A point cannot link

There is one thing a knitted fabric does that nothing else does, and it can be said in a sentence with a number in it. Every needle loop is drawn through the loop below it. A tube of n wales therefore has a linking number of n between every pair of adjacent courses, and that number is not a model of anything: it is what the machine does, counted.

This collection has a model of that fabric. It computes the fabric’s thickness from first principles, its contact force, its bending rigidity, its warmth, its extension curve and the pressure a cuff applies to a wrist. Asked the linking number of two adjacent courses, it returns minus thirteen parts in a million.

Not one per wale. Not one. Nothing.

Where the zero comes from #

It comes from one sentence, written two ladders ago for good reasons and with no visible cost until now.

The three-dimensional loop was built by taking the planar solve and letting it climb through the fabric’s thickness. A course’s crest is a needle loop’s head and its trough is the feet resting on the head below; the head lies half a yarn diameter behind the fabric’s mid-surface and the feet half a diameter in front; so a half period climbs a whole diameter as it descends a course spacing, and the fabric comes out two diameters thick.

Every part of that is right, and the thickness it predicts is the collection’s most falsifiable result. But it places the interlacing at a point: two centre lines passing, one diameter apart, neither going round the other.

Two curves that pass beside one another are not linked, however close they come. A near miss is not a knot. The distance can be made a diameter, a micrometre or a nanometre and the linking number stays at nought, because linking is not a matter of proximity.

Which is why proximity was the wrong thing to get right #

The model was built to get the distance right, and it does. One diameter at the interlacing is exactly what two threads in contact are, and the thickness that follows is 0.334 millimetres for an ordinary jersey with nothing fitted.

The property that makes a knitted fabric a knitted fabric is not a distance. It is an arrangement, it takes an integer value, and it is invisible to every gate this collection has: nothing anywhere asks whether a loop goes through another loop, because until this ladder there was no quantity to ask it with.

The four symptoms #

The reason this is worth a rung rather than a footnote is that the collection had already recorded four things it could not explain, in four different ladders, and they are all this.

The extension ceiling is three times too high. The model says a jersey can be pulled to three hundred and twenty-two per cent along its courses before the yarn runs out. Real jerseys stop at about a hundred, which is what a knit gives when it is pulled reports from measurement. The recorded diagnosis was that the model does not stop adjacent courses passing through one another — which is true, and is this.

The wale-direction curl cannot be computed. Bending the fabric about its wale direction turns a thread’s end tangents out of the fabric, and a free thread with turned ends is unstable in its own plane: the solve leaves the fabric altogether, by three times the fabric’s own thickness, for a third off the energy. The recorded diagnosis was that the configuration is one the thread’s neighbours would not allow, and a branch jump agrees rather than failing, which is why it took a shape check rather than an energy check to find. The neighbours in question are the loops that ought to be threaded through it.

The course has no writhe. Zero, as an identity, because the model’s course is achiral — and it is achiral because a near miss has no handedness. That is the previous rung.

And two half periods run alongside one another at every crest within a fiftieth of a diameter, for more than a millimetre of arc, because what holds them apart in a fabric is the needle loop of the next course drawn between them, which is what holds a crest apart.

Four findings, four ladders, four diagnoses written independently, and one cause.

What a linked model would have to do #

The repair can be stated precisely, which is worth something even if it is not done here.

At each interlacing, the arriving yarn must pass round the yarn already there rather than beside it: leave the mid-surface on one side, travel past, and return on the other, so that the two centre lines wind about one another once.

Three things follow immediately and none of them is small.

It costs yarn. Going round rather than past adds roughly half a circumference of the standing yarn at every interlacing, which at two interlacings a stitch is about half a diameter of extra length. At a three and a half millimetre loop and a 0.167 millimetre diameter that is around two and a half per cent of the loop length, and every dimension the fabric has is computed at a fixed loop length.

It costs bending. The extra path is bent to a radius of about one diameter, which is the tightest bend anywhere in the fabric, and the bending energy goes as the square of curvature.

