A knot is nothing but contact

Every other way of joining two threads adds something. A sewn join adds stitches. A spliced one interweaves the two. A welded one melts them, a glued one bonds them, a whipped one wraps a third thread round both.

A knot adds nothing at all. A thread is led round itself in a particular order and pulled tight, and what holds it is that the thread presses on itself hard enough for friction to carry the load.

That makes a knot the cleanest available specimen of what contact does, which is why it belongs on a ladder about a fabric that occupies the same space twice.

The two things that happen #

Follow a thread into a knot and two quantities change along it, and they change in opposite senses.

The tension falls. A thread wrapped round something and pulled at one end has less tension at the other, by the capstan relation: the tension drops by e to the minus mu theta, where theta is the angle wrapped and mu is the coefficient of friction. That is the same arithmetic this collection uses for a thread held in a seam and for a yarn withdrawn from a cloth, and it is exponential in the wrap.

The curvature rises. The thread is bent to a radius of a few of its own diameters, and the outside of a bend carries a strain on top of whatever the tension supplies.

So the fibre at the outside of the bend is carrying a tensile strain from the load and a bending strain from the geometry, and it breaks when the sum reaches its own breaking strain.

Why it holds #

The holding is entirely the first of those, and the arithmetic is worth doing because the answer is not obvious.

A coefficient of friction of a third and a wrap of one full turn gives a tension ratio of e to the minus two point one, which is about an eighth. So one turn of wrap leaves an eighth of the tension at the far end.

Two turns leave a sixty-fourth. Three leave a five-hundredth.

That is why a knot holds and why it holds so decisively. The load at the standing part is carried, and by the time the thread has gone round twice there is almost nothing left for the tail to resist. A knot does not have to be tied tightly to hold; it has to wrap enough.

It is also why a knot in a slippery thread is a different proposition. Halve the friction coefficient and one turn leaves a third rather than an eighth, and the number of turns needed to reach the same security nearly doubles — the same sensitivity a seam’s grip has.

Why it costs #

The cost is entirely the second, and it is the part that makes a knot expensive rather than free.

A thread bent to a radius rho has its outermost fibre stretched by r over rho, where r is the radius of whatever is bending. That strain is spent before any load arrives, so the thread has less breaking strain left to give.

For a yarn bending as a solid section, r is half the yarn’s diameter. At a bend radius of one diameter that is a strain of a half — fifty per cent — against a cotton’s breaking strain of about seven per cent. Such a yarn could not be knotted at all; it would break in the tying.

For a yarn whose fibres bend individually — the free end of the bracket — r is half a fibre diameter, which for a twenty tex cotton is thirty times smaller. The bending strain is three and a half per cent, and about half the breaking strain survives.

Ropes are knotted, and knots hold. So the second is what happens, and a knot is a third reading of this collection’s oldest bracket.

What a knot is not #

Three things a knot is often described as and is not, and each misdescription leads somewhere wrong.

A knot is not a jam. Nothing is wedged. The thread is not trapped in a narrowing space, and a knot in a perfectly frictionless thread would slip apart no matter how tight it looked. Friction is not incidental to a knot; it is the whole of it.

A knot is not a topological object, in the sense this work has been using. A knot in a piece of rope with two free ends is topologically trivial — anybody can untie it — and its holding is not protected by any invariant. That is a real difference from a knitted loop, where the holding is topological and friction is optional.

And a knot is not a stress concentration in the usual sense. The strain at the outside of the bend is not a concentration produced by a notch or a corner; it is the ordinary strain of bending, and it is the same everywhere along the bend.

Which is why knots differ #

The two mechanisms explain the whole of why one knot is better than another, and it is worth being explicit because the folklore is not.

Security — whether a knot holds without slipping — is decided by the total wrap. A knot with more turns is more secure, and the dependence is exponential rather than linear, which is why one extra turn often converts an unreliable knot into a reliable one.

