A tuck is the one stitch that links twice

Weft knitting has three stitches and every fabric it makes is a pattern of them. A knit, where the needle takes new yarn and draws it through the loop it was holding. A miss or float, where the needle takes nothing and the yarn passes by. And a tuck, where the needle takes new yarn and does not cast off, so it now holds two loops at once.

This collection has enumerated them, drawn them, computed their yarn consumption and censused the fabrics they make. What it has never said is what each does to the fabric’s linking.

The three, in linking terms #

A knit stitch links once. The new loop passes through the old one, which is a linking number of one between the two courses at that wale.

A miss links nothing. The yarn passes across the face of the fabric without engaging the needle, so no loop is formed and nothing is threaded. A float’s contribution to the fabric’s linking is nought.

And a tuck links twice — but not in the way the phrase first suggests. A tuck does not make two links; it leaves two loops in one head, so when the next knit stitch is finally made, it is drawn through both of them at once.

So a tucked wale has a course whose loop is threaded through two courses’ loops rather than one.

What that does to a run #

A run is a chain: loop k was held by loop k+1, loop k+1 is gone, loop k is free. It propagates because each step creates the condition for the next.

A tuck breaks the chain, because the loop above a tuck is held by two loops rather than one. Lose one and the other is still holding.

That is the whole mechanism, and it is why a course of tuck stitches is the standard run-stop in a stocking, why a tucked structure is more run-resistant than a plain one — the same asymmetry fraying and running fall out of, and why the resistance is a property of the arrangement rather than of the yarn or the tension.

It also predicts something sharper than “more resistant”: a run should stop at a tuck rather than being slowed by one. A friction mechanism slows a run; a topological one stops it, because there is nothing left to propagate through.

What a miss does #

A float is the opposite case and it is worth stating because it is the one that weakens rather than strengthens.

A missed needle holds its old loop for another course. That loop is now spanning two courses’ worth of fabric and is held, above, by whatever the next knit stitch does.

So a float does not add a link and it does not remove one; what it does is make one loop carry more of the fabric. If that loop goes, more goes with it.

That is why float structures — single-jersey plating, some jacquards, some mesh fabrics — are less run-resistant than plain jersey rather than more, and it is the opposite direction from a tuck.

Three stitches, three effects on the linking: one adds, one is neutral, one doubles.

Which explains the fabrics #

The three fabrics this collection already knows about fall into place.

Plain jersey is all knit, so its linking is one per wale per course and its chain is unbroken everywhere. It runs freely, and a run is a race between two energies once the topology has permitted it.

A tuck fabric — a piqué, a lacoste, a cardigan — has tucks scattered through it, so its chain is broken wherever a tuck sits. It runs poorly and stops at the first tuck.

A float fabric has loops carrying more than their share, and runs readily.

And an interlock does something different again: it is two rib fabrics knitted into one another, and the two balance for different reasons, so every course is bound on both faces and every loop is held from two directions. It does not run at all.

That last is not a tuck effect and it is the same principle: a loop held by two things does not free anything when one of them goes.

Why the model cannot compute any of this #

The uncomfortable half, and it is the same defect the whole of this work has been circling.

This collection’s knitted geometry places its interlacing at a point where two centre lines pass a yarn diameter apart. Two curves passing beside one another are not linked, so the model’s adjacent courses have a linking number of nought where a fabric’s have one per wale.

A model with no links cannot distinguish a stitch that adds one from a stitch that adds two.

So everything on this rung is a statement about knitting rather than a computation on this collection’s own machinery, and that is stated rather than hidden. The linking numbers quoted are what the fabric has; the model has nought for all three stitches.

What a repaired model would give #

It is worth saying what the computation would be, because it is a good test of a repaired model and it is cheap once the repair exists.

Build the threaded interlacing. Lay out three fabrics — a plain jersey, a fabric with a tuck course, and one with a float course — and compute the linking number between adjacent courses at each wale.

The prediction is exact rather than approximate: one for a knit wale, two for a tuck wale, and nought for a float wale with the link carried to the course above.

