Why a knit runs and a weave frays

Cut a woven cloth and the threads at the edge slide out one at a time, a few millimetres, and stop. Cut a jersey and a wale unfastens itself from the cut to the far edge in a second, taking a stripe of the fabric with it.

Both failures are famous, both are named, and the two are so unalike that they are hard to think of as two answers to one question. This collection described both in an early essay, in an essay about ravelling, fraying and running, and described them well; what it could not do was say why the two fabrics have different failures, other than by pointing at the difference between them.

The two now have a cause, and it is a single number computed the same way for both.

The two numbers #

A linking number counts how many times one closed curve passes through another. It is an integer, it cannot change while the curves stay unbroken, and it does not depend on any material constant.

For two adjacent courses of a knitted tube it is the number of wales: every needle loop is drawn through the loop below, once, in the same sense.

For any two threads in any woven cloth it is nought: weaving takes a thread over its neighbour and brings it back, and a round trip through nothing is nothing.

Two fabrics, one instrument, two answers, and the answers are as far apart as an integer quantity can put them.

What each number implies about holding #

The implication runs in a direction that is worth spelling out slowly, because the two fabrics need opposite things and it is easy to get the sense backwards.

A structure with linking is held whether or not it is pressed together. Nothing needs to be tight. A knitted fabric at any density, from a fine machine jersey to a hand crochet with a centimetre of air between its loops, is a fabric, because what holds it is threading rather than friction.

A structure without linking is held only by friction, and friction has to be paid for with contact — a normal force, a coefficient, and enough crossings along the length being resisted. A woven cloth therefore has a minimum density below which it stops being a cloth, and this collection has a whole field about where that number comes from.

That is the asymmetry. The knitted fabric is held by an arrangement, which is an integer; the woven cloth is held by an exponential, which has a scale.

Fraying, derived #

A cut across a woven cloth severs every weft pick it crosses. Each severed pick is now a free end held by friction against the ends it crosses.

How hard is it to pull out? The grip at each crossing is a normal force from the crimp times a coefficient of friction, and the forces accumulate along the length — not additively but through a capstan relation, because each crossing wraps the thread through an angle and drops the tension by a factor. So the resistance grows roughly exponentially in the number of crossings the thread still has left in the cloth.

That gives fraying its whole character in one sentence: the first few crossings hold weakly and every crossing after that holds much better, so a cut edge sheds a short fringe and then stops. The fringe length is where the exponential takes over, and it is a few millimetres for an ordinary shirting and considerably more for a satin, because a satin has fewer crossings per centimetre.

Nothing propagates, because pulling one pick out does not weaken the ends it crossed. They are still crossed by every other pick.

So fraying is local, self-limiting and slow, and all three properties come from friction being the mechanism.

Running, derived #

A knitted fabric’s failure is the opposite in every one of those respects.

Break one loop and the loop threaded through it has lost the only thing holding it. It is now free — genuinely free, not weakly held — and the tension in the fabric pulls it out of the loop below, which is then free in its turn.

That is a propagating failure, and it propagates because each step of it creates the condition for the next. There is no accumulation to overcome and no exponential to run into: what held loop k was loop k+1, and loop k+1 is gone.

It is also fast, and fast for a reason worth having. The energy released is the loop’s own bending energy returning as its yarn straightens, and the fabric does not have to be pulled for it to happen — a jersey runs on the table. This collection has computed that race: a run is a race between two energies, the bending energy released against the friction resisting the yarn’s withdrawal, and it runs when the first wins.

And it is directional. A run goes along a wale, because that is the direction the threading chains in, and it goes in the direction the loops open rather than both ways.

The two remedies are different, and that is the check #

A derivation is worth more when it predicts what has to be done about the thing it explains, and here the two remedies are as unalike as the two failures.

To stop a cloth fraying, add friction or add crossings. Overlock it, hem it, fuse it, mill it, resin-finish it, weave a leno at the selvedge, or weave the whole cloth tighter. Every one of those works and every one of them is about grip.

