A third bar has no say in a warp knit's first give

Opposed bars even a warp knit by locking it ended by reading its own formula more closely than its argument had needed to. A course’s spacing at a stated width is the tighter of the two circles its underlaps allow, and the tighter one is always set by the shortest underlap, when the fabric is narrowed, or by the longest, when it is widened. Its middle never decides anything. On a two-bar course that says nothing, since there is no middle. On a three-bar course it says something rather large.

A real patterned warp knit very often has three bars: a ground of two, and one on top to carry the pattern. If the middle of a course’s shogs has no influence on how the course gives, then the third bar is mechanically free. It can lap what it likes and the cloth keeps its give. That turns out to be exactly true, and exactness is the start of this account rather than the end of it, since a warp knit has other ways of charging for a lap.

Why the middle never binds #

The shog is the anisotropy reduced a warp knit’s first give to one object. An underlap of shog $s$ has a fixed length $u$, a wale spacing $w$ and a course spacing $c$, and they lie on a circle, $(sw)^2 + c^2 = u^2$. At the knitted width every circle passes through the same knitted course spacing, $c_0$, which is the same whatever the shog. So substitute, and away from the knitted point

$$c^2 = c_0^2 + s^2\,(w_0^2 - w^2).$$

That is the whole of it. At a stated width, a course’s spacing under one underlap depends on its shog only through $s^2$, and it does so monotonically. Narrow the fabric, $w < w_0$, and the second term is positive: a longer shog opens further, so the shortest shog is the tightest, and it is the one that limits the opening. Widen it, and the term is negative: a longer shog closes faster, so the longest is the tightest.

Every course takes the minimum across its bars. So on narrowing it reads its smallest shog, on widening its largest, and a shog strictly between the two is above the minimum on both sides, at every width, and never reaches it. That holds for any ground at all, and for any number of extra bars. It holds with no term in it for friction or stiffness, since none of these first-give circles has one.

The largest shog also sets the fabric’s widest possible state, the first underlap anywhere to lie flat, and a middle shog lies flat later. So nothing in the fabric’s limits moves either.

The checks run the sum every way it can be run: chain and satin ground, tricot and cord middle, every width from six tenths to 1.05 by hundredths, in floating point, with the two spacings differing by nothing above rounding. Add a shog of four to that course and it does bind: the rule has an edge, and a pattern that reaches past its ground’s own longest lap takes over the course’s widening. So the free alphabet is exactly the integers between the ground’s two shogs. A chain-and-satin ground allows tricot and cord and either end. A tricot-and-satin allows only cord. Two tricots allow nothing but tricot.

A ground therefore sets how much freedom it offers, as a range, and locking a fabric in both directions, which the opposed-bars account found to be what makes it even, is also what opens the alphabet. The stiffest cloth has the freest middle bar.

What bowing does, and why it stays out of it #

The open question also named a thing the first give might not survive. A middle bar’s underlap has other bars’ underlaps stacked with it, and lying over them it bows, rather than running straight between its needles.

That does not disturb the result, and the reason is a sign. A bowed underlap is longer than the straight line between its ends. Pull the fabric and that extra length is the first thing to come out, so a bowed underlap’s circle has a larger radius than its straight run implies, and it reaches its limit later. Bowing can only add slack, and a bar that was already not binding has only further to go. The same reasoning applied to a ground bar would move the answer, since a bowed ground underlap is a real change in the limit. How far it moves one is a question about the stack’s heights, which nothing here computes, and it would shift every number in this account by the same proportion without restoring the middle’s influence.

A bow still has consequences this first-give model cannot show, as a yarn lifted over its neighbours is more exposed at the surface, and it is one reason a top bar’s pattern reads as a pattern at all. That is a virtue rather than a cost.

A lap still costs a beam #

Free in the give is not free. A patterning bar’s underlap changes length from course to course, and a patterned warp knit needs a stroke as long as its stripe: a beam delivers at one rate, the bar’s demand swings around it, and each thread runs a shortfall that has to be paid back every repeat. Two bars cannot share a beam, so a third bar has its own beam and carries that stroke alone.

