Knit, tuck and miss

Everything on this site rests on a weave being a binary matrix. Warp up or weft up, one bit per intersection, and every question about a fabric becomes a question about the matrix — how long its runs are, whether it is connected, how many of them there are.

Knitting has looked like the exception. The loop is the object, and the arguments about it are about a thread bent through the thread below it: why a knit extends, why stockinette curls, why a run runs. None of that is an array.

It can be. At every needle on every course a weft-knitting machine does exactly one of three things, and it is those three and no others.

Three symbols, and the machine that has exactly three #

A weft-knitting needle rises to take yarn and falls to form a loop. How far it rises decides everything.

Knit. The needle rises fully, so the old loop slides down past the latch and off; it takes the new yarn and pulls it through the old loop, which is cast off. A new loop exists and the old one is now part of the fabric below.

Tuck. The needle rises partly. The old loop stays on the hook and the new yarn joins it there. Nothing is cast off and the old loop is held for another course, with two thicknesses of yarn now in it.

Miss. The needle does not rise at all. It keeps its loop, takes no yarn, and the yarn floats straight past it across the back of the fabric.

That is the whole vocabulary and it has been the whole vocabulary since the latch needle. Every weft-knitted structure the trade names is a small array over those three: plain jersey is all K; half-cardigan alternates K and T on one bed; single piqué — lacoste — is a tuck on every other needle of every other course; a float jersey is a miss on the same schedule.

So a weft-knitted structure is an array over {K, T, M} in exactly the way a weave is an array over {0, 1}, and the repeat is toroidal in both directions for exactly the same reason: a structure that is only right in its interior describes a fabric that is wrong wherever the repeats meet.

The question survives the translation #

The site exists to ask one question of a draft that the drawing cannot answer: does this describe one cloth. A weave falls apart when its above-and-below relation is not strongly connected, and the failure is invisible — a two-layer draft looks exactly like a one-layer draft.

The knitted version of that question is not connectivity. A weft knit is one continuous thread travelling across the needles, so its fabric is connected by construction and the criterion has nothing to work on. Something else replaces it, and it is one line.

A needle that never knits never casts off.

The old loop stays on the hook. Next course it is joined by more yarn and stays; the course after that, more again. What happens next is decided by the machine: either the hook cannot close over the accumulated yarn and the needle jams and breaks, or the yarn breaks first and the loop drops — which is a ladder running the length of the piece, the same failure the site has already traced to a knit being one thread rather than many.

There is a second condition and it is the other way round. A course that never knits is never caught by anything. Every needle in it either tucks or misses, so no loop is formed anywhere along it, and the yarn of that course lies in the fabric held by nothing. It is a length of yarn inside a fabric rather than part of one.

Both conditions are properties of the array. The first is a column with no K in it; the second is a row with no K in it. Neither is visible on a loop diagram of any size, because a loop diagram of more than about six needles cannot be read at all — which is the reason the array exists.

The third condition belongs to the machine #

There is a limit that is not geometric and it has to be stated separately, because it is the only number in this essay that comes from a manual rather than from a count.

A needle can hold so many courses of yarn and no more. Each held course adds a thickness in the hook, and at some point the latch will not close over them — so a structure that tucks the same needle five courses running is knittable in the array’s sense and unknittable on any machine. Four is a generous limit; three is what most machines are set for; two is what most structures use.

The site’s convention for a number like that is to make it an argument rather than a constant, and to say which value produced which figure. Every census here states its hold limit.

What was counted, and how #

Three symbols on w × c positions is 3^(wc) arrays. That is 81 on a two-by-two and 19,683 on a three-by-three: small enough to walk exhaustively and large enough that the fractions mean something.

About one array in five is a fabric. The fraction does not fall steadily with the repeat, which was the shape expected and is not what the counting says: the two conditions bite in opposite directions as the array grows longer one way or the other, and the middle two rows are lower than either neighbour.

The failures do not weigh the same either. On the three-by-three, 12,824 arrays fail because some wale never knits and 2,808 because some course never knits — a ratio of nearly five to one, and it is not because one condition is stricter than the other. It is because a wale that never knits is also a needle that holds for ever, so those arrays fail the machine limit as well and are counted once under the first heading.

Why the failing share does not fall with the repeat #

The census’s most surprising column is the share, because it does not behave the way an enumeration’s share usually does: 21.0 per cent at two by two, 15.0 at the two rectangular sizes, and back up to 20.6 at three by three.

A share that rose or fell steadily would have one mechanism behind it. Two conditions acting on rows and columns separately do not give one mechanism, and the shape falls out of how each condition scales.

A wale fails when its whole column is free of K. With three symbols the chance a given position is not a K is two thirds, so a column of c positions is free of K with probability (2/3)^c — which falls fast as the array gets taller. A three-course array is much safer per wale than a two-course one.

And there are more wales to fail as the array gets wider. The chance that no column fails is roughly the per-column chance raised to the number of wales, so widening the array makes failure likelier while deepening it makes each column safer.

The same argument runs the other way for the course condition. So depth protects the wales and threatens the courses, and width does the reverse, and the two effects are of comparable size because the two conditions are symmetric under transposing the array — which is exactly why the two rectangular rows are equal to the last unit.

That equality is worth noticing on its own. Two courses by three wales and three courses by two wales give identical counts, and they do so because the pair of conditions is symmetric under exchanging rows for columns even though the two failures are physically nothing alike: one is a needle that jams and one is a length of loose yarn. The arithmetic cannot tell them apart and the machine can, which is the same division this site keeps meeting between a count and a mechanism.

