A jersey's course has no writhe

A jersey knitted from a hard-twisted yarn leans. The wales run off the vertical, a T-shirt’s side seam spirals round the body, and the effect is large enough that the trade has a word for it and a specification limit on it. It is not a defect of one machine: a fabric leaning is a property of its yarn rather than of its knitting.

The accepted explanation is a trade. The yarn arrives with a residual torque; the fabric’s own path can absorb some of that twist by winding about itself; and the fabric distorts to whatever arrangement lets the yarn shed most twist for least deformation. This collection has said so, and priced the effect at nine degrees per unit of twist factor above balance — with the honest note attached that the nine was fitted rather than derived, because deriving it needs a loop’s torsional compliance and there was none.

There is one now. The compliance exists, the identity that governs the trade exists, and the quantity the trade is made of is computable from this collection’s own solved fabric.

It is zero.

What was asked #

A thread whose ends are held has a fixed total of twist plus writhe. So a fabric can relieve its yarn of twist only by acquiring writhe, and the amount available is a property of the fabric’s own geometry rather than of the yarn.

The writhe of a curve is the Gauss double integral taken over the curve twice, and it counts, in a precise sense, the average number of times the curve crosses itself when viewed from all directions. It is a number about a shape and nothing else.

So the question is: how much writhe does one course of a plain jersey have?

Take the solved half period from the collection’s own three-dimensional loop, lay two of them end to end to make a stitch, lay twelve stitches round a tube so that the course closes, and integrate.

The answer is minus six parts in a million per wale. At double the sampling it is minus three. At double the wales it is the same per wale to the last figure it can be read to. Whatever is being measured, it is not a shape.

Why it is exactly zero rather than very small #

A number that comes back at six parts in a million is usually a small quantity computed correctly. This one is not: it is an identity, and knowing that it is an identity is worth much more than knowing that it is small.

Writhe is chiral. Reflect a curve in any plane and its writhe changes sign — that is what the signed crossing count means, and it is the reason writhe can tell a left-handed helix from a right-handed one.

So any curve that a reflection carries to itself has writhe equal to minus its own writhe, and therefore none.

The solved course is such a curve. The model builds it out of half periods that are exact mirror images: a half period descends from a crest to a trough and the next ascends from that trough to the next crest, and the second is the first reflected through the fabric’s own mid-surface. Lay them end to end and the whole course is carried to itself by a reflection in the plane containing the course direction and the thickness.

Hence no writhe, exactly. The six parts in a million are the closure and the sampling reporting themselves.

The trouble with that answer #

A real jersey is not achiral. It is one of the most obviously handed structures in the whole subject.

Look at a stitch. The needle loop’s head passes through the head of the loop below it, and it passes through in a definite sense: the leg that arrives from the left goes behind, and the leg that leaves to the right comes in front, or the other way about, and which it is depends on which way round the machine was threaded. A knitted fabric has a face and a back that look nothing alike. Its loops lean the same way as one another, which is what makes a lean visible at all — and the asymmetry of a single loop is a thing this collection has measured rather than assumed.

A structure that visibly differs from its own mirror image cannot have zero writhe for a symmetry reason. So the model has a symmetry the fabric does not.

Finding out where the model acquired that symmetry is the whole content of this rung, and the answer is one line.

Where the symmetry came from #

The three-dimensional loop was built by taking the planar solve and letting it climb. A half period leaves a crest along the course direction and arrives at a trough along the course direction, and between them it climbs one yarn diameter through the fabric’s thickness — because the head lies half a diameter behind the mid-surface and the feet half a diameter in front.

That is correct, and it is what makes the fabric two diameters thick. But it places the interlacing at a point: the two centre lines pass, one diameter apart, and neither goes round the other.

Two curves that pass beside one another are mirror images of two curves that pass beside one another the other way. There is no handedness in a near miss. The model’s crossing has no sense, so the model’s course has no chirality, so the model’s course has no writhe.

The fabric’s crossing has a sense, because the yarn actually goes through.

The one place the model does have a handedness #

It is worth saying what the model does carry, because it is not nothing and it is not enough.