Efficiency — how much of the rope’s strength survives — is decided by the tightest bend radius anywhere in the knot. A knot that leads the loaded part round a gentle curve costs little; one that puts a hard bend in the standing part costs a great deal.

Those are independent. A knot can be secure and inefficient, which is what an overhand is; or efficient and insecure, which is what a poorly dressed bowline is. The folklore treats them as one axis and calls knots “good” or “bad”, and the arithmetic says there are two.

Where the two meet #

There is one interaction between them and it is the reason a knot has to be dressed rather than merely tied.

A knot that has been pulled tight without being arranged has its bends in the wrong places: the standing part takes a sharper turn than it needs to, because the tail has been pulled through and dragged it round. That costs efficiency without buying security, since the total wrap is the same however the turns are arranged.

Dressing a knot is moving the sharp bends off the loaded part and onto the tail, where they cost nothing because the tail is not carrying load. That is a real mechanism with a number attached, and it is why the same knot tied by two people can differ in strength by twenty per cent.

The comparison with a splice #

A splice is the alternative to a knot and the comparison is instructive, because it attacks exactly the term a knot cannot.

A splice interweaves the two ropes’ strands so that the load is transferred over a long length with no sharp bend anywhere. The tension is carried by friction along that length, exactly as in a knot, but the bending term is nearly absent: the strands follow gentle curves at radii of many diameters.

So a splice keeps ninety per cent or more of the rope’s strength, and a knot keeps about half, and the difference is entirely the bend radius.

That is a clean confirmation of the account rather than a separate fact. Two joins with the same holding mechanism, differing only in curvature, differing in efficiency by exactly the amount the curvature term predicts.

It also says what a better knot would have to do: lead the loaded part round the largest radius available. That is precisely what the knots with the best measured efficiency do — a figure-of-eight bend, a double fisherman’s — and it is why they are bulkier than the knots they replace.

What a fabric borrows from this #

The reason a knot belongs in a ladder about knitted fabric is not only methodological, and it is worth making the connection explicit.

A knitted loop is bent to a radius of about one yarn diameter at its tightest, which is exactly the radius a tight knot puts in a thread. So every stitch in a knitted fabric has the bending strain of a knot in it, and the arithmetic that says a knot halves a yarn’s strength says the same about a loop.

That is not a new claim — this collection has computed a knitted fabric’s strength as far below its yarn’s for several ladders — and it is a new reason for it. The usual account is that the yarn in a loop is at an angle to the load and only its component along the load counts, which is the obliquity argument applied to a fabric. The bending account is additional and is of the same order.

Which of the two dominates has never been asked here, and the two are separable: obliquity depends on the loop’s shape and bending strain on its radius, and the two move differently with the tightness factor.

What was counted, and how #

The capstan relation is the collection’s own and is used unchanged. The coefficient of friction is the mid-point of the range the site’s fibre table carries for the fibre in question.

The bending strain is computed at both ends of the stiffness bracket, because the two differ by the ratio of the yarn’s diameter to a fibre’s — a factor of thirty for a twenty tex cotton — and the difference is the whole result.

The breaking strain comes from the two specific quantities the collection’s own table holds: a tenacity in newtons per tex divided by a specific modulus in the same units is a strain, and the conversion is written once so no result carries a second version of it.

The check is that the maximum of the tension is at the entry, which is where a knot is observed to break, and that the capstan drops the tension through the wrap rather than raising it. Both would fail if the exponential had the wrong sign, which is a slip that would otherwise produce a plausible-looking curve.

Why the tail matters and the loop does not #

A consequence of the exponential that is worth stating because it settles a practical argument.

The load in a knot is carried at the standing part and falls through the wrap. By the time the thread reaches the tail there is almost nothing left — an eighth after one turn, a sixty-fourth after two. So the tail carries no load, and its only job is to be long enough that the thread cannot pull back through the wrap as the knot works.

