, the form appears.

Are there natural forms for each of the mathematical knots and links? We saw some as the minimum ropelength positions in Chapter 7. Is there some way we could automatically find such a form? That is, could we tie the knot loosely in a string, and have it find its best form without us manipulating it?

Anything that is string-like can be knotted…… To talk about the energy of a knot we assume the string has some charachteristics we can measure. An energy is just a number that is calciulated for a knot. The ropelength is an energy, as is the stick number. For ropelength we assumed the knot was made of an ideal rope — a uniform bendable cylinder. For stick number we assumed it was an ideal chain of sticks.

Next we imagine the knot is a kind of charged infinitely flexible (but not stretchable) wire. The idea is that because of the charge it would self repell. So if it is the unknot, it would self-repell into the shape of a circle. The circle would be the minimum energy position (we hope).

It seems like it would relatively straightforward to do a rough computer simulation of this. Place some number of charges along the string, and model them all pushing away from one another. The usual way to do this is with an inverse distance force between the charges — the closer they are the stronger the push away. But if that is all we do, the charges just all run off toward infinity in different directions, like we did when Alex Mackensie showed up in the schoolyard. A way to keep it stringlike is to require that successive charges along the string remain a fixed distance apart while the string changes shape. A physical reading of this is that the string can bend, but it cannot stretch.

But still, there is a problem: how is the string a string between charges? There is nothing to prevent segments between charges from passing through each other, because thus far they are just ghosts: we imagine they are there but we haven’t given them any physicality.

We could try to fill in by adding more charges. Clearly this makes the energy greater because there are more charges and because sucessive and nearly succxessive charges are closer together, which makes an inverse distance term larger.

It also makes the energy larger because there are more charges. But maybe we can fix that by making the charge strength lesser, perhaps try to keep the total a unit — make the charge strength 1/n, where n is the number of charges. A problem is that even when we do this the energy goes to infinity as the number n of charges goes to infinity, so a single segment of this sort of string is a singular object. You can’t simply smear discrete charges into a continuous distribution in one dimension. (You can smear discreet people with plausible charges, but that is a different story.)

Our ways forward are at least two. We could modify the inverse distance law in a small way to somehow to prevent the segments from passing through one another.

An example is the box energy, also known in some places as the Buck energy. The only reason I’m hestitant to call it that here is that I’ve actually spawned at least three knot energies and I dont want to be accused of favoritism.

This energy has the nice property that it is becomes infinite as two segments approach each other, and reaches infinity before they actually cross. So if we use this energy the knot-type will not change as we follow the evolution of minimizing the energy.

Morwen Thistlethwaite used this energy to make some beautiful pictures.

So this does a good job for computation, but leaves us with a couple avenues for continuation. It would be nice to find something well-behaved in the continuouus limit. Or better, find some sort of inverse distance energy that is well-defined on a continuous curve to begin with.

Or we could use an inverse law with a different structure.

What if the string was self-repelling? That is, if it wanted to push away from itself, but could only do that so much, because the knot is in a closed loop, and we don’t let the string stretch.