A middle bar striping six courses of tricot against six of cord has a stroke of 2.39 millimetres a thread, and striping chain against satin has 6.30. Those are the same in every way as the single-bar account’s figures, as they must be, for the same arithmetic. The third bar’s freedom is about the cloth; the machine has not heard of it.

And it costs a weight. A course with a longer underlap in it holds more yarn, and six such courses together are a band of the kind a thick place makes. On this ground a tricot course and a cord course differ by 17 per cent of the underlap in them and a chain and a satin course by 49 per cent, the loops left out, which the earlier account bracketed and which would lower both. So a length-patterned middle bar leaves the give even and puts a weight stripe into a cloth that was otherwise uniform. The first-give arithmetic that says it costs nothing cannot see that, and a viewer holding the fabric up to a window can.

That is the same division the opposed-bars account made, from the other side. There the total underlap was constant and the give was not what the total said. Here the give is constant and the total is not. Between them the two results take a course’s yarn and a course’s give fully apart. Neither says anything about the other, and a three-bar fabric can be designed on the two as separate questions.

Which plans are free on all three counts #

Only a constant underlap length has no stroke and no weight swing, and the lapping notation has room to say that and still pattern. A bar lapping tricot can step left or right on each course, and both are a shog of one with the same length. Direction is a pattern the beam cannot see. That is what makes an atlas lapping possible: the thread climbs across wales one needle at a time and comes back, with a constant shog throughout.

How much room is that? A bar’s plan is a list of signed shogs, one per course, that comes back to where it began. The count is exact.

Of 398,567 closed plans at eight courses, every one leaves the first give untouched; 22,555 have a stroke under a millimetre; 211 have no stroke at all. That 211 is one chain plus seventy direction patterns at each of the three other shogs: choose which four of eight courses step left. It is an exact count of what a beam need not hear about. Ninety-one of those have no run of three laps one way, by the same count with that one further condition: one chain, and thirty at each shog.

A designer also has threading, and the opposed-bars account’s weight stripe has no way to appear in it. A bar’s thread colours, one guide at a time, stripe a cloth with no change of lap at all, and on that route every plan above is free in all these respects. So the same arithmetic says something simple about how a warp knit’s design should sit. Lapping is the expensive way to make a pattern and threading the cheap one, and if lapping is to carry it, direction costs less than length.

A wider alphabet is bought in the ground #

The free alphabet is the integers between a ground’s two shogs, so the ground decides how many plans are on offer, and that is a real design decision. The two-bar account of a warp knit treats a ground as the choice of what a fabric’s stiffness will be. It is also the choice of how much a top bar can do.

Put a chain under each case, which holds every course against narrowing at its knitted spacing, and vary only the long bar. A chain and a tricot allow a middle bar shogs of nought or one, and at eight courses that is 1,107 closed plans. A chain and a cord allow up to two: 38,165. A chain and a satin allow up to three: 398,567. The alphabet multiplies by about thirty and then ten.

The same choice moves the cloth. Pull each of those grounds a fiftieth wider and its courses close to 0.97, 0.86 and 0.64 of their knitted spacing. The longer the long shog, the more freedom a top bar has and the harder the fabric’s courses shut when it is pulled wide. A designer cannot spend one without the other: the bar that opens the alphabet is the bar that sets the width-wise lock. Since a warp knit that does not shut its courses much is usually what is wanted from how a warp knit differs from a weft knit, a chain-and-tricot net and a chain-and-satin ground are very different starting points for one top bar.

Nothing about that competes with the theorem. Whichever ground is chosen, the middle bar’s freedom inside it is complete. What changes is only how large a room it is free in, and how much the walls have cost.

