Chapter 17: Breaking Strings and Stopper Knots
A trick sometimes used by seamstresses and tailors is to tie a knot in a thread that needs to be cut or broken. This serves two purposes — the knot weakens the thread, making it easier to break. It also determines where the thread will break, since it will break at the knot.
Exactly why it breaks at the knot is a bit of an open question. Most threads stretch a little under tension. Once the knot is tightened it is difficult for the thread to stretch through the knot. Also, as the string enters the knot the force vector from the pulling no longer lines up with tangent vector of — the direction — of the string. In many knots, the tangent vector takes an abrupt turn entering the knot. And the string usually breaks at the entrance of the knot. This has led some to theorize that the knot breaks the string by squeezing it. This seems unlikely in the sense of the string cutting through itself. But the fact that the rope cannot slide through the tightened knot does mean that in a sense it is being pulled around against a tight edge at the entrance to the knot. This has led some to theorize that the knot weakens the string in an amount proportional to the angle of turn the string takes as it enters the knot. This model works reasonably well for some common knots. However, it is possible to tie a knot so that the first bend is deep inside the knot, and have the string break there, so it appears that the situation is a bit more subtle than the simple model of bend at the entrance describes.
There is a history of people testing the strengths of various knots (or rather how much they weaken the line) in various filaments — fishing line, rope, polymers, etc. This is done with force meters or weight drops or even with a relative measure: tie two different knots in a line, then see where the line breaks when force is applied.
A knot can frequently be an obstruction, presenting a difficulty in handling a line. But other times this obstruction is exactly what is called for. Usually this is when the line slides through an aperture of some kind, and we would like to prevent the line from sliding through completely. In rigging, such as on a sailboat, a line may be threaded through a pulley, and when it is let loose we don’t want it to come out. In climbing, a safety rope may slide through a cinch or a closed hand as the climber moves, but we certainly don’t want to slide all the way through, which could result in a fatal fall. In the cell, protein strings are often formed inside the cell, but work their way through cell walls to get to their destination. To do this they pass through pores, which are holes in the wall. In many cases the protein string can pass through the pore before it folds into its final 3D shape, but not after it folds. We know that proteins, DNA, and RNA can be knotted, and so this general question of whether a knotted or folded polymer can pass through a pore, is a question with biological applications. Finally a seamstress, tailor, surgeon or cobbler might tie a knot as an anchor — to prevent a stitching thread from being pulled through a surface.
A knot that is tied to create such an obstruction is called a stopper knot. A good stopper knot will not come undone when the string is pulled so that it is held against the surface. And of course it presents a cross-section that is wider than the string, wider than the aperture. The first requirement means the the overhand knot, the trefoil, is not a good stopper knot, because it is relative easy to move along the string, coming undone when the end is pulled through. In the class of simple knots, the figure eight is much better, because as we saw in Chapter 2, it is much less likely to walk along the string.
The second requirement is a little subtle, because as we know knots can change shape, so what if the knot can adopt a long thin profile and therefore slide through the aperture?
Of course if we keep the knot pulled tight, perhaps it has to remain a certain width. But let’s try approaching this as a combination topological and geometrical problem. That is, say we had an ideal rope, such as we used in the chapter on ropelength. So the rope has diameter 1, is perfectly flexible, and keeps a perfectly circular cross-section. Now say we have a circular hole in a plane, of a given diameter, say 2. Is there some way to tie the rope so that we can be sure the rope will not slide through the hole? What if the diameter of the hole is larger?
Now this is perhaps not so interesting as a strictly topological question as stated, because if we tie the knot in an open length of rope we could always untie it and it will slide through. But we could assume the knot is in a closed loop, and even allow that the rope be a link of some number of closed loops. And in the interest of generality we put no restrictions on the lengths of the loops.
It is clear that there are entanglement patterns which have long and thin configurations, such as a product of many trefoils, or the simple chain of many links. Adding another trefoil component to the product or another link to the chain does not increase their efficacy as stopper knots, so here we have entanglements of arbitrarily high complexity which can fit through the same aperture as a single trefoil.
So, is there an ideal stopper knot or link pattern? That is, if we have an aperture of a given diameter, is there a pattern we can tie in unit radius ideal string that we know will not fit through the hole?
The answer is yes. We make a link of N closed loops, where each loop is linked with every other loop.
Now say our aperture is a circular hole in the horizontal plane. Let our stopper link start completely below the plane, and we will try to pull it up through the hole. Because we have ideal rope loops, each of our loops has a center curve that is the center of the cylinder, just as in the ropelength chapter. Now as we pull through, consider the very first moment that a loop has its center curve on the horizontal plane or above, and none below.
Such a loop must have a point of its center curve on the horizontal plane, because this is the first moment of being on the plane or above, and we are assuming a continuous motion from below to above. But if an ideal rope of diameter 1 has a point of its center curve in a plane, it must intersect that plane with area at least Pi. So this loop takes up at least Pi area of the hole.
But it is all the other loops that finish the story. The other loops are all linked with this one, and since this one has no part of its center line below the plane, any loop linked with it must have a part above the plane. But since this is the very first moment that a loop has its center curve on the horizontal plane or above, and none below, this linking loop must also intersect the plane with area at least Pi.
This is because at this moment there are two possibilities. The first is that this loop also has its center curve on the horizontal plane or above, and none below, and so must have a point of its center curve on the horizontal plane. The second is that some of the center curve of this loop is below the plane and some above, which means of course that it intersects the plane.
So all of the loops intersect the plane in area at least Pi. And since they cannot inhabit the same space, this means that if the aperture has area less than N*Pi, the ideal stopper link cannot fit through.