To stop a knit running, break the chain. A run is stopped by anything that makes one loop unable to be pulled out of the next: a tuck stitch, a change of structure, a locked selvedge, a fusible tape, an interlock construction where each course is bound on both faces, or a fabric felted enough that the yarn cannot slide at all.

Notice that only the last of those is about friction, and it is the one everybody reaches for last. Notice too that tightening a knit does not stop it running, which is exactly what the derivation predicts and is not obvious: a very tight jersey runs perfectly well, because tightness raises the friction and the run is not held back by friction alone.

That asymmetry is the check on the argument. If both failures were friction failures, both would be helped by tightening, and one of them is not.

Why interlock does not run and jersey does #

The cleanest test case is a knitted structure that fails to run, and there is one.

Interlock is two rib fabrics knitted into one another, so that every course is bound on both faces. A loop pulled out of an interlock does not free the loop above it, because that loop is also held by a loop on the other face. The chain has two strands and breaking one leaves the other.

Interlock is famously run-resistant and is used where a jersey would be unacceptable. The linking picture says why: the fabric’s linking graph is not a chain but a lattice, and a lattice does not unfasten from a single break.

A rib sits between the two and behaves accordingly: it runs, but less readily than a jersey, because a run has to cross from one bed’s wales to the other’s and a run cannot cross a bed.

Three structures, three degrees of run resistance, and the ordering follows from the shape of the linking rather than from any measurement.

What a cut does to each, in one sentence each #

The two failures can now be stated in a form that makes them obviously the same question.

Cutting a woven cloth removes friction from a thread that had nothing else. The thread comes out, over the length where the remaining friction is small, and stops where it is not.

Cutting a knitted fabric removes a link from a loop that had nothing else. The loop comes out, and the loop that depended on it comes out after it, and so on to the edge.

Both are the removal of the only thing holding a thread. In one case the only thing is a force and the removal is partial; in the other the only thing is an arrangement and the removal is total.

The third failure, which belongs to neither #

There is a third way fabric comes apart and it is worth putting beside the other two, because it is the one that is neither.

Seam slippage is threads sliding away from a line of stitching until a gap opens beside the seam. It happens in woven cloth, it is a friction failure like fraying, and it is not fraying: nothing comes out of the cloth, and the cloth is intact on both sides of the gap. What has failed is the cloth’s ability to resist a shear along a line.

The linking picture puts it exactly where it belongs. A woven cloth’s resistance to any relative motion of its two thread systems is friction and nothing else, so a load applied along one system and reacted by the other is carried entirely by the crossings. Enough load and the crossings slide. This collection has computed that force and found it is what holds a thread in a seam, and the number is not large.

A knitted fabric does not slip at a seam in the same way, and the reason is that it has no two systems to shear against one another: it has one thread, and shearing it means deforming loops rather than sliding one thread past another. It fails differently and mostly by stretching.

So of the three failures, two are friction and one is topology, and knowing which is which decides what to do about each.

Why a knit’s remedy has to be structural #

The practical consequence of the derivation is worth stating as advice, because it inverts the intuition.

A woven cloth that is failing can usually be finished out of trouble: mill it, resin it, set it, or beat it up harder on the loom. All of these raise the friction and friction is the mechanism.

A knitted fabric that is running cannot be finished out of trouble in the same way, and attempts to do so are mostly disappointing. Raising the friction slows a run and does not stop it, because a run is not held back by friction — it is held back by nothing at all once the link is gone, and friction only decides how fast the freed loop comes out.

What stops a run is a change of structure: a tuck that binds a loop to a second neighbour, an interlock that binds every course on both faces, a locked edge, or a course of a different stitch. Every one of those adds a second link to a chain that had one.

That is why run-stop courses exist in stockings, why interlock costs more than jersey and is used anyway, and why a laddered stocking is stopped with nail varnish — which is not a friction treatment but an adhesive one, and works by removing the yarn’s ability to move at all.

What was counted, and how #

Nothing on this rung is new arithmetic. The two linking numbers are computed on the previous two rungs, by the same Gauss double integral, on this collection’s own solved geometry, and checked against four arrangements whose answers are known by inspection.

