A tuck decides whether a third of two-bed fabrics lean

A jersey leans because its yarn still turns, and a rib knitted from the same yarn does not. The reason was a count: a loop knitted on the front bed and one on the back are mirror images, so their torques oppose, and a fabric with as many of each nets to nothing whatever the yarn is doing. Single jersey’s imbalance is one, a rib’s is nought, and a half-cardigan, knitting four loops on the front and two on the back in its repeat, sits at a third.

That count had one line in its method that is easy to read past. “A tuck is not a loop and a miss is nothing.” A miss is certainly nothing: the yarn floats past a needle and wraps nothing. A tuck is not so obviously nothing. It is a loop of the same yarn, laid over a needle of one bed and held there, drawn off one course later along with the next loop. The yarn’s residual torque acts on it exactly as it acts on any length of yarn bent round a needle.

How much of a knit loop’s lean a tuck carries is not measured here, and it may be anything from nothing to all of it. That is ordinary ignorance about a small quantity — a yarn’s residual torque is itself described by one angle and a fitted slope — and it would not be worth an essay if the answer changed nothing. It changes a great deal: counted over every two-bed fabric a small frame can make, the share a tuck carries decides whether a third of them lean.

Counting a tuck as a share of a loop #

The balance count is kept exactly as it was, with one parameter added. Each wale of a two-bed structure contributes its knitted loops to its bed’s count, as before, plus a share w of its tucks. The imbalance of a fabric is the difference between the two beds’ counts over their total, taken for each fabric the structure separates into and reported for the worst.

At w = 0 that is the original count, loop for loop. At w = 1 a tuck leans as much as a knit loop. Structures with no tucks do not care what w is.

The half-cardigan is the case that moves furthest. Its front bed knits on every course, four loops in the repeat; its back bed knits two and tucks two. Counting tucks as nothing it is 4 against 2, an imbalance of a third. Counting them as full loops it is 4 against 4, and balanced. In between, its imbalance is exactly (1 − w)/(3 + w).

A full cardigan does not move at all, because it tucks equally on both beds and its tucks cancel whatever they carry — the same mirror argument that makes a rib and an interlock straight, applied to held loops. A rib that both tucks and floats falls from a third to a seventh, because its tucks sit on the bed that was short but do not make up the whole of the difference.

What a third of a jersey’s lean is in degrees #

The lean itself is the measured relation the earlier account used: nine degrees for every unit of twist factor above a balanced 2.4, fitted rather than derived. At a twist factor of 4.0 a single jersey leans 14.4 degrees.

A half-cardigan in the same yarn then leans 4.8 degrees if a tuck carries nothing and none at all if it carries a full loop’s torque. A rib that tucks and floats leans 4.8 degrees or 2.1. A knitter choosing a half-cardigan precisely because it is a double fabric — firmer, thicker, less inclined to curl than a jersey — is also choosing a fabric whose spirality nobody here can predict to within its own size.

That is the practical form of the result. The count was a permission: a balanced structure cannot lean, whatever the yarn, and that is exact. For any structure with tucks on unequal beds, the permission itself now depends on an unmeasured number, and the exactness the count was valued for is lost precisely where the tucks are.

Every two-bed fabric of a small frame #

The named structures are a handful. The question worth asking is how common the dependence is, and the two-bed census answers it.

The frame is two needles on each bed over two courses, with rib gating, the same frame on which a second bed changes what a float is: every assignment of knit, tuck or miss to each needle of each bed on each course, 6,561 of them, of which 1,135 are fabrics a machine could make under the knittability rules — every wale knits somewhere, every course knits somewhere, and no needle holds longer than four courses. Of those 1,135, 910 contain at least one tuck.

135 fabrics are balanced whatever a tuck carries — the ribs and the interlocks, and every structure whose knits and whose tucks are each divided equally between the beds of every fabric it makes. 612 lean whatever a tuck carries. And 388 are decided by it: 236 are balanced only if a tuck carries nothing, and would lean if it carries something, and 152 are balanced only if a tuck carries a full loop’s torque.

The two decided classes run in opposite directions, which is what makes the dependence impossible to hedge. There is no single assumption about tucks that errs on the safe side: counting tucks as nothing calls 236 fabrics straight that may lean and 152 leaning that may be straight, and counting them as full loops reverses both errors.

