Prove that an infinitely long rope can only form slipknots

Question

I've heard that an infinitely long rope can only form slipknots, is that true, and is there a simple proof/obvious counterexample? Answers requiring no preliminary knowledge about topology would be preferred. Thanks!

Answer 1

If you start with an infinitely long rope and "form" a knot without moving the ends (assumed infinitely far away), then the resulting "knot" can certainly be untied by performing these orerations backwards and hense also without involving the ends (so that in the sense of knot theory this knot is naught, it's not a knot).

Whether actually doing this with a physical rope (with friction and other physical constraints, such as loops "clamping" other parts of the rope) means that actually pulling at the ends results in slipping is however a different question. For example, you could form a loop in the middle and make a "real" knot of it by treating it as a short rope with a freely moveable end. It is not clear that a real rope with friction would always cause this construct to "slip".

Answer 2

Well first, what you call a knot isn't what a knot theorist would call a knot, although it can be made equivalent. A mathematician's knot is a closed curve in space. The space we choose is normally $\mathbb{R}^3$ or, more suitable in this case, the $3$-dimensional sphere $S^3$ which we can think of as $\mathbb{R}^3$ with a point at infinity - the point that the two ends of the rope are connected to. In this setting, we can think of an infinitely long rope (whose ends go off to infinity) as a closed curve in the sphere $S^3$ which goes through the point at infinity - these two notions are equivalent so we can freely move from one setting to the other.

So now we also need to tackle what it is you mean by 'pulling on the ends of the rope'. I don't think this has much meaning when we're in the realm of knot theory where all our strings are stretchable and there is no friction (which in the real world would stop us being able to 'pull on both ends' to untie the knot). Do you mean that starting with an infinitely long rope which starts off as 'straight', you want to show that we can't knot it in a way such that it is unknottable? This follows almost immediately from the definition of 'ambient isotopy' which is what we say to mean a smooth action that is performed on the rope to knot it - and for our purposes we would say we also want the isotopy to fix the point at infinity, the two ends of the rope. It should be pretty clear that if we can perform an ambient isotopy on the straight rope to get a 'knotted rope', we can also reverse that isotopy to get from the knotted form to the unknotted form.

Framed in the language of knot theory, this statement has almost no context. It just says that if a pointed knot $K$ in $S^3$ is pointed-ambient isotopic to the pointed unknot, then there exists a pointed-ambient isotopy of $S^3$ taking $K$ to the unknot. This is basically the definition.