What a second guide bar is for

A warp-knitting machine gives every wale its own thread. Each thread makes a loop on its own needle, swings sideways by some number of needle spaces — the shog — and makes the next loop there, and it is that sideways swing that ties one wale to another. A thread that never leaves its wale makes a chain, and a fabric of chains is a set of independent cords with nothing holding them together at all.

The single-bar case has an exact answer: the wales split into $\gcd(\text{shog}, \text{wales})$ independent groups, so a lapping makes one fabric exactly when its shog is coprime with the width. It is the satin theorem in a knit, arrived at from a completely different direction.

Real machines have two guide bars, or four, or more, and every bar contributes its own ties. What happens then is the question this essay answers, and the answer has a consequence worth having.

The single bar, restated #

A bar with shog $m$ ties wale $w$ to wale $w + m$, for every $w$, taken round the width because a fabric is wider than one repeat.

So the ties available are the multiples of $m$ modulo the number of wales $N$, and two wales are in the same fabric exactly when they differ by such a multiple. The multiples of $m$ modulo $N$ are the multiples of $\gcd(m, N)$, so the wales split into $\gcd(m, N)$ classes and each class is a separate fabric.

A shog of one always works, because one is coprime with everything. A shog of two on an even number of wales makes two fabrics interleaved: all the odd wales in one, all the even in the other, and the two lift apart.

Two bars, and the generalisation #

With two bars each thread is knitted at each course by whichever bar is laying it, and the ties made by the two bars are superposed. So the graph on the wales gains edges from both, and the question is what the union of the two tie-sets connects.

The answer follows from the same argument. The differences available are the multiples of $m_1$ and the multiples of $m_2$, and any combination of them, so the reachable differences are the multiples of $\gcd(m_1, m_2)$ — and taken modulo $N$, the classes are the multiples of

$$\gcd(m_1, m_2, N).$$

The divisor is taken over both shogs and the width together, and it extends to any number of bars in the obvious way.

That single line has the consequence the figure shows. A shog of two on twelve wales gives two fabrics; a shog of three gives three; and $\gcd(2, 3, 12) = 1$, so a machine running both makes one. Neither bar can do it alone and together they can, which is the structural argument for a second bar rather than a decorative one.

The reverse also holds and is easy to overlook. Shogs of four and six on twelve wales share the factor two with each other and with the width, so $\gcd(4, 6, 12) = 2$ and the pair is no better than either alone. Adding a bar does not automatically help; it helps exactly when it brings a new factor.

What was counted, and how #

Nothing here is taken on the algebra’s word.

The wale graph is built explicitly — one node per wale, an edge from every wale to the wale a shog along, for each bar — and its connected components are found by the same strongly-connected-components routine that decides whether a woven draft is one cloth. Weaves, braids and warp-knit wales are all decided by one implementation on this site, which is why a bug in it would be caught by three different families of figure rather than by none.

Then the count from the search is asserted against the greatest common divisor. Search and closed form, in that order, because a closed form nobody tested is a guess with confidence attached and a search nobody checked is an implementation reporting on itself.

Two further assertions run. Some lapping in the figure must fail to make one fabric, so a table of all-green rows cannot pass as a demonstration of a condition; and some pair must rescue a shog that fails alone, since that is the claim the figure exists to make. If a change to the parameters ever removed both, the figure would refuse to draw rather than quietly become a list.

The width is part of the condition, and it moves #

A weaver’s repeat is a property of the cloth. A knitter’s wale count is a property of the machine, and it changes.

That asymmetry has a consequence with no equivalent in weaving. A lapping that makes one fabric on a machine of a hundred and twenty-seven needles may make several on one of a hundred and twenty-eight, because the arithmetic depends on the factorisation of the width. A shog of five is safe on any width not divisible by five and fatal on the widths that are.

In practice machine widths are large and the shogs are small — one, two, occasionally three — so the condition is almost always satisfied and almost never thought about. The exception is at the edge of the fabric and in narrow-width work, where the effective width is small enough for a shared factor to be likely, and where a fabric that separates into two is a real and reported failure mode.

The condition also constrains what a machine can do when a bar is threaded partially. Leaving guides empty — threading one in two, one in four — is how patterning is done, and a partial threading changes which wales a bar can reach. The connectivity question then has to be asked about the actual set of ties rather than about the nominal shog, and the answer can be a good deal worse than the shog suggests.

Why real two-bar fabrics do not need this #

The honest limit of the argument is that it explains a possibility the industry does not exploit.

Tricot’s shog is one. Locknit, which is the commonest two-bar fabric there is, runs a back bar with a shog of one and a front bar with a shog of two. Since the back bar’s shog is coprime with every width, the fabric is already one fabric before the second bar is considered, and the connectivity condition is satisfied without it.

So the second bar in a real fabric is not there for connectivity. It is there for what the first bar cannot supply: cover, because a single-bar tricot is thin and open; stability, because two bars lapping in opposite directions cancel the sideways skew a single bar imposes; and surface, because the front bar’s longer underlaps lie on the technical back and give the fabric a face and a reverse.

That is a case of a mathematical condition being real, checkable and not binding, and it is worth saying so plainly rather than implying that machinery was designed around it. The value of the condition is as a check — it says which lappings are impossible, which is a smaller and more reliable service than saying which are chosen.

It becomes binding in one place, and that place is worth naming because it is where new warp-knit structures are actually invented. A patterning bar threaded one-in-two or one-in-four is a bar whose ties reach only some of the wales, and combining several partially threaded bars is how spacer fabrics, nets and open-work structures are built. There the connectivity is genuinely in doubt, the divisor argument no longer applies unchanged — the ties are not all the multiples of a single shog — and the graph has to be walked. A net is a fabric that is deliberately full of holes and is nonetheless one piece, and the difference between full of holes and in several pieces is precisely a component count.