What is new is the pairing, and the pairing is what makes the two derivations possible: a failure mode is a statement about what holds a structure together, and a linking number is a statement about what holds a structure together that survives every constant.

The capstan arithmetic behind the fraying length is this collection’s own, from the ladder on what holds a thread in a seam, and it is used unchanged. The run’s energy balance is likewise the collection’s own and is not recomputed here.

Where the model stops #

The run derivation describes a fabric the model does not have. This collection’s own knitted geometry has a linking number of nought, so the mechanism above is a statement about knitting rather than a result computed from the site’s own machinery. That is an honest and uncomfortable position and it is stated rather than hidden: the argument is right and the model cannot make it.

Neither derivation gives a number. How long a fringe a given cloth sheds, and how fast a given jersey runs, are quantitative questions with parameters in them, and the linking number has none. It says which mechanism, not how much.

And the middle cases are not covered. A warp-knitted fabric — tricot, raschel — is made by threading, so it links, and yet most warp knits do not run, because the loops are threaded sideways as well as vertically and a single break does not free a chain. Nothing here computes that, and it is the obvious next structure to ask about.

Nor are the finishes. Milling, resin and heat setting change a cloth’s failure completely and change no linking number at all, so anything they do is outside this account entirely.

The generalisation #

The habit this rung is an instance of is worth naming, because the collection has now done it twice and did not notice the first time.

When two behaviours look unalike, look for one quantity that takes different values. Fraying and running are so different in speed, direction, extent and remedy that treating them as two answers to one question requires an act of will. They are two answers to one question, and the question is what holds the thread.

The first instance was the integrity check: a great many apparently different ways for a draft to be bad turned out to be one connected-components count. The second is this. Both times the unifying quantity was an integer over an arrangement, both times it was cheap to compute, and both times the behaviours it explained had been correctly described for a long time without being explained.

The next place to look is probably wherever this collection carries a list of distinct-seeming failures — a catalogue of weaving faults, a list of the ways a seam gives way — and asks whether they have a common instrument.

The two pictures either side of this sentence are the same comparison at two densities, and the point of drawing both is that only one of the two fabrics has a density at which it stops working.

The last picture is the mechanism with everything removed from it except the mechanism.

What this says about designing a fabric to fail well #

Every fabric fails eventually and a designer chooses which way, which is a decision this framing makes explicit rather than intuitive.

A woven cloth fails gracefully at an edge and catastrophically at a tear. Its threads are independent, so a small damage stays small; but a tear propagates along a thread line because the threads have nothing binding them across it. Tear strength is therefore a woven preoccupation and this collection has a whole rung on where the tearing happens.

A knitted fabric fails gracefully at a tear and catastrophically at a break. Its loops share load, so a tear has to break many of them and mostly does not propagate; but a single broken loop unfastens a wale.

So the two are complementary, and the choice between them for a given use is often a choice between those two failure profiles rather than between drape, weight or cost. A sail is woven because a broken thread must stay a broken thread. A bandage is knitted because it must conform and because a small hole must not become a tear.

That is a design rule with a derivation behind it rather than a preference, and it comes out of two integers.

Who found it, and when #

That knitted fabrics run and woven fabrics fray is knowledge older than any literature. That the difference is topological has been said in passing by several authors since the 1960s, usually in the course of explaining why knitted fabrics recover better.

What is this collection’s own is computing both numbers on its own geometry with the same instrument, and then using them to derive the two failure modes rather than to describe them — including the two predictions that make the derivation testable: that tightening a knit does not stop it running, and that a structure bound on both faces does not run at all.

Where the ladder goes next #

The knitted answer is a defect rather than a fact, and it is worth taking one more rung to see how far the defect reaches. Four things this collection had recorded separately as unexplained turn out to be the same omission seen from four directions, and putting them side by side is five symptoms of one omission — five, because the writhe makes a fifth.

After that the ladder leaves topology for a mechanism that is available in the model as it stands: what a twisted thread does when nothing is holding it straight, which is why a slack yarn snarls.