A tuck held two courses is a second unknown

A tuck decides whether a third of two-bed fabrics lean gave a tuck a share $w$ of a knit loop’s torque and counted every two-bed fabric of a small frame against it. The result was sharp: unless $w$ is nought, one, or one of a few simple fractions, only the structures that divide both their knits and their tucks equally between the beds hang straight, and on two needles by three courses that is 9,581 of 117,951 fabrics.

That count had a premise of its own, and it is the same kind of premise the one before it had. A tuck was one thing. A needle that tucks on one course and knits on the next was counted the same, loop for loop, as a needle that tucks on two courses running and holds both loops until it knits. The essay ended by naming that as the next question: whether a tuck’s share depends on how long it is held, and so whether the census needed a third class.

It does, and the reason is not a measurement. It is a count. Give a held loop its own unknown share, and the immune class itself depends on what that share is.

A held tuck is several loops in one hook #

A tuck lays yarn into a needle’s hook without casting off the loop already there — the third of the three things a needle can do, beside knitting and missing. If the needle tucks again on the next course, a second loop of yarn goes into the same hook, on top of the first, and both stay there with the old loop until the needle next knits. Then all of them are cast off together, into the head of the new loop. That is what a needle holding a tuck for two courses means, and it is common: a tuck held over two or more courses is how the fuller tuck textures are made.

A miss changes nothing in a hold. The needle takes no yarn, so a sequence tuck, miss, tuck still leaves two tuck loops in one hook. That is exactly how the knittability rule already reads a hold — round the repeat, misses passed over — and the count below reads it the same way, so a needle’s holds are whatever that rule says they are.

On a three-course repeat a needle must knit at least once, so it can hold at most two courses. Every tuck in the frame is therefore one of two things: a single tuck, held for one course, or a loop of a held tuck, two loops in one hook. Call their shares of a knit loop’s torque $s$ and $h$. The account that came before set $h = s$ without saying so.

Every structure hangs straight on a line #

A fabric’s lean is its imbalance: front-bed loops against back-bed loops over their total, per fabric, with the worst fabric reported. With two shares, the front-minus-back count of one fabric is

$$\Delta K + s\,\Delta T_1 + h\,\Delta T_2,$$

where $\Delta K$ is its knits’ difference between the beds, $\Delta T_1$ its single tucks’ and $\Delta T_2$ its held loops’. The fabric hangs straight where that vanishes. In the square of $s$ and $h$ that is the whole square when all three differences are nought, nowhere when only $\Delta K$ is not, and otherwise a straight line. A structure of several fabrics hangs straight where all of their lines meet.

So the picture changes shape. On one unknown a structure balanced at every share, at one share, or at none. On two, the middle case has become a line, and a line can lie anywhere: across the square, along an edge, or through a corner and nowhere else. The figure draws five, each traced from its own count rather than ruled in beside it.

The half-cardigan of the earlier count knits every course on its front bed and on its back bed knits one course in two and tucks the other. It holds no tuck longer than one course, so it never sees $h$: it hangs straight on the vertical line $s = 1$, which is the earlier result with the other unknown added and found not to matter. A held half-cardigan — the same front bed, a back bed knitting one course in three and holding a tuck over the other two — holds only held loops, and hangs straight on the horizontal line $h = 1$.

The other three are the interesting ones, because they hold both kinds of tuck, and the lines they hang straight on run through the inside of the square.

Two accounts of a held tuck, and neither is measured #

There are two simple things a held loop could do, and they are worth stating before any fabric is counted, because each is a line in the square.

Per loop. Every tuck loop carries the same share, however many others are in its hook: $h = s$. That is the diagonal, and it is the account the earlier count used. Its reasoning is that torque acts on yarn bent round a needle, and two loops in one hook are two lengths of yarn bent round it.

Per hold. A hold carries one tuck’s share however many loops it holds, so each of the two loops in a held tuck carries half: $h = s/2$. Its reasoning is that the loops in one hook are bent round the same needle at the same place and cast off as one, so they may act on the wale as one bend rather than two.

Neither is derived here and either may be wrong; the truth could be anywhere in the square. They are useful because they are the two natural guesses, and because the census can say exactly how much hangs on the difference between them. What a tuck costs in yarn is a known length per loop, which is a per-loop quantity; that is no evidence about torque, since the torque acts through how the yarn is held as well as through how much of it there is.