An integer prediction at every wale is the sharpest possible test of a repaired geometry, much sharper than any energy or dimension, because there is no tolerance in it. A repaired model that returned 1.8 for a tuck would be wrong rather than imprecise.

That makes this rung’s main output a specification for a check rather than a result, which is the honest way to describe it.

The two-bed cases #

The three stitches are single-bed statements and a two-bed fabric has more arrangements available, so it is worth extending the count.

A rib knits alternate wales on alternate beds. Each wale’s chain is unbroken along its own bed and no chain crosses from one bed to the other — which is why a run cannot cross a bed and why a rib runs along one wale and stops there.

An interlock knits every wale on both beds, so each loop is held from both faces. That is not a tuck and it is the same mechanism: two things holding where a jersey has one.

A purl fabric alternates the direction of successive courses, so a loop is drawn through the loop below from the front and the loop above is drawn through it from the back. The linking is still one per wale per course, and the fabric’s run behaviour is closer to jersey’s than to interlock’s — which is what the count predicts and is what is observed.

So the two-bed structures do not add a new mechanism; they add ways of arranging the same one. What decides run resistance in every case is how many things hold each loop, and the answer is one for jersey and purl, one per bed for rib, and two for interlock and for a tuck.

What was counted, and how #

Nothing here is computed on this collection’s own geometry, for the reason above.

The linking numbers are read off the stitches’ definitions, which are unambiguous: a knit draws through one loop, a tuck accumulates a second, a miss draws through none.

The instrument that would compute them exists — the Gauss linking integral, checked against four arrangements whose answers are known by inspection — and is applied to the model’s own courses elsewhere in this work, where it returns nought.

The fabric behaviours quoted are trade knowledge and are not measurements made here.

What a tuck costs #

A run-stop is not free and the costs are worth listing, because they explain why a whole fabric is not made of tucks.

Yarn. A tucked loop holds two loops in one head, so the head has to be larger and takes more yarn. A tuck fabric is heavier than a plain one at the same loop length and gauge.

Width. A tuck spreads its wale, because the accumulated loops sit side by side rather than stacked, so a tuck fabric is wider and less extensible across its courses.

Length. The same accumulation shortens the fabric wale-wise, so a tuck fabric is shorter and thicker.

And appearance. A tuck is visible: it makes the characteristic cellular texture of a piqué, which is the reason most tuck fabrics are made and is a constraint on where a tuck can be hidden.

So a run-stop course in a stocking is a visible band, and putting one in every course would make a completely different fabric. The trade’s practice of scattering tucks — a few per repeat — is the compromise between the topology and the four costs.

Where the model stops #

Everything. This rung is an argument about knitting made with a vocabulary this work built, on a model that cannot support it.

And the tuck’s own geometry is absent too. A tuck arrives at its needle at an angle rather than at an extreme point of its wave, which breaks the symmetry the collection’s three-dimensional loop solve rests on — so even the tuck’s shape is outside what this collection computes, let alone its topology.

That was recorded as an open item several ladders ago and remains one.

Nor is the run’s dynamics here. A run stops at a tuck topologically, and how far it gets before it reaches one, and whether it can restart past it, are questions with friction and energy in them that this collection has partly answered for a plain fabric and not at all for a patterned one.

Why three stitches and not more #

A question the linking view answers neatly and that the usual account leaves as a fact about machines.

A needle in a weft knitting machine can do exactly three things with an incoming yarn: take it and cast off, take it and not cast off, or not take it. There is no fourth.

In linking terms those are: add a link and release the old one, add a loop to the head without releasing, and do nothing.

So the three stitches are not an arbitrary vocabulary that happens to have three members. They are the complete set of things that can be done to a held loop, and the completeness is why every weft-knitted fabric that exists is a pattern of these three.

That is a satisfying place for an enumeration to land and it is the kind of statement this collection likes: a set that is complete for a reason rather than by convention.