A larger frame, the same third #

Two courses is a small repeat, and a share found on it could be an accident of its size. Three courses tests that.

On three courses there are 117,951 fabrics and 97 per cent of them contain a tuck. 42,138 — 36 per cent — are decided by the tuck, against 34 per cent on two courses. The balanced-either-way class shrinks, from 12 to 8 per cent, because a longer repeat has more ways to put its tucks unevenly, and the decided share holds.

With interlock gating, where the two beds’ needles share needle spaces and only 120 structures are fabrics on the smaller frame, the tuck decides 16 of them, all eight in each direction. The share is smaller and the dependence is still there.

How the imbalances move, not only whether #

A decided fabric is one that is exactly balanced under one account. The census also shows how the rest of the distribution shifts, and the shift is not just at nought.

Counting tucks as loops empties the balanced class by 84 fabrics and fills the imbalance of a seventh from 16 to 208, because a tuck added to one bed’s count most often turns a balanced repeat into a slightly unbalanced one. It also moves fabrics the other way, filling an imbalance of three fifths from 16 to 64. Only the fully unbalanced class, 80 fabrics, is untouched, and those are exactly the structures knitting on one bed alone, where there is no second bed for a tuck to make up — including the tube whose two beds are two fabrics.

At almost every share, only the ribs’ class hangs straight #

The four classes describe the two ends of the interval, and the truth about a tuck is very probably neither end. A tuck is a loop and is not knitted off, so it is natural to expect it to carry some of a knit loop’s torque and not all. What happens in between is not an interpolation of the ends, and it is sharper than either.

Each fabric’s difference between its beds is its knit difference ΔK plus w times its tuck difference ΔT. That vanishes for every share when both differences are nought, for no share when only the tuck difference is, and otherwise at exactly one share, w = −ΔK/ΔT. So each two-bed fabric either hangs straight whatever a tuck carries, or hangs straight at one particular share, or never.

On the two-course frame the balancing shares are nought, a half and one, and nothing else. At any other share — a third, a tenth, nine tenths — only the 135 fabrics balanced at every share hang straight, and the other 1,000 lean. The 236 that the original count called straight lean at every share above nought; the 152 that counting tucks as loops would call straight lean at every share below one; and 580 lean whatever a tuck carries.

On the three-course frame the shares multiply to a quarter, a third, a half, two thirds and three quarters besides the ends, 8,010 fabrics balancing at a half alone. Still, at a generic share only 9,581 of the 117,951 fabrics hang straight: eight per cent, against 27 per cent that the knit-only count would have called balanced.

That turns the census into a sharper statement than a third. Unless a tuck carries exactly nothing, exactly a full loop’s torque, or one of a few simple fractions, a two-bed structure with tucks unequally divided leans, and the count that treated a tuck as nothing overstated how many straight fabrics there are by a factor of about three.

A half-cardigan weighs a tuck against a knit loop #

The census says the share matters. It also says, through the half-cardigan’s closed form, how to measure it without the fitted slope that turns an imbalance into degrees.

Knit a single jersey and a half-cardigan from the same singles yarn at the same twist, relax both the same way, and measure both leans. The jersey’s imbalance is one and the half-cardigan’s is (1 − w)/(3 + w), so the ratio r of the two leans is that expression, and it can be turned round:

$$w = \frac{1 - 3r}{1 + r}$$

No fitted constant enters. The slope of lean against twist factor, the yarn’s fibre, whether it was steamed a week or a year ago — everything that makes a single spirality measurement uncertain — divides out of the ratio, because both fabrics carry the same yarn in the same state. The half-cardigan is a balance with a knit loop on one pan and a tuck on the other.

The measurement has a practical difficulty the arithmetic does not remove. A lean of a few degrees is read off a relaxed tube to perhaps a degree, so at a twist factor of 4.0 the half-cardigan’s reading of 0 to 4.8 degrees gives w to within a fifth or so. A livelier yarn widens the range and sharpens the reading, which is the one place in this account where a harder-twisted yarn is the better choice.

Why the balance count could not see this #

The count’s silence about tucks was not carelessness, and it is worth saying why the question stayed hidden.

