What a closed thread cannot choose

This collection is built on counting. A weave is a matrix and its floats are enumerated; a satin exists at a given number of ends or it does not, and which is which is settled by a division; a cloth is one cloth or several, and the answer is a connected-components count over an explicit graph rather than an opinion. The habit is deliberate and it is the reason the site exists — the draft is a matrix before it is anything else.

There is a quantity about a thread that is counted in exactly that sense, that nothing in this collection has ever used, and that turns out to be the sharpest available statement of what separates the two ways of making cloth.

The number #

Take two closed curves in space that do not touch. Count the number of times one passes through the other, with a sign for the direction. The answer does not depend on how the curves are drawn, only on how they are arranged, and it cannot change while they move continuously without passing through one another.

That is the linking number, and it is the oldest invariant in the subject. It has a formula — a double integral over both curves, due to Gauss — and the formula’s most useful property is not that it computes the number but that it computes an integer. A numerical method that returns 0.87 for a linking number has not found a slightly linked pair of curves. It has a bug, or a curve that is not closed, or a sampling too coarse, and the answer says which by how far off it is.

That is a rare and valuable thing in a collection full of quantities whose correctness cannot be read off their own output. A bending energy of 24,395 nanojoules is either right or wrong and the number does not say, which is why every force here is quoted with the bound it was computed at. A linking number of 0.9997 says it is one and that the sampling is four hundred segments.

Twist and writhe, which add to it #

The linking number describes two curves. Applied to a single thread, it says something more interesting, and the statement is one of the more elegant results in the subject.

A thread is not a curve; it is a curve with material around it. Draw a line down its length — a painted stripe, or the path of one of its fibres — and there are now two curves: the centre line and the stripe. Close the thread into a ring, so that the stripe joins up with itself, and the two curves have a linking number.

That number splits into two parts, and neither part is an integer while the sum is:

Lk = Tw + Wr

The twist is how many times the stripe winds about the centre line, which is what a torsional rigidity resists. The writhe is how many times the centre line winds about itself — a property of the shape alone, with no material in it, computed by the same Gauss integral taken over one curve twice.

The identity says the two can trade freely, at no cost to the topology, and that their sum cannot change at all.

Why an exact conservation law is worth having #

The reason to care is that this collection is short of them.

Almost everything computed here is an equilibrium: a shape that minimises an energy subject to constraints, whose answer depends on constants nobody knows to better than a factor of three hundred. The results are honest and they are bracketed, and the bracket is carried into every downstream number.

A conservation law is a different kind of statement. It does not depend on a modulus, on a friction coefficient, on a packing factor or on whether the fibres slide. It says a quantity is what it is because of how the thread is arranged, and any model that violates it is wrong for reasons that have nothing to do with its parameters. That is a stronger footing than anything else on this site stands on, including the bracket every force here carries.

So the identity is usable in a way an energy is not: as a check on a model. If a model of a fabric reports that its yarn has relieved some twist, the model had better be able to say where the writhe went. If it cannot, either the twist did not go anywhere or the model has lost track of the ends.

That is exactly the check this ladder is about to fail, on this collection’s own model, and the failure is the most useful thing on the ladder.

The sign, which is a fact about the fabric #

A linking number carries a sign, and the sign is chirality: the arrangement and its mirror image have opposite signs.

Everything follows from that one sentence. A quantity that changes sign under a reflection is zero for anything that is its own mirror image, and that is not an approximation or a small number — it is exactly zero, as an identity, for the same reason that the average of a function and its negative is nought.

So the first question to ask of any structure is whether it is chiral. If it is not, its writhe is zero and it has nothing to trade. If it is, the sign of its writhe is decided by the handedness of the structure, and a yarn of the opposite twist direction laid into the same structure will behave in the opposite sense.

That is a strong and testable prediction about fabric. It is also the reason the trade cares which way a yarn is twisted at all, in a way that has nothing to do with appearance.

What a knitted fabric is, said in this language #

A knitted fabric is made by drawing a loop of yarn through the loop below it. Every needle loop of every course passes through the loop of the course beneath, once, in the same sense.

So two adjacent courses of a knitted tube — and a tube is where the question is askable, because a course of a tube is a closed curve while a course of a flat fabric is not — have a linking number equal to the number of wales. That is not a model of anything and it is not approximate. It is what the machine does, counted.

Everything a knitted fabric famously does follows from that number being nonzero. It can be made from a single thread and taken apart by pulling that thread, because nothing but the threading holds it. It runs when a loop is broken, because the loops below have lost the thing that was holding them, and a run is a race between two energies only after the topology has permitted it. It needs no sett, no beat-up and no reed, because it is not held by friction between crossing systems and does not need them pressed together.

And what a woven cloth is #

A woven cloth is made by passing one thread over and under another. Neither ever passes through the other.

So the linking number of any two threads in a woven cloth is zero, at any crimp, for every weave that exists or could exist. That is not a limitation of any model and it is not a number that a better calculation would change. Crimp moves a thread up and down across its neighbour, and a curve that goes over and comes back has done nothing a linking number can see.

Everything a woven cloth famously does follows from that. It frays when it is cut, because nothing holds a thread in but friction against its neighbours. It needs a sett, because the friction has to be enough — which is what how close threads can be set is about, and why the question has no knitted equivalent. Its edges have to be finished, and a selvedge is a woven-in solution to a topological absence — the same absence ravelling, fraying and running were described by before there was a word for it. And it does not run, because there is nothing threaded through anything to come undone.

Two fabrics, one number, and the number is not a description of them but the reason for almost everything else about them.