Every two-bed fabric of three courses, by its line #

The frame is the same one the earlier census walked: two needles on each bed over three courses, rib gating, every assignment of knit, tuck or miss, admitted when every wale knits, every course knits somewhere and no needle holds past four courses. That admits 117,951 fabrics, and 53,346 of them — 45 per cent — hold a tuck for two courses somewhere. The held tuck is not a corner case of the frame. It is nearly half of it.

The eight classes are the whole of the answer, and the arithmetic behind them is exact: each fabric’s condition is a line with integer coefficients, two lines meet at a point given by fractions, and nothing is sampled. 7,793 fabrics are straight whatever either kind of tuck carries — 6.6 per cent of the frame — and 45,406 lean whatever either carries.

The 24,018 straight at one single-tuck share split three ways: 11,376 at $s = 0$, 10,494 at $s = 1$ and 2,148 at $s = \tfrac12$. Most hold no held tuck at all and are the earlier census’s decided classes, unchanged; 2,256 do, with their held loops already balanced between the beds. The 13,344 straight at one held share sit at $h = \tfrac12$ (5,406), $h = 0$ (4,392), $h = 1$ (3,174), and a few hundred at a quarter and three quarters.

And every fabric that hangs straight at a single point does so at a corner. 7,464 fabrics balance at exactly one pair of shares, and in every one of them that pair is a corner of the square: 3,216 at $(1, 0)$, 2,628 at $(0, 1)$, 1,152 at $(0, 0)$ and 468 at $(1, 1)$. No fabric on this frame hangs straight at a single pair strictly inside the square. If both shares are somewhere between nothing and a full loop’s — which is where any reasonable guess about a real yarn puts them — a fabric hangs straight along a line or not at all.

The safe class splits #

The earlier census’s most useful class was the one immune to the question: structures balanced at every share, which a designer could knit without knowing what a tuck carries. On this frame it held 9,581 fabrics. Every one of them is still immune if a held loop carries what a single tuck does. Not every one is immune otherwise.

1,788 of the 9,581 — very nearly a fifth — are immune only on the diagonal. They balance single tucks on one bed against held loops on the other, loop for loop, so the two kinds cancel exactly when they carry the same and not otherwise. They are the structures that looked safest and are in fact betting on an unmeasured equality.

The per-hold account reorganises the class rather than shrinking it. Under it 13,337 fabrics are immune, of which 5,544 are immune under no other account: they hold a hold of two on one bed against a single tuck on the other, which balances if a hold is one bend and not if it is two loops. Altogether 7,332 fabrics hang straight under one of the two natural accounts and lean under the other, against only 7,793 that are straight under both. On this frame the immune class is not a property of the structures alone. It is a property of the structures and of an account of the held tuck that nobody has measured.

Two structures, each straight under one account #

The census says which fabrics would settle the question. The smallest of them make the experiment almost as simple as the half-cardigan’s.

Two loops against two knits as many loops on each bed — four apiece in the repeat — and holds a tuck for two courses on one front needle against a single tuck on each back needle, both on the first course. Its front carries two held loops and its back two single tucks, so its imbalance is

$$\frac{|h - s|}{4 + s + h},$$

nought exactly when every loop carries the same, whatever that share is.

One hold against one has the same front and a back bed that tucks on one needle only, missing on the other. Its front carries one hold of two loops and its back one single tuck, and its imbalance is

$$\frac{|2h - s|}{8 + s + 2h},$$

nought exactly when a hold carries one tuck’s torque however many loops it holds.

Knitted side by side from one yarn, at most one of the two can hang straight, unless a tuck carries nothing at all. The first is straight on the diagonal and the second on the half-slope line, and the two lines meet only at the corner where both shares are nought. So the pair is a choice between the two accounts, made by eye: whichever hangs straight names the account, and if neither does, the truth is somewhere else in the square and the shares have to be read off separately.

How large the leans are, and what that asks of a yarn #

The two null structures are delicate instruments, and it is worth being plain about how delicate. Each knits equally on its beds, so its whole imbalance comes from a difference between tucks, and tucks are a small part of a structure’s loops.