A tuck is the one stitch that links twice: it holds two loops in one head, which is why it stops a run and why it changes a fabric’s width. Those are properties of how it links. The balance count asks a different question — where yarn is bent round a needle and on which bed — and a tuck bends yarn round a needle as surely as a knit does. The count asked a question about loops and answered it about knit stitches, and the named structures it was checked on happened to put the difference where it did not show: ribs and interlocks have no tucks, and single jersey has only one bed.

What a tuck costs in yarn is known exactly, since a tuck takes its loop’s length without the loop being knitted off. What it costs in torque is a separate quantity, and there is no reason to think the two shares are the same: the yarn in a held tuck is bent round the needle but is not drawn through the loop below, and the torque that leans a wale acts through that interlooping as well as through the bend.

What a knitter can do without the number #

The share is unmeasured, and a structure does not have to wait for it. The census’s own classes say which designs are safe from the question altogether.

A structure is immune to the tuck’s share exactly when every fabric it makes divides both its knits and its tucks equally between the beds. That is the class of 135 on the small frame and 9,581 on the larger one, and it has a simple reading at the machine: whatever a design does on the front bed with tucks, it does the same number of times on the back. A full cardigan is the named example. A half-cardigan, which tucks on one bed only, is the named counter-example, and every design derived from it inherits the question.

The rule costs something, and it is worth being plain about what. A half-cardigan’s one-sided tucking is what gives it its character — a plump face on one side and a flatter back — and a design that tucks symmetrically to escape the torque question gives that asymmetry up. Spirality and a two-faced tuck structure are bought together, and until the share is measured a designer who wants one is gambling on the other.

For the designs that must keep one-sided tucks there is a second route, and it is the one the account below already found works for single jersey: take the torque out of the yarn. A set yarn has no torque, and a structure knitted from one does not lean whatever its loops and tucks are doing, so the share stops mattering at the same moment the lean does. The twist a fabric gives back is the warning attached: a set that is undone in washing restores the torque, and with it the question.

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, admitted when every wale knits at least once, every course knits somewhere, and no needle holds a loop for more than four courses. For each admitted structure the wales are grouped into the fabrics the connectivity criterion finds; each fabric’s front and back counts are its wales’ knits plus a share of their tucks; the imbalance is the difference over the total, and the structure’s is its worst fabric’s.

A share of nought was confirmed to reproduce the earlier balance count for every named structure it was run on; structures without tucks were confirmed not to change with the share; the half-cardigan was confirmed at a third with a share of nought and at nought with a share of one; and on the two-by-two frame every one of the four classes was confirmed to occur, to account for every fabric between them, and to leave more than a quarter of the fabrics decided by the share.

Where the count still stops #

The torque acts on a whole loop, not a number. The account treats a loop’s lean as a unit that adds across a fabric, and a tuck’s as a fraction of it. A real tuck’s contribution may depend on what it is held with — how many courses, next to what — so w may not be a single number for all structures.

Only the imbalance is counted. Two fabrics with the same imbalance lean by the same fitted amount here, and a fabric’s own stiffness against shear, which resists the lean, is not modelled. A firm double fabric may lean less than its imbalance says.

The machine’s own helix is set aside. A circular machine leans its courses whatever the yarn, and a tube’s measured spirality is the helix and the torque together. The half-cardigan measurement has to be taken on flat-bed fabric, or corrected for the helix on both fabrics alike, for the ratio to mean what it says.

Still open: the share itself #

Everything above narrows to a single number that has not been measured: the share of a knit loop’s torque a tuck carries. The experiment is the one described — a jersey and a half-cardigan in one lively singles, knitted flat, relaxed alike, leans read — and its ratio gives the share directly. Repeated with a steamed yarn, both leans should vanish together, which is the check that what was measured was torque. Repeated with tucks held for two courses rather than one, it would say whether a tuck’s share depends on how long it is held, and so whether the census’s two classes need a third.

Who found it, and when #

Spirality, cardigan structures and the use of rib-based double fabrics to suppress it are ordinary knitting knowledge, and the balance count per fabric was worked out in the account below. Asking what share of a knit loop’s torque a tuck carries, finding over the two-bed census that the share decides the lean of a third of the fabrics, and turning a half-cardigan into the instrument that measures it, were done here.