And it changes the constraint, from an endpoint condition to a non-penetration condition. That is a different class of problem: the current solve minimises an energy subject to three endpoint equalities, and a contact problem minimises it subject to an inequality that is active on part of the domain and inactive on the rest.

What the zero does not invalidate #

It would be wrong to read this as saying the collection’s knitted results are worthless, and it is worth being specific about which ones survive.

Every quantity computed from the loop’s shape at fixed geometry is unaffected. The bending energy of a stitch, the contact force, the fabric’s thickness, its bending rigidity, its thermal resistance, its fibre fraction: all of these are integrals over a curve whose shape is decided by where its two ends are and how much yarn runs between them. Threading that curve through another one does not change any of those to first order, because the extra path is short and local.

Every quantity computed from a comparison at fixed topology is likewise unaffected. The ratio of a rib’s warmth to a jersey’s is a comparison between two fabrics whose interlacings are equally unmodelled, and the error cancels.

What the zero invalidates is anything that depends on the fabric being held together — the extension ceiling, the run, the curl about the wale axis, the spirality — and it invalidates those completely rather than by a factor.

That division is the useful output of this rung: it says which of the collection’s own results to keep and which to stop quoting.

What the number would be worth if it were right #

It is worth asking what a linked model would buy, because the answer decides whether the repair is worth anybody’s afternoon.

A hard extension ceiling. Two courses that are threaded through one another cannot separate beyond the length of yarn in the thread joining them, and that is a much tighter constraint than “the yarn between two interlacings must reach”. It is the obvious candidate for the missing two thirds, and it is not available in any model where the courses are merely near one another — as the contact ladder finds, stopping the courses overlapping barely moves the ceiling at all.

A run with a mechanism. A run is a loop that has lost the loop threaded through it, propagating. In an unlinked model a broken loop is a broken loop and nothing follows.

A chirality, and therefore a writhe, and therefore the twist–writhe trade that the standard account of spirality is made of.

And a curl about the wale axis, because the buckled configuration the solver keeps finding is one that a threaded neighbour physically blocks.

Four results the collection does not have, from one change. That is an unusually high return for a piece of modelling, and it is worth saying so plainly rather than leaving it as an implication.

Why nobody noticed for four ladders #

The uncomfortable question is how a model of a knitted fabric went four ladders without anybody asking whether its loops were threaded, and the answer is instructive rather than embarrassing.

Every quantity the collection computed was a local one. A bending energy, a contact force, a thickness, a rigidity: each is an integral over one half period, or a comparison between two. A half period does not know whether it is threaded through anything. It knows where its two ends are and how much yarn runs between them, and the model got both of those right.

So the model was right about everything it was asked, and it was asked only local questions — because those are the questions a solve of one segment can answer, and building the solve was the work. The first non-local question anybody put to it was the writhe, and it failed immediately.

That is a general hazard and it has a name in this collection already: a model is tested by what it is asked, and a collection that asks one kind of question builds a model that answers that kind and no other. The gates that follow figures into files had the same shape — three of them went blind and all three reported green, because each was still answering the question it had been written to answer.

What was counted, and how #

The two courses are the same solved curve, one course spacing apart, wrapped onto a cylinder so that each closes. The Gauss double integral is taken over the pair, at forty samples a half period and twelve wales — about a thousand segments a curve, so a million terms, which takes a moment.

The check that makes the number trustworthy is not its size. It is that the same integral returns one for two rings threaded through one another, nought for two rings side by side, nought for two rings stacked, and nought for two rings in one plane nearly touching, each to four parts in ten thousand at four hundred segments.

That fourth case is the one that earns its place. Two curves passing very close together is exactly where an integral of an inverse cube is most likely to misbehave, and it is also the shape a woven crossing makes — the same near-contact geometry Peirce’s cloth is built out of. A method that reported a link there would have reported one for every crossing on this site.

