A woven cloth is not linked at all

Pull a thread out of the edge of a woven cloth and it comes. It does not have to be untied, unhooked or unthreaded; it slides, against friction, and when it is out there is a cloth with one fewer thread in it and a fringe where the thread used to be.

Everybody knows this and nobody has said why in a form that can be checked. The reason turns out to be a number, the number is an integer, and it is zero.

The claim #

A linking number counts how many times one closed curve passes through another, with a sign. It cannot change while the curves move continuously without cutting, and it takes an integer value.

For a woven cloth’s two thread systems it is nought, and the reason is one sentence: weaving takes a thread over its neighbour and brings it back. Going over and coming back is a round trip through nothing. It moves the thread up and down across the plane of the cloth and leaves it, topologically, exactly where it started.

Crimp does not change it. Crimp is how far up and down the thread goes, and however far up it goes it comes back down. A five per cent crimp and a fifty per cent crimp are the same arrangement with a different amplitude.

The weave does not change it. A plain weave takes the thread over one and under one; a five-end satin takes it over four and under one; a matt takes it over two and under two. Every one of those is a sequence of round trips.

And the number of threads does not change it. A cloth of a thousand ends has a thousand threads, each unlinked from each of the others, a thousand times over.

Why this is not a limitation of a model #

It is worth being emphatic about this, because the previous rungs were about a model failing to reproduce a fabric’s topology and this one is not.

Here the model and the fabric agree, and they agree because there is nothing to disagree about. A woven cloth’s threads really are unlinked. No refinement of any calculation will find a link in one, and a calculation that reported one would be wrong.

That makes this the more useful half of the comparison. The knitted result — a linking number of zero where the fabric’s is one per wale — is a diagnosis of a model. The woven result is a fact about cloth.

What holds a cloth together instead #

If nothing is threaded through anything, something else has to be doing the work, and this collection has spent several ladders on what it is.

Friction. A thread in a cloth is gripped by every thread crossing it, over a contact patch, under a normal force supplied by the crimp. This collection computes that force from the thread’s own bending: an unloaded cloth on a table has no tension in it and its threads still press on one another hard enough to decide whether a seam slips or a cut edge frays. That is the relaxed cloth’s contact force, and it is the entire mechanism.

Geometry. The friction has to be enough, which means the threads have to be close enough and there have to be enough crossings. That is what a sett is, why a cloth has one, and why how close threads can be set is a founding question of the subject.

And the interlacing pattern, which decides how many crossings a given length of thread gets. A satin with four-end floats gets one crossing where a plain weave gets four, which is why the float decides so much and why a satin frays where a poplin does not.

None of those is topological. All three are quantities with brackets on them.

The consequences, one at a time #

Almost every distinguishing behaviour of woven cloth follows from the number being nought, and it is worth walking through them because the list is longer than anybody expects.

A cut edge frays. Cut across a cloth and every severed weft pick is held only by friction against the ends it crosses, for whatever length of it remains. Pulled, it slides out. There is nothing to undo, because nothing was done up.

A cloth needs a selvedge. The two edges parallel to the warp are the ones where the weft turns round, so the weft is continuous there and holds itself; the cut edges have to be hemmed, overlocked, fused or woven with a leno to stop the fraying. A selvedge is a woven-in answer to a topological absence.

A cloth does not run. Break one thread in the middle of a cloth and one thread is broken. Nothing propagates, because nothing downstream was depending on it in the way that a knitted loop depends on the loop threaded through it. That is the good half of the same fact.

A cloth has to be beaten up. The friction that holds it needs the threads pressed together, so a loom’s reed drives every pick against the fell with a force, and the cloth’s density is set by that force rather than by geometry alone.

And a cloth can be unwoven thread by thread, which is what makes fringing a finish, drawn-thread work a technique, and a frayed cuff a familiar sight.

Why a leno is the exception that proves it #

There is one woven structure whose threads do wind about one another, and its existence is an argument for the rule rather than against it.

A leno or gauze weave crosses two warp ends over one another between picks, using a doup that carries one end from one side of its partner to the other. The two ends therefore wind, and the structure is famously stable at very open setts where an ordinary weave would slide apart.

That is exactly what the linking number predicts. An open sett means few crossings and little friction, so a cloth held by friction alone falls apart; a structure whose threads wind about one another is held by something friction cannot supply. Leno is the weaver’s answer to running out of friction, and it works by borrowing the knitter’s mechanism. What a leno twists that a weave only crosses is a rung of its own.

Why braiding is the other exception #

The second structure that escapes is a braid, and it escapes differently.

A braid’s strands pass over and under one another repeatedly and, crucially, they do not return to where they started: a strand travels across the braid from one edge to the other and back, so consecutive crossings with a given neighbour are not round trips. Braids are held together with no beat-up, no reed and very little friction, and a braid does not fray when it is cut in the way a cloth does — it unlays, which is a different failure and a slower one.

Two exceptions, and both are structures famous for holding together where an ordinary cloth would not. That is what an explanation looks like when it is doing work rather than being restated.

What the number costs a cloth, in force #

The linking number says friction is doing all the work. This collection can say how much work that is, and the figure is worth having because it turns an abstraction into a load.

A thread in a relaxed shirting is gripped at every crossing by a normal force the crimp supplies — a few millinewtons, computed from the thread’s own bending rather than from any tension, because a cloth on a table has no tension in it. Multiply by a coefficient of friction of about a third, and by the number of crossings along the length being pulled, and the resistance to withdrawal grows very fast with how far into the cloth the thread goes.

That is why fraying is an edge phenomenon. The first centimetre of a cut weft is held by a handful of crossings and slides; the same thread two centimetres in is held by twice as many and does not. The transition is not sharp and it is not a threshold; it is a length over which an exponential in the number of crossings takes over, and it is the same capstan arithmetic that decides what holds a thread in a seam.