The solved loop tilts. Its plane is the fabric’s turned about a dozen degrees, and the tilt has a sign: the crest lies behind the mid-surface and the trough in front. Reflecting the fabric through its own mid-surface reverses that tilt, so the tilt is a handed quantity and the model has it.

What the tilt does not do is produce writhe, because every half period tilts one way and every other half period tilts the other. The handedness alternates, so it cancels along the course as fast as it accumulates. A quantity that alternates in sign along a curve contributes nothing to an integral over the whole of it.

That is the difference between a local handedness and a global one, and it is exactly the distinction the writhe is sensitive to. A fabric whose every crossing has the same sense accumulates; a fabric whose crossings alternate does not. A real jersey is the first kind. This model is the second.

So the mechanism is not available in this model #

The consequence is worth stating flatly, because it is a negative result about a mechanism this collection has published.

In the model as it stands, a jersey cannot relieve its yarn of any twist by writhing, at any size, in any state, at any tightness. There is nothing to relieve it with. The writhe is zero and stays zero.

It follows that the nine degrees per unit of twist factor cannot be derived from this model, and not because the arithmetic is hard. The quantity it would be derived from does not exist here.

That is a stronger statement than “the model is missing torsion”. Torsion is a term that could be added to an energy. This is a missing topology, and no term added to any energy will produce it: the shape has a symmetry, and only changing the shape will remove it.

The skew that does not help either #

The obvious next thought is that the fabric does not have to writhe as a course. It could skew — lean its wales, which is the deformation actually observed — and a skewed fabric’s course is a different curve with, perhaps, a different writhe.

That is checkable directly, by shearing the solved course and integrating again, and it was checked before this rung was written.

Skewing the course by a tenth of a radian changes its writhe by about five parts in a hundred thousand per wale. At any residual torque a real yarn carries, the energy that buys is smaller than the fabric’s own bending energy by several orders of magnitude, and the equilibrium lean it predicts is a small fraction of a degree against an observed five to fifteen.

So the answer survives the obvious repair: the model is not merely unwrithed at rest, it is unable to acquire writhe by the deformation the effect is made of.

What a null result is worth #

It would be easy to read this rung as a failure, and it is not. It is the most useful thing on the ladder, for two reasons.

The first is that it rules out a mechanism. Before this, “the fabric writhes to relieve the yarn” was a plausible story with a fitted constant in front of it. It is now a story this collection can say its own model has no room for, which is a much more precise position: either the mechanism operates through something the model omits, or the mechanism is wrong.

The second is that it locates the omission. The writhe is zero because the crossing is a point. That single sentence turns out to explain four other things the collection had recorded separately as unexplained, and finding out that four problems are one problem is worth more than solving any of them.

Which four #

They are worth listing, because the pattern is the finding.

The linking number of adjacent courses is zero where a fabric’s is one per wale, for the identical reason and in the identical words. That is the next rung.

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

Two adjacent courses occupy the same space at rest, approaching to four fifths of a diameter, because the model places them at a diameter at the crossing and nothing stops them coming closer either side of it. That is the fabric that does not fit, and it is the whole of the next ladder.

And the wale-direction curl cannot be computed, because a thread with its end tangents turned out of the fabric runs away to a configuration its neighbours would not allow — and the neighbours are exactly the loops the model does not thread.

One defect, four symptoms, and none of them was recognised as related until the writhe came back at zero.

What was counted, and how #

The course is the collection’s own: a twenty tex cotton at a three and a half millimetre loop, fully relaxed, solved as a three-dimensional elastica with twelve terms in its basis — the same solve that produced the fabric’s thickness and every force on it, laid end to end and wrapped onto a cylinder so that it closes.

The closure is not a detail and getting it wrong cost an afternoon. The first attempt joined the course’s two ends with a straight chord across the fabric. That chord is longer than everything else in the picture, it lies beside the next course’s chord, and the pair contributed a linking number of minus 0.037 to an arrangement whose answer is nought. A closure is part of the curve, and a number that should be an integer and comes back at a few hundredths is reporting its closure rather than its topology.