So the share at a square repeat is high because both conditions are as weak as they get for that number of positions, and the share at a rectangular one is lower because one of the two conditions is being made stricter faster than the other is relaxed. Nothing about that is a fact about knitting; it is a fact about two coverage conditions on a rectangle.

What the array is for #

Setting the notation up this way is not only bookkeeping. Three things become computable that were not.

Whether a structure is a fabric, which is the whole of the above, and which the trade settles by knitting one.

How much yarn it uses, which is a count over the array once the yarn in each of the three actions is known — and it is, exactly, for two of the three. That is the next rung.

Which structures exist at all, in the sense the site means: the four thousand knittable three-by-threes are not four thousand fabrics, because a structure and its translate are the same fabric started at a different needle, and reducing by that is the same orbit counting the weave censuses need. That reduction is recorded here as not done.

What the three actions do to the fabric #

The array is a notation and not a prediction, but the three actions have three characters that every knitter knows and that are worth setting down before the arithmetic reaches them.

A tuck opens the fabric out. The held loop has two thicknesses of yarn in it and is pulled wide by both, so it pushes its neighbouring wales apart. A tuck fabric is wider and shorter than a plain one of the same yarn at the same setting, thicker in the hand, and more open — which is why single piqué is a summer shirt fabric and why cardigan structures are used for the bulky parts of a knitted garment.

A miss pulls it in. The float lies across the back between two loops and there is no loop between them to hold them apart, so the two wales close up. A float fabric is narrower and longer than plain, denser, less extensible across the width, and it has a smooth face and a floated back. It is the structure behind every two-colour jacquard jersey, because a colour not wanted on the face has to go somewhere and the back is where.

And a knit does neither, which is why plain jersey is the reference every one of those comparisons is made against.

The site’s own float is the connecting object and the analogy is closer than it looks: in both fabrics a float is a length of yarn on the surface with nothing holding it down, and in both it decides lustre, snagging and how far the fabric can be pulled. Where it stops is a separate rung, and the differences are large enough to need one.

Both effects are dimensional and neither is computed here. What is computable exactly is how much yarn each action costs, and that turns out to be enough to settle the weight question without settling the dimension one — which is the awkward and honest position the next rung ends in.

Where the analogy holds and where it stops #

The array is a real translation of the site’s central object and it is worth being precise about how far it goes.

It holds for the repeat. Both are periodic arrays with a toroidal repeat, both have a minimal unit that may be smaller than the writing, and both have counts — floats, runs, symmetries — taken cyclically.

It holds for the integrity question, in the sense that both fabrics have a condition that is invisible in the drawing and decides whether the object exists. It does not hold in the form of that condition: the weave’s is connectivity of a relation between threads, and the knit’s is a coverage condition on rows and columns. They are not the same theorem wearing different clothes.

It does not hold for the symmetry. A weave’s two symbols can be exchanged — turn the cloth over and warp-up becomes weft-up — and the site counts that exchange as part of a draft’s symmetry group. The knit’s three symbols cannot be permuted at all. K, T and M are three different physical actions with three different consequences and no operation on a fabric maps one to another, so there is no two-colour bookkeeping and no forty-six groups.

And it does not hold for the geometry. A weave’s matrix decides floats and interlacings exactly and says nothing about yarn. A knit’s array decides which needle does what and says nothing about the loop, which is where the knit’s own mechanics live — its extension, its curl, its recovery. The array is a layer above the loop, not a replacement for it.

Where the model stops #

The array knows nothing about the bed. Everything here is single jersey — one needle bed. A rib or an interlock has two beds facing each other and a needle on each, so the structure is an array over a larger alphabet and the conditions are more complicated: a wale on the back bed that never knits is held by the front bed’s fabric and does not ladder the same way. Rib and interlock are treated on this site as a geometry rather than as a notation, and joining the two is not done.

The hold limit is one number standing for a mechanism. How many courses a needle can hold depends on the yarn’s thickness, the hook’s size, the machine’s cam setting and how tightly the previous loop was drawn. Four is a convention.

And nothing here is a fabric until it is knitted. A structure can satisfy both conditions and the hold limit and still be unmakeable for reasons the array cannot see — a float too long for the machine’s yarn carrier, a tuck that drops on a particular needle timing, a tension that cannot be balanced across the repeat. The conditions are necessary and the essay does not claim they are sufficient.

Who found it, and when #

The three actions are as old as the latch needle — Matthew Townsend’s, 1849 — and the notation is older than that in spirit: knitters have always written structures as grids of symbols, and industrial knitting has used a needle-by-course diagram since the trade had drawing offices.

What the trade’s notation does not carry is the closure. A weaving draft is understood to be an arbitrary binary matrix, which is why somebody eventually asked how many there are; a knitting diagram is understood as a way of writing down structures somebody already has. The pattern books list half-cardigan, cardigan, piqué, and a few dozen more, and none of them asks what fraction of the writable arrays are fabrics, because the question does not arise if the arrays are always transcribed from something that already exists.

The condition itself is not a discovery either. Every knitter knows that a needle must clear its loop, and every knitting-machine manual says so in the section about needle timing. What is new here is only that it is stated as a property of the array — a column with no K in it — where it is checkable before anything is threaded, and where it can be counted.

Where the ladder goes next #

The array says which structures exist. The next rung asks what they cost, and finds that two thirds of the question is exact: a knitted loop is about four wale spacings of yarn and a float across one needle is exactly one, so replacing a knit with a miss removes three quarters of a loop and the arithmetic has no free parameter in it at all.