# Is friction required for knots (not hitches or bends) to hold fast to a rope?

Using rope with idealized coefficient of surface friction approaching zero, that is in all other ways typical (typically stretchable, compressible, flexible and twistable), is it possible to tighten a true knot, defined here as a “prime knot” according to knot theory, (which excludes hitches such as the clove hitch covered in Baymans Theory of Hitches (1977) and excludes bends such as the square knot and sheepshank addressed by Maddocks and Keller (1987)) to a point that non frictional forces, such as forces from pulling or twisting, will suffice to allow the knot to hold as a stopper knot? To simplify, are the tightening forces (and / or torsional forces ) that occur when we try to pull a trefoil (aka “overhand knot”) through an opening too small for it to fit through, enough to keep a it from unwinding?

• Maybe I am missing something here, but isn't a knot always held tight, as long as the rope is taut? Regardless of friction? How could a knot ever unwind if the ends of the rope are fixed? And in the case of prime knots where there are no ends, it requires the rope to break somewhere for the knot to unravel, if I'm not mistaken. Regardless of friction. Aug 21, 2020 at 20:20
• Yes but the question is when only ONE end of the rope is being pulled. So only part of the rope is taut... the part between the knot and the end being pulled. Take a piece of string tie a knot in it. Hold the string between your thumb and finger and pull until the knot is stopped by your fingers. At that point (assuming your fingers are strong enough to hold the knot in place) either the string will unwind or it will break as you pull harder and harder. so if there is no friction in the string can the knot hold, or is there something other than friction that creates the knot. Aug 21, 2020 at 20:31
• There's one way to find out. Grease a rope, tie several knots in it, pull on it, and see if the knots come undone. Aug 22, 2020 at 17:23
• David, so funny you should say that. Just about 10 seconds ago I thought "What a stupid question I have asked... based on my spending hours sifting through literature, and seeing how little study has been done regarding forces on knots, I would bet nobody knows the answer, so I guess I will have to experiment to find out. I am still holding out for someones keen insight but I will start experimenting while I wait. Aug 22, 2020 at 17:30
• If I had to bet I would bet that modulus of elasticity can make a knot hold in a rope with a surface Cof near zero. Aug 22, 2020 at 17:37

Imagine a frictionless rope in a tube.

Now make an overhand knot out of tube with the rope inside.

Pull the rope, holding the tube. It is frictionless, OK ?

Now remove the tube. The rope-only knot differs in a sense that the rope contact with itself (just like with the tube) or has free surface (that has even more freedom).

• Good way to visualize it, but I think it doesn't address the question. In the example you give the rope is un able to "squeeze" itself. Hence there is really no knot in your example as I see it. When I "tighten" a knot I am using one part of the rope to apply a constricting pressure on another part of the rope (constriction is not allowed in your example). Friction-less doesn't bring with it un-compressible. If I can compress a friction-less rope enough, will that halt the motion? Maybe that is a better way to frame the question. Aug 21, 2020 at 22:47
• @AAnderson You ask:"If I can compress a friction-less rope enough, will that halt the motion?" That depends. If the rope has internal friction then a force is required to shift the position of the contriction along the length of the rope. If there is no internal friction then the rope is like a tube filled with a friction-less fluid. Then any constriction of the rope is free to shift along the length of the rope. Aug 22, 2020 at 7:23
• Cleonis, Great nuance. I meant to say any avg rope (it can 1. stretch, 2. be compressed, 3.be twisted, and 4. has a stiffness) except it has a CoF approaching zero. And each of the other 4 properties may be contributing to a knots ability to resist slipping apart. IOW, what ultimately holds a knot in place?. If it is all friction then then the material becomes more important that the knot, A typically "reliable" knot in a slippery rope is worse than an a typically "unreliable" knot in a rope with a high CoF. This could have a big impact in real world engineering. Aug 22, 2020 at 15:04
• @AAnderson If the rope has stiffness - that is, it resists bending to some extend - then that stiffness alone will untie the knot already. The rope will try to straighten out, pulling the free end back through the knot. Jan 18 at 21:31

Yes, friction is required to hold a knot fast. This is why stronger knots have more turns. Each turn is adding frictional forces to keep the knots shape.

No. To be frictionless would mean that there is no electromagnetic interaction, so what force is left to hold it together? Perhaps a gravitational rope...or one that depends on the strong or weak force...

• So are you saying that without friction knots are impossible. That would mean (I think) that you are saying you cant ever squeeze a frictionless object tight enough to hold it? (like trying to grab a pumpkin seed - it always squirts out) But what if you could squeeze the pumpkin seed enough to create little dimples. At that point, even if friction-less, you could squeeze and hold the seed cuz the sliding forces would be in offsetting directions. So, if as the knot tightens it crates a sort of hourglass in the rope, could it hold, even without friction? Aug 21, 2020 at 20:55
• @AAnderson your example is what I thought at first, but I thought that would essentially the same interaction as friction, except on a macroscopic scale. The dimples in your example play the same role that very small irregularities would have for a rope with friction. Sep 1, 2020 at 11:52