At a twist factor of 4.0, where single jersey leans 14.4 degrees on the fitted slope, neither null structure ever leans more than 2.9 degrees, and on the other account’s line it leans 1.3 at most, reached only when a single tuck carries a full loop’s torque. A lean is read off a relaxed fabric to about a degree, and the yarn’s torque behind it is itself described by one angle and a fitted slope, so on that yarn the reading separating the accounts is barely above the reading’s own scatter.

A livelier yarn widens it in proportion. At a twist factor of 5.0 the same jersey leans 23.4 degrees, the larger separation becomes 4.7 and the smaller 2.1. That is the same conclusion the half-cardigan reached, and for the same reason: an instrument that measures a share of the yarn’s torque wants as much torque in the yarn as it can get, which makes this one of the few places a knitter would choose a hard-twisted singles on purpose. A yarn that has been set has no torque, so the yarn must be knitted fresh, and both null structures leaning nothing in a steamed yarn is the check that what they read was torque.

A held half-cardigan weighs a held tuck #

The null structures say which account is right without saying what the shares are. Reading the shares themselves takes two fabrics, each of which sees one share only, and the census supplies both.

The held half-cardigan holds only held loops, two in every back-bed hook, and its imbalance is $(1 - h)/(2 + h)$. Its lean $r$ as a fraction of a jersey’s in the same yarn turns round to

$$h = \frac{1 - 2r}{1 + r},$$

with no fitted constant in it, exactly as the half-cardigan’s $(1 - 3r)/(1 + r)$ gives $s$. And it is the better instrument of the two. A half-cardigan’s lean runs from a third of a jersey’s to nothing as $s$ goes from nought to one; a held half-cardigan’s runs from a half. At a twist factor of 4.0 that is 7.2 degrees of range against 4.8, half as much again to read a share off.

Three readings, two unknowns, and a check #

Put the instruments together and the experiment is overdetermined, which is the most useful thing about it.

A half-cardigan beside a jersey gives $s$. A held half-cardigan beside the same jersey gives $h$. Those two readings fix a point in the square, and that point predicts the lean of every other structure — in particular of the two null structures, which now have a predicted lean rather than an open one. Two loops against two should lean $|h - s|/(4 + s + h)$ of a jersey and one hold against one $|2h - s|/(8 + s + 2h)$.

If they do, the account has survived a test it could have failed. If they do not, the failure is more interesting than either share, because it says the count itself is wrong: that a fabric’s lean is not a sum of front loops against back loops with each kind of loop carrying a fixed share. That is the premise every count of spirality on this account rests on, from a jersey leaning because its yarn still turns onwards, and nothing so far has given it a way to be wrong.

That premise has a concrete failure in view. A tuck loop’s torque may depend on what surrounds it — the loops knitted next to it, the bed opposite it — and not only on how long it is held. Four structures cannot rule that out, but they can catch it: a dependence on surroundings would show as null structures leaning by amounts no single point of the square predicts.

Longer holds, on a narrower frame #

Three courses allow a hold of two and no more, so the question of a hold of three cannot be asked on that frame. A frame one needle wide on each bed can run to six courses and stay small enough to walk, and there a needle can hold a tuck for two, three or four courses.

On that frame 15,463 structures are fabrics and 6,456 of them hold a tuck for more than one course. The split has the same shape as on the wider frame: 897 immune under both accounts, 84 more only per loop, 444 more only per hold. The per-hold account again calls more fabrics straight than the per-loop one, 1,341 against 981.

One further thing was counted there, because it decides whether the second unknown is the last. Give every hold length its own share — one for a hold of two, another for three, another for four — and count the fabrics immune to all of them. It is the same 897. On this frame every fabric that is immune with single and held tucks given separate shares stays immune when the held shares are separated again by length. The line that matters falls between one course and more than one, and nowhere after it — on this frame, which is a count and not a proof that it holds on wider ones.

What a knitter can do without either number #

The census’s rule for safety tightens, and it is still a rule a knitter can apply at the machine.

A structure is straight whatever its tucks carry exactly when every fabric it makes divides its knits, its single tucks and its held tucks — each kind separately — equally between the beds. The earlier rule asked for knits and tucks; this one asks for single and held tucks to be balanced each in its own right, so that a held tuck on one bed is matched by a held tuck on the other rather than by two single ones. A full cardigan, which tucks single courses equally on both beds, passes it, for the same mirror reason a rib and an interlock hang straight. Any design that answers a held tuck with single tucks on the opposite bed does not, however even its loop count looks.