That is why a knot’s security is quoted as a minimum tail length rather than as a force, why a short tail is dangerous and a long one is merely untidy, and why a knot that has been cycled loose is a knot whose tail has crept rather than whose friction has failed.

It also settles what “tightening” a knot does. Pulling a knot tight does not raise the friction — the capstan relation contains no normal force, because the normal force and the friction it produces both scale with the tension. What tightening does is seat the knot: it moves the turns into their final positions, so that further working does not shorten the tail.

Two mechanisms that are usually described as one, separated by an equation that has no normal force in it.

Where the model stops #

The thread is treated as wrapping a rigid cylinder. It is wrapping itself, and the thing it wraps is deforming under the same pressure. That flattens both, which raises the contact area and changes the friction, and none of it is computed.

The friction coefficient is a single number. A yarn’s friction is directional, load-dependent and different against itself than against anything else, and this collection has carried two coefficients rather than one for exactly that reason.

The bend radius is an input. Which radius a given knot takes is a question about the knot’s own equilibrium, and computing it is a contact problem with self-contact throughout — the hardest kind. Every number here is quoted at a stated radius.

And nothing is dynamic. A knot that has been shock-loaded is a different object from one that has been pulled slowly, and the fibres in a knot creep and reseat over hours.

The generalisation #

The rung is here for a reason that is about method rather than about knots.

This ladder is about a fabric whose parts occupy the same space, and the fabric is a complicated object: hundreds of contacts, all of them mutual, none of them isolable. A knot is the same physics with one contact region and two free ends, and it can be reasoned about completely.

When a mechanism is hard to see in the object of interest, find the simplest object that has only that mechanism in it. A knot has nothing in it but contact — no interlacing, no sett, no crimp, no topology, no second thread system — so anything a knot does is something contact does.

That is a general move and this collection has made it before without naming it. A single crossing was the simplest object with an interlacing in it; a single half period was the simplest object with a loop’s bending in it. Both were built as instruments and both turned out to be more informative than the fabrics they were abstracted from.

What a knot says about the fabric it is tied in #

One last connection, because it is the reason a knot appears in a collection about cloth rather than about rope.

Every fabric this collection computes is held together at contacts, and every one of those contacts is the same physics as a knot’s: a thread wrapped through an angle, pressing on a neighbour, held by friction that grows exponentially in the wrap.

A woven cloth’s thread wraps its neighbour through a small angle at each crossing — a few degrees of crimp — so each crossing holds weakly and the security comes from having many. That is why a cloth needs a sett.

A seam’s thread wraps through much more, which is why a seam holds better than the cloth around it.

A knot’s thread wraps through a full turn or more, which is why a knot holds better than either.

And a knitted loop’s thread wraps through nearly half a turn at each interlacing, which is a great deal — and yet a knitted fabric’s holding is not friction at all, because the loops are threaded. The friction is there and is not what is doing the work.

So the same arithmetic runs through the whole subject at different wrap angles, and the one structure where it does not decide the answer is the one where the topology takes over.

Who found it, and when #

The capstan relation is Euler’s, from 1775, and is the oldest piece of mechanics in this collection.

Knot strength has been measured since the nineteenth century and the rule that a knot halves a rope’s strength is older than any of the measurements. The explanation in terms of bending strain at the outside of the first curve is standard in the rope literature.

What is this collection’s own is computing the bending strain at both ends of its own stiffness bracket, and observing that the coherent end forbids knots entirely — which turns a familiar rule of thumb into evidence about a quantity the collection has never been able to pin down.

Where the ladder goes next #

Two questions follow immediately and both have sharp answers.

The first is where a knot breaks, which the two mechanisms settle between them: the tension falls monotonically and the bending strain is fixed, so the maximum is at the entry, before the knot has done any gripping at all. Where a knot breaks.

The second is how much it costs, which turns out to be about half — and the arithmetic that gives a half is the same arithmetic that says a yarn’s fibres slide. A knot halves a yarn and says why.