The closure is part of the curve, and getting it wrong is the mistake this rung nearly shipped. The first attempt joined a flat course’s two ends with a straight chord. The chord is longer than everything else in the picture and lies beside the next course’s chord, and the pair contributed a linking number of minus 0.037 — small enough to look like convergence and large enough to be a hundredth. A quantity that must be an integer and comes back at a few hundredths is reporting its closure.

Where the model stops #

Nothing here computes what a linked model would give. It says what the current one gives and what the fabric gives, and it prices the repair in yarn and in bending. It does not solve the repaired problem.

And the tube is not a flat fabric. A tube’s courses are closed and a flat fabric’s are not, so the quantity is defined for one and not the other. That is a real restriction and it is handled by choosing an object where the question is well posed rather than by extending the question. A flat fabric’s courses are linked in exactly the same way and the invariant that says so is a relative one, which this collection does not carry.

The comparison assumes the fabric is plain. A rib, an interlock or a purl fabric has a different linking structure — a purl course links courses on both faces — and none of that is computed. What is computed is the plain case, where the answer is one per wale and the model’s is nought.

The generalisation #

Two things generalise, and the second is larger.

The first is about idealisations. An idealisation that gets a distance right can still get an arrangement wrong, and the two failures look nothing alike. A distance that is out by ten per cent produces results that are out by ten per cent. An arrangement that is wrong produces results that are out by everything, on the quantities that depend on it, and exactly right on the quantities that do not — so the model reports a mixture of excellent numbers and nonsense with no marker separating them.

The second is about how this collection checks itself. Every gate here reads a fabric and asks whether something about it is right: whether a label fits, whether a weave is one cloth, whether a figure’s options are read, whether a caption’s number matches the picture. Not one of them asks whether a structure has the topology its name implies, and the reason is that until this ladder there was no number to ask with.

The connected-components check that decides whether a cloth hangs together is the nearest thing, and it is the same kind of question asked of a weave: an integer over an arrangement, invariant to every constant. The linking number is its knitted counterpart, and the collection went eighteen phases without it.

What a reader should do with a model that has this in it #

A model with a known structural omission is not a model to throw away, and the collection has been in this position before. What it needs is a rule for reading it, and the rule here is short.

Trust a number that is an integral over one segment. The thickness, the contact force, the bending energy, the rigidity, the fibre fraction, the thermal resistance: every one of these is decided by where a half period’s two ends are and how much yarn runs between them, and all of that is right.

Distrust a number about the fabric coming apart. The extension ceiling, the run resistance, the curl about the wale axis, the spirality. These depend on the loops being threaded and the loops are not threaded.

And treat a zero with suspicion. Two of the collection’s knitted results are exactly nought — the curling moment about the course direction and the writhe of a course — and only one of them is nought for a reason the fabric shares. Telling them apart needed a symmetry argument in one case and a threading argument in the other, and neither is the kind of thing a gate catches.

That third rule is the one worth keeping, because it is the rule that found this. A model’s null results are where its omissions live: an omitted mechanism contributes nothing, and contributing nothing looks exactly like a converged calculation of something negligible.

Who found it, and when #

The fact that knitted loops interlink and woven threads do not is as old as the two crafts and is not a discovery. Grosberg and others made the point in the 1960s in the course of arguing about why knitted fabrics recover and woven ones do not.

What is this collection’s own is asking the question of its own model rather than of the fabric, computing the answer, and finding that the model has been describing an unlinked object throughout. That is not a fact anybody could have learnt from the literature, because nobody else has this model.

The date it became askable is the date the third dimension arrived. Before that the loop was a plane curve in a plane fabric, no two curves could pass one another at all, and the question had no meaning.

Where the ladder goes next #

The woven side of the comparison is the reverse case and it is not a defect: a woven cloth’s threads really are unlinked, at every crimp, for every weave, permanently. That is what makes it fray, and it is the cleanest available statement of what separates the two ways of making cloth.

Put the two together and the two failure modes fall out of the two numbers, which is why a knit runs and a weave frays — a pair of behaviours this collection has described accurately for its whole life without being able to say what causes them.