The cost is the one already found. A one-sided tuck structure is chosen for its asymmetry, and balancing every kind of tuck on both beds gives that up. What is new is that the balance has to be kept kind by kind: two designs with identical counts of tuck loops on each bed can differ in whether they are safe, because one pairs held with held and the other pairs held with single. For a structure that must keep its asymmetry, the set yarn remains the answer it was, with the warning the twist a fabric gives back attached: a set undone in the wash restores the torque, and with it both questions.

Why one share looked like enough #

The single-share account was not careless, and the reason it looked complete is worth saying. A tuck is the one stitch that links twice: it holds two loops in one head, which is a statement about how it links. A held tuck holds three or more, and the linking count simply adds them. Torque does not obviously add the same way, because it acts through the bend round the needle and through the way the loops are drawn off, and a second loop in a hook is bent and drawn off differently from the first — it sits on top, and it is cast off with the others.

On two courses, which is the frame the first census was built on and the half-cardigan was checked against, the question cannot arise at all: a needle that must knit once in two courses can hold for one at most. The structures on which the account was checked were exactly the ones on which it could not fail. The second unknown is invisible on the frame that found the first, and it appears the moment the repeat is long enough to hold a tuck twice.

What was counted, and how #

The structures are the two-bed census’s: every assignment of knit, tuck or miss to each needle of each bed on each course of a two-needle, three-course repeat with rib gating, admitted when every wale knits at least once, every course knits somewhere, and no needle holds past four courses. For each structure the wales are grouped into the fabrics the connectivity criterion finds — the one that decides whether a double jersey hangs together. For each wale the holds are read round the repeat from a knit, misses passed over, and every tuck loop is assigned to the hold it belongs to; a fabric’s three differences between the beds are its knits’, its single tucks’ and its held loops’.

Each fabric’s condition is a line with integer coefficients and is classified exactly: two distinct lines are intersected by Cramer’s rule in fractions, and a single line is cut to the unit square by the signs of its four corners. Equal shares were confirmed to reproduce the one-share imbalance on every named structure at three shares; the one-share immune class was confirmed to be the earlier census’s balanced-at-every-share class, 9,581, on the same frame; on two courses, where no hold is longer than one, the two counts were confirmed to agree exactly; the eight classes were confirmed to account for every fabric; and the two null structures were confirmed never to hang straight together at four pairs of shares off the corner where both are nought.

Where the count stops #

The shares are constants. Each kind of tuck loop carries one share in every structure, whatever is beside it. That is the premise the three-reading check exists to test, and it may fail; a tuck loop next to a held one, or opposite a knit, could carry something different.

Only the imbalance is counted, as before. A fabric’s resistance to shear is not modelled, so two structures of the same imbalance lean the same fitted amount here, and a firm double fabric may lean less.

The frames are small. Three courses allow a hold of two; six courses on one needle allow a hold of four, and the second-order result about hold length was found there and nowhere wider. An eight-course repeat on two needles is 3^32 assignments and is not walked.

The machine’s own helix is set aside, and a circular machine leans its courses whatever the yarn. Every instrument here has to be knitted flat, or corrected for the helix on every fabric alike, for the ratios to mean what they say.

Still open: where in the square a real yarn sits #

Four knitted structures and a jersey, all from one fresh singles, would place it: the half-cardigan gives the single share, the held half-cardigan the held one, and the two null structures check both the point and the count that put it there. The one result here that does not wait on the knitting is the census’s — that on three courses nearly a fifth of the fabrics that looked safe from the question are safe only on the diagonal, and that a per-hold account would make 5,544 others safe instead. Whichever line a real tuck lies on, a few thousand of these fabrics change sides.

Who found it, and when #

Held tucks, cardigan structures and the use of double fabrics against spirality are ordinary knitting knowledge, and the share a single tuck carries was set up as a measurement in the account below. Giving a held loop its own share, sorting every fabric of a frame by the line it hangs straight on, finding that every single-point balance falls at a corner, splitting the immune class between the per-loop and per-hold accounts, and designing the held half-cardigan and the two null structures that would read and test both shares, were done here.