The cost that is left over #

Direction is not altogether free, and it is worth being plain about what is unpriced. A run of underlaps all one way pulls the wales sideways, by a component the first give’s circles take no account of, since they treat a course as a thing with a spacing and not a thing with a lean. A lapping’s walk ties its wales together with those same underlaps, and an atlas is known to lean in zigzag bands, for that reason, and a thread’s two legs sit unequally in a stitch that has been lapped in the same direction twice. That is why the census gives a separate row to plans that avoid triple runs, as a proxy: it is a limit on how far a lean can accumulate before it turns back.

The same plan can be mirrored on another bar. Two bars lapping opposite directions on every course leave the fabric no net sideways pull, and that is what a two-bar atlas does in practice. The first-give rule says it can be done with the lean cancelled and the give still untouched, provided both bars’ shogs lie inside the ground’s range. So at the cost of one more beam and one more stroke of nothing at all, direction patterns become free of lean as well, at least to the order computed here. Whether that fully removes the lean nobody can say from this; a warp knit’s loops and underlap crossings carry their own moments, which a first-give model does not hold.

How it was computed #

A course’s spacing at a width is the minimum of each bar’s circle, at 28 gauge and 14 courses a centimetre, with the loops held at their knitted length, as in the earlier accounts of this cloth. A plan’s stroke is the range of the running sum of each course’s underlap less the repeat’s mean. A course’s swing is the ratio of its largest and smallest total underlap, ground included and loops left out.

The census walks every sequence of eight signed shogs from −3 to +3, abandoning one whose sum can no longer return, so a plan is counted once for each place it can start, out of an upper limit of $7^8$ sequences. Its two middle counts are closed forms that the walk reproduces: one chain and 3 × 70 flat plans, $\binom{8}{4} = 70$ being the ways to choose the left-going courses.

Required of it, not shown: that a tricot or cord added to a chain and satin course move no spacing at any width in steps of a hundredth from six tenths to 1.05; that a shog of four added to it do so; and that a constant plan have no swing. Each of those fails if the circle’s law is changed.

What the circles do not model #

The first give is only the first. Past it, the loops have to open or close, and the earlier work on these fabrics has held them fixed throughout. A middle bar’s yarn in a loop is extra yarn in that loop, and the loop’s own give is therefore not exempt. So this is a theorem about how a fabric starts to give. It is silent about the second half of the same load–extension curve, where a middle bar very probably does count.

Friction is absent. Underlaps stacked against each other press and slide, and a third layer in the stack changes the force at every crossing on the two beneath it. The circles describe where a course can go, not how much force it takes to send it there.

The loops’ bracket is carried from the earlier account, and the swing figures quote underlap only, as an upper bound on the swing in a course’s whole yarn. A weight swing is not yet a visible stripe: opacity and sight decide that.

And there is more than one kind of top bar. Lace warp knits use a partial threading, and bars that miss some needles entirely behave differently from a full-threaded middle bar. Everything above is about a fully threaded one.

Who worked out which part #

Multi-bar warp knits, atlas lappings and the use of a top bar for pattern are ordinary practice, known to any warp-knitting mill and written up in the trade’s texts. The underlap circle and its rule, and the stroke and weight accounts, are the earlier essays’ own.

What is added here is the exact statement that a shog between its course’s shortest and longest cannot bind, by the $s^2$ form of the circle; the alphabet that a ground therefore sets; the sign of the bowing argument; the separation of give from weight and stroke, as three questions with three answers; and the exact count of plans free on each, with direction and threading picked out as the only free ways to pattern.

Still open: what a middle bar does once the loops begin to give #

The whole of this holds while the loops are held still, and first give is a small part of a warp knit’s working range. A middle bar’s yarn does not sit only in underlaps: it passes through the same needle loops as the ground, and so it changes the amount of yarn there is in each loop to be pulled out, and the contact between legs.

So the second stage of a warp knit’s extension should show a middle bar where the first stage cannot. A chain-and-satin fabric knitted with and without a tricot top bar, in both directions, ought on this account to have an identical start to its load–extension curve and part company at a stated strain. The start is a prediction with no adjustable part. Where they part, and by how much, is the first number that would price what a loop does for a warp knit, which so far nothing here has.