The tube fixes it because a tube is a real fabric rather than a device. A circular machine makes them, a sock is one, and every needle loop in a tube is drawn through the loop below exactly as it is in a flat fabric.

The integral itself is checked on four arrangements whose answers are known by inspection, and it returns them to four parts in ten thousand at four hundred segments a curve.

Where the model stops #

Nothing here says how much writhe a real jersey has. It says the model has none. Computing the real figure needs the crossing drawn as a crossing, which is a geometry this collection does not have.

And the symmetry argument is about the model’s construction rather than about knitting. A different solve — one that let the two half periods differ, as they do in a fabric where one is a needle loop and the other a sinker loop — could be chiral without any contact being modelled at all. That is the cheapest available repair and it has not been tried.

The skew test used the free shape at every skew rather than re-solving the loop under the imposed shear, so it is an estimate rather than an equilibrium. The estimate is four orders of magnitude short of what the effect needs, which is a large enough margin that a proper solve is very unlikely to close it — but it is an estimate, and this collection has been wrong before about an estimate that added a term to a functional rather than re-minimising it.

The generalisation #

The habit worth taking from this rung is about what a symmetry in a model means.

A model with more symmetry than its subject is not slightly wrong. It is exactly wrong about every quantity the symmetry forbids, and it is exactly wrong in a way that looks like a small number rather than like an error. Six parts in a million reads as a converged calculation of something negligible. It is a calculation of something forbidden.

So the question to ask of any result that comes back at nought is not “is it small enough to ignore” but “is it zero for a reason, and does the subject share the reason”. Here the reason is a mirror symmetry, and the subject does not share it: a knitted fabric is chiral and the model is not.

That test is cheap and this collection has not been making it. There are other quantities here that come back at zero — the curling moment about the course direction is exactly nought for a reason worth more than the number — and each of them deserves the same question asked of it.

The curling moment is the instructive comparison, because there the answer is different. It is zero because the fabric’s mid-surface bisects every segment, and a real jersey’s mid-surface very nearly does — so that zero is a genuine statement about the fabric, and the small eccentricity that produces the observed curl is a correction to it. Here the zero is not a statement about the fabric at all. Two identical-looking null results, one of which is physics and one of which is an omission, and telling them apart took a symmetry argument rather than a bigger computation.

What would have to change #

The repair is not a term and it is not a constant. It is a geometry, and it can be stated precisely enough to be someone’s afternoon.

The model’s half period runs from a crest to a trough and climbs one diameter on the way. To make the crossing a crossing rather than a near miss, the yarn arriving at the interlacing has to pass round the yarn already there: to leave the fabric’s mid-surface on one side, travel past, and come back on the other, so that the two centre lines wind about one another once rather than approaching and separating.

That costs bending energy, and how much is exactly the question. It also costs an extra half diameter of yarn at every interlacing, which has to come out of the loop length, which changes every dimension the fabric has. So it is not a decoration on the existing solve; it is a different boundary-value problem with an extra constraint in it, and the constraint is a non-penetration condition rather than an endpoint.

What makes it worth doing is that the same change fixes all four symptoms at once, and that nothing smaller fixes any of them.

Who found it, and when #

The identity that makes writhe worth computing is Călugăreanu’s. The observation that a knitted fabric is chiral is as old as knitting. The mechanism by which residual torque produces spirality has been the standard account since at least the 1950s.

What is this collection’s own is the negative: computing the writhe of its own solved course, finding it zero, and identifying the symmetry that forces it. That the symmetry traces back to a modelling decision made two ladders earlier — that an interlacing is a point where two centre lines pass a diameter apart — is the part worth carrying, because that decision was made for good reasons and had no visible cost until now.

Where the ladder goes next #

The same integral applied to two courses rather than one gives a much larger statement, because the answer there is not merely zero but known to be wrong: a knitted tube’s adjacent courses link once per wale, and this model’s link not at all. A point cannot link, and a model whose fabric is unlinked is not modelling a knitted fabric in the only respect that distinguishes one.

And the woven side of the comparison is the reverse case, where zero is the right answer permanently and for every weave — which is what makes a woven cloth fray where a knitted one runs.