The third fabric, which has neither answer #

Two ways of making cloth have been named and there is a third, and it is worth putting beside them because it separates the two properties that have so far travelled together.

A nonwoven is a web of fibres held by entanglement, by adhesive or by heat. Ask it the linking number question and the answer is not zero and not the number of wales: it is a distribution, because two fibres in a needled web may pass through one another’s loops any number of times or not at all, and nothing decides it but the process.

That matters because it separates being held together from being linked. A woven cloth is held together and unlinked; a knitted fabric is held together and linked; a nonwoven is held together by an amount of linking nobody chose. This collection has argued before that what holds a nonwoven together is a different question from what holds a cloth together, and the linking number is the sharpest available form of that difference.

It also explains a practical fact. A nonwoven does not fray and does not run, and it is the only one of the three that does neither — because fraying is what happens to an unlinked structure and running is what happens to a linked one, and a structure with a random amount of linking has too little of either to propagate.

What the number does not decide #

It is worth being explicit about what a topological quantity cannot tell anybody, because the temptation with an exact invariant is to lean on it too hard.

The linking number does not decide strength. A woven cloth is far stronger in tension than a knitted one of the same yarn, and the reason is that its threads run straight between crossings while a knitted fabric’s run round loops. Nothing topological knows about that.

It does not decide whether the fabric holds together under load. A knitted fabric can be pulled apart by making its loops slip through one another, which changes no linking number at all, because slipping is a deformation and the ends were never held.

And it does not decide dimensions. Every number this collection has computed about a knitted fabric’s spacings comes from an energy minimisation against measured constants, and the topology enters none of it.

What it decides is the failure mode and the assembly, and those are exactly the two things that most distinguish the two fabrics in use: how they come apart, and how they were put together in the first place.

What was counted, and how #

The Gauss double integral is evaluated over segment midpoints, with the self term dropped for a writhe and nothing dropped for a linking number. It is a double sum over a few hundred segments a curve, which is nothing, and the reason not to be cleverer is that a closed form for this quantity is exactly the sort of thing that returns a plausible number for a wrong reason. The double sum is the definition.

Four arrangements check it, and every one of them has an answer known by inspection: two rings threaded through one another, two rings side by side, two rings stacked one above the other, and two rings in a single plane nearly touching. The answers are one, nought, nought and nought. The worst departure at four hundred segments a curve is four parts in ten thousand, and it falls as the segment count rises — which is what a discretisation converging on an integer looks like.

The third of those four is the one that earns its place, because it is the shape a woven cloth makes and it is the case where an incorrect implementation would be most likely to return something nonzero. Two curves passing very close together are exactly where a numerical integral of an inverse cube blows up, and a method that reported a link there would have reported one for every crossing in every cloth on this site.

Where the model stops #

A linking number needs closed curves and cloth is made of open ones. Every thread in a woven cloth ends somewhere, and a course of a flat knitted fabric ends at the selvedge. The way round it here is a tube, which is a real fabric — a circular machine makes them and a sock is one — rather than a device. For woven cloth the threads are closed far outside the crossing, along paths that contribute nothing, and the contribution of a closure is checked rather than assumed.

And the identity is about an unbroken thread with held ends. A yarn in a fabric is held by friction rather than clamped, so its ends can rotate slowly. Over a wash, a yarn can shed twist by rotating in the fabric, and the sum of twist and writhe is then not conserved because the ends were not really held. That is not a failure of the identity; it is a statement that the object it applies to is a thread on a timescale short enough that friction has not let go.

Nothing here is a measurement of a real fabric. Every number above is a count over an idealisation. What makes the counts worth having is that they are the same counts whatever the idealisation gets wrong about diameters, stiffness or crimp — which is the property an integer invariant has and an energy does not.

The generalisation #

This collection already had one invariant and did not recognise it as a member of a family.

The integrity check asks whether a weave’s threads form one connected component, and that too is a topological quantity: it does not change when the cloth is stretched, it does not depend on any material constant, and it takes an integer value. It is the reason this site exists, and it has been described for its whole life as a graph question rather than as a topological one.

The linking number is the second member, and naming the family is worth something. A topological quantity is a claim about arrangement that survives every deformation and every constant. In a collection whose besetting difficulty is that its constants are bracketed, such quantities are the ones to look for first, and there are probably more of them: the number of layers in a compound cloth, the crossing number of a braid, the parity of a satin’s step.

The rule to take away is that if a claim can be made as a count over an arrangement, it should be, because a count over an arrangement is the only kind of claim in this subject that does not have a bracket on it.

Who found it, and when #

Gauss wrote the linking integral in a notebook in 1833 and did not publish it. Călugăreanu proved the identity that splits it into twist and writhe between 1959 and 1961, working on closed ribbons; White and Fuller gave it the forms now used in the early 1970s, and it entered common use through molecular biology, where the object being twisted is a strand of DNA and the enzymes that change the linking number are the point.

Nothing in this rung is new. What is this collection’s own is only the application: asking the question of a weave and a knit side by side, and computing the answer for its own model rather than quoting it for an idealisation.

Where the ladder goes next #

The identity says a fabric can relieve its yarn of twist only by giving its path a writhe. So the question is how much writhe this collection’s own solved course has, and the answer is available with the machinery already built.

It is none. The model’s course is carried to itself by a reflection and writhe changes sign under a reflection, so a jersey’s course has no writhe — exactly, as an identity, rather than approximately. And the same sentence said about two courses rather than one is a much larger statement, because it means a point cannot link and this collection’s fabric is not knitted at all in the only sense that counts.