So a woven cloth is held by an exponential rather than by an integer, and the practical difference is that an exponential has a scale. A fabric can be too open to hold, and a knitted fabric cannot be.

Which is why sett is a woven idea #

The word sett means nothing to a knitter, and that is not an accident of vocabulary.

A woven cloth has to decide how many threads per centimetre it carries, and the decision is a mechanical one: too few and the cloth slips, sleazes and frays; too many and it will not weave at all, because the threads jam before the reed can beat them up. This collection has spent a whole field on where that number comes from — the cover factor, the jamming condition, the bearing curve, the beat-up force.

A knitted fabric has no such decision. Its density is set by the loop length the machine draws and by nothing else, and the fabric holds together at any density whatever, from a fine jersey to an open crochet lace with a centimetre between its loops. There is no knitted equivalent of sleaziness caused by insufficient friction, because friction is not what is holding it.

That asymmetry has been visible in this collection’s own structure for eighteen phases — a field called setting with thirty-nine essays in it, none of which is about a knit — and the reason for it is one integer.

What was counted, and how #

A warp end and a weft pick are drawn at right angles with the cloth’s own crimp on both, sampled at forty points each, and closed along paths that run far outside the crossing so that the closure contributes nothing. The integral is taken over the pair.

The answer is four parts in a hundred thousand, and it is the same at every crimp from three per cent to twenty and at every number of interlacings from one to eight.

The check that matters is not that number. It is that the same integral, on the same code, returns one for two rings threaded through one another — and returns nought for two rings side by side, two rings stacked, and two rings in a single plane nearly touching. That last case is the woven geometry stripped to its essentials, and it is where an integral of an inverse cube is most likely to misbehave. A method that reported a link there would have reported one for every crossing on this site.

The closure was the thing to get right. A first attempt on the knitted side joined a course’s ends with a chord across the fabric and produced a linking number of minus 0.037 out of nothing but the closure. A closure is part of the curve. Here the two closures run at six times the crossing’s own span, in perpendicular directions, and their contribution is below the sampling.

Where the model stops #

This is a statement about two threads, and a cloth is many. Nothing here computes anything about the whole cloth’s topology as a system, which is a much harder question with a much richer answer — a woven cloth is a link of thousands of components with every pairwise linking number zero, and pairwise linking numbers do not determine a link. Two threads can be unlinked pairwise and inseparable together, and whether that ever happens in a cloth is not settled here.

And unlinked does not mean unheld. A thread pulled from the middle of a tightly set cloth takes a great deal of force to move, and pulling it out of a metre of cloth is not something anybody does by hand. The linking number says nothing about how hard it is; it says the difficulty is entirely friction, and therefore that anything which changes the friction changes it. A resin finish, a heat set, a felted wool: all of these hold a thread far better and none of them changes the topology.

And the crimp is drawn as a sinusoid. A real thread’s path between crossings is an elastica and this collection has one, but the linking number does not care about the shape and using the right one would have added nothing except a longer computation.

The generalisation #

The rule worth carrying is about which properties of a structure survive its parameters.

This collection is full of quantities that depend on a sett, a count, a crimp, a modulus or a coefficient of friction, and every one of them is quoted with the assumptions that produced it. A topological quantity is not one of those. It is the same for a cheesecloth and a duck, for a cotton and an aramid, wet or dry, before finishing and after.

So when a question is about a class of structures rather than about one, the first thing to ask is whether it has a topological answer, because a topological answer applies to the whole class at once and needs no bracket.

Two questions in this collection have turned out to be of that kind. Whether a draft makes one cloth or several is a connected-components count over an explicit graph, invariant to everything, and it is the reason the site exists. Whether a fabric’s threads are linked is the second. Both are integers, both are cheap, and both answer questions that would otherwise take a great deal of measurement to approach.

What a finish does, in this language #

A finish changes a cloth’s behaviour a great deal and changes its topology not at all, which makes the linking number a good instrument for saying what a finish actually is.

Milling a wool felts the fibre surfaces together across the crossings, so the threads are no longer free to slide relative to one another. A well milled cloth can be cut and will not fray at all — a melton is cut and left raw for exactly that reason. Nothing has been threaded through anything; the friction has been replaced by something closer to an adhesive.

A resin finish on a cotton does the same thing chemically, cross-linking cellulose across the contacts.

A heat set on a thermoplastic does it by melting and refreezing the contact.

All three convert a cloth held by friction into a cloth held by adhesion, and all three leave the linking number at nought. So the number is not a complete account of what holds a cloth together; it is an account of what holds a cloth as woven together, before anybody has done anything to it.

That is worth saying because it bounds the claim. The argument here is that a woven cloth’s structure supplies no topological hold. It is not that a woven cloth is weakly held — a milled melton is held very well — but that whatever holds it is not the weaving.

Who found it, and when #

That woven threads interlace without interlocking is the oldest observation in the subject and belongs to nobody. The linking integral is Gauss’s, from 1833. Applying an invariant to textile structures is a small and scattered literature, mostly about knitted and braided structures and mostly from the last thirty years, and most of it is interested in what can be made rather than in what holds.

What is this collection’s own is the pairing: computing the number for both fabrics with the same integral, on its own geometry, and reading the two failure modes off the two answers.

Where the ladder goes next #

The two numbers are now in hand — one per wale for a knit, nought for a weave — and the two failure modes everybody knows fall straight out of them. Why a knit runs and a weave frays is a pair of behaviours this collection has described accurately for its whole life without being able to say what causes them, and the cause is one integer each.

And the exception named above is worth its own rung, because it is the one place a weaver reaches for the knitter’s mechanism on purpose: a leno twists what a weave only crosses.