Chapter 3: Open and Closed Loops
The same knot can look very different in different circumstances. It could be tied loosely or pulled tight.
It could have extra wrappings or crossings that do not contribute to the knottedness.
It could be tied in yarn, or string, or rope, a leather strap, a ship’s hawser. In what way are these all the same knot? There is also the question of the end of the string. At what point, when tying a knot, as you pass an end through a loop, does it become a knot?
One answer to these questions is the topological knot. Say we tie a knot in a string, and then fuse the open ends to one another. The string is now a closed loop, and the knot is caught in the loop — it can’t be undone without cutting the loop open. This is a good technique for studying knotting and tangling patterns. Since the knot can’t escape, we can manipulate it, study it, to try to discover its essential nature.
We don’t usually close the loop in real life — like when we tie our shoes — in fact we often want to remove the knot or tangle after it has served its purpose — like when we tie shoes or braid hair
The thing about strings is that they are flexible but strong: you can move them about, but they are usually difficult to break. We can use these two qualities to define loop equivalence. For instance, the two loops to the right are equivalent, because we could make the first loop into an approximate circle like the second loop without breaking it.
On the other hand, these two are not equivalent because we cannot turn the circle into the wavy line without cutting it open, nor could we turn the wavy line into a continuous circle without joining the open ends.
It matters a great deal whether a string is open —- meaning that it has two ends, or closed, meaning that it is a continuous loop with no end. If a knot tied in an open string — we can simply pull one of the open ends through the knot, and so undo it without cutting.
But a knot in a closed loop cannot be pulled out, unless you cut the string. So the two closed loops pictured above are not equivalent. This is easy to believe from experience — tie a knot in an open string, tape together the open ends to make a closed loop, and try to take out the knot. It is easy to believe, but the currently known mathematical proofs of this simple fact are complex. Maybe you can find a better one.