Warp knitting is the exception and it is exactly where the enumeration breaks. A warp knitting machine’s guide can move sideways as well as swinging, so it can lay its yarn into a different needle from the one it came from — which is a fourth thing, and is why a tricot’s structure cannot be written in the same notation.

The generalisation #

The rung is an instance of something worth naming about what a new vocabulary does.

A quantity is worth having when it lets a question be asked, even before it can be answered.

Before this work, “why does a tuck stop a run” had no crisp form in this collection. It could be described — the loop is held twice — and the description was not a quantity, so it could not be compared with anything, checked, or computed.

The linking number gives it a form: one, two, nought. That form is not yet computable here, and it is now stateable, and a stateable claim is a claim somebody can test.

That is most of what an invariant is for in a subject like this one. It is rarely the number that matters; it is that a family of vague statements about holding, running, fraying and unravelling collapses into arithmetic on integers.

The count a fabric could be specified by #

The three integers suggest a fabric descriptor this collection could compute and the trade does not use, and it is worth setting out because it would be cheap.

For any stitch pattern, count for each loop how many loops hold it. A plain jersey gives one everywhere. A tuck fabric gives one for most loops and two for the loops above a tuck. A float fabric gives one for most and nought for the floated ones, with the load carried forward.

Take the minimum over the fabric and that is the weakest link in the chain — a number between nought and two that says whether a run can propagate at all.

Take the mean spacing between loops held more than once and that is how far a run can travel before it is stopped.

Two numbers, both integers or counts, both computable directly from a stitch notation without any geometry at all. They would say more about a fabric’s run behaviour than any measurement currently made, and the input is a pattern the knitter already has.

That is a proposal rather than a result, and it is the kind this collection is well placed to make: the site already parses stitch notations and censuses them, so the machinery is a walk over an existing structure.

A prediction about where a run actually stops #

The two conditions together give a prediction about the shape of a stopped run, and it is observable on a laddered stocking.

If a run is stopped topologically — by a tuck — it stops abruptly, at a definite course, and the fabric above the stopping course is undisturbed.

If it is stopped frictionally — by running out of energy — it stops gradually, and the last few loops are partly pulled rather than fully released.

Those look different. A tuck-stopped run has a clean edge; a friction-stopped one has a ragged one with distorted loops above it.

Anybody with a laddered garment and a magnifier can tell which happened, and the distinction says whether the run-stop in the fabric did its job or whether the run happened to run out.

That is a small observation and it is the kind that turns an argument into something checkable without an instrument, which is what this collection tries to produce wherever it can.

Who found it, and when #

The three stitches and their effects on fabric behaviour are the foundations of weft knitting and are in every text.

That a tuck resists a run is universal trade knowledge and is why run-stop courses are knitted.

What is this collection’s own is the framing: reading the three stitches as three linking contributions, which turns a set of practical facts into three integers, and specifying the check a repaired model would have to pass.

What the count does not capture #

Two things the linking view leaves out, and both matter to a fabric’s actual run behaviour.

Friction. A loop that is topologically free still has to be pulled out through its neighbours against friction, and this collection has computed that race: a run happens when the bending energy released beats the friction resisting. A tightly knitted fabric of a hairy yarn runs slowly and a slack one of a smooth yarn runs fast, at identical linking.

And the load. A run needs something pulling. A fabric lying on a table with a broken loop in it may not run at all until it is picked up.

So the linking count says whether a run can propagate and the friction and load say whether it does. Two independent conditions, and a fabric is run-proof if either fails.

That is why the trade’s remedies split the way they do: a tuck attacks the first, and a tight construction of a hairy yarn attacks the second. The first is reliable and visible; the second is invisible and can be undone by a wash that smooths the yarn.

Where the ladder goes next #

This collection’s contact half has one measurement left to use rather than to report. A fabric’s yarn is pressed on a fifth of its length and free on the rest, so a fabric’s friction lives in a fifth of its yarn — and everything this collection computes about withdrawal, slippage and fraying assumes it lives everywhere.

A fabric is a population of contacts.