Reading a lapping notation #

The condition above is stated in terms of a shog, and a machine is set up in terms of a chain notation, so the translation is worth one section.

A guide bar’s movement over a repeat is written as a pair of needle positions per course: 1-0/1-2 for tricot, 1-0/2-3 for a bar with a longer underlap. The first pair is the swing over the needle that forms the loop, and the second is the underlap that carries the thread across to where the next loop will be formed. The shog is the underlap, and it is the second number of each pair that matters for connectivity: the overlap forms the loop and ties nothing new.

That is why the notation reads confusingly to somebody looking for the shog. A tricot notation contains a 2 and a tricot’s shog is one, because the 2 is a position rather than a distance. Taking differences rather than reading numbers is the whole of the translation, and the wale graph above is built from differences for exactly that reason.

Where a bar walks several needles before returning — an atlas lapping — there is no single shog, and the ties are all the differences along the walk. The divisor is then taken over the whole set of them, which is the same statement as before with more terms in it. An atlas whose walk includes a step of one holds together on any width, however far it travels.

Why a partial threading breaks the divisor argument #

The essay’s closing note says the divisor argument no longer applies unchanged to a partially threaded bar, and the reason is worth setting out, because it is the difference between an arithmetic answer and a search.

A fully threaded bar with shog m ties every wale to the wale m along, so the set of reachable differences is closed under addition — which is exactly the property that makes the answer a subgroup and therefore a divisor. Adding two ties gives a third; the differences form a group, and a group of integers modulo N is the multiples of a divisor.

A bar threaded one guide in two ties only the wales its guides occupy. Wale zero is tied to wale m, and wale one is tied to nothing at all by that bar. The reachable differences are no longer closed under addition, because a path is only available where a thread exists, and the structure on the wales is a graph rather than a group.

That is why the machinery walks the graph. A graph’s components are not a divisor of anything in general, they need not all be the same size, and two partially threaded bars can leave a fabric with components of unequal width — which a divisor answer cannot even express, since divisor classes are always equal in size.

And unequal components are the practically dangerous case. A fabric in two equal halves is an obvious failure that shows itself on the first metre. A fabric in one large component and one narrow strip along part of its width is a fabric with a loose selvedge or a running band in it, which reads as a machine fault rather than as a structural impossibility — and which no amount of adjusting will cure, because the strip was never tied to anything.

So the divisor is the special case and the graph is the general one, and the special case covers exactly the lappings the industry runs by default.

The one thing the graph route keeps from the divisor route is the check. A component count from a search is still asserted against a closed form wherever a closed form exists, so the partially threaded cases are exercised by machinery that has been verified on the cases where the answer is known independently. The structures where the condition becomes binding are the ones built out of partial threadings, and those are precisely the ones where the shortcut is unavailable.

What the condition cannot see #

Three things, and the first is the same limitation the woven integrity check has.

It is topological, not mechanical. A fabric whose wales are all in one component holds together in the sense that no subset lifts off. It may still be feeble, sleazy, or so loosely tied that it distorts under its own weight, and none of that is visible to a graph. The check answers whether the structure exists, not whether it is any good — exactly as it does for a woven draft.

It says nothing about the loops. Everything above treats a wale as a node and a lapping as an edge. Whether the loops themselves are sound — closed or open, well formed, of a length the yarn will take — is a different question with a different machinery behind it, and this site’s loop model is the place it is asked.

And it does not describe what happens at the selvedge. The whole argument takes the width cyclically, because that is what makes it arithmetic. A real fabric has two edges, and a shog that runs off the edge is a guide with no needle under it. The behaviour there is a machine question rather than a structural one, and it is where several of the failures that this condition would predict actually appear.

Who worked it out #

The single-bar condition appears in warp-knitting texts as a practical rule about threading and shogging, usually without the arithmetic, and the multi-bar generalisation is the sort of thing that is obvious to anybody who states the single-bar case algebraically and has therefore probably been noticed many times without being written down.

The underlying fact is elementary group theory: the subgroup of the integers modulo $N$ generated by a set of elements is generated by their greatest common divisor with $N$, and the cosets of that subgroup are the components. The same statement decides which satins exist, which twill steps visit every end, and which lappings make cloth, and the recurrence is a fact about the arithmetic rather than about textiles — three periodic structures visited by a fixed step, which is a very common shape.

What this site adds is the enumeration and the check. The condition is computed by walking a graph and then verified against the divisor, and both the failure case and the rescue case are asserted to occur, so the figure cannot become a table of agreement with nothing being tested.

It is also worth recording where the single-bar version of this came from on this site, because it was not read anywhere. An earlier essay here arrived at the coprimality condition for a warp-knit lapping by searching the wale graph, noticed that the answers were exactly the divisors, and only then wrote the closed form and asserted the search against it. That is the reverse of the usual order and it is the better one: a search that finds a pattern is evidence, and a formula that predicts the search is a claim the search can refute. This essay’s generalisation was arrived at the same way, and the assertion that the two agree is the same assertion, with more shogs in it.

Where the ladder goes next #

Below this rung is warp knitting itself, and the sideways argument is the satin theorem, which is the same coprimality condition in a woven cloth.

The knit ladder proper runs through the loop, curling, running and recovery, all of which are about weft knitting and none of which transfers to a warp knit unaltered.

What the pictures here cannot show. Every lapping diagram on this page draws a few wales and a few courses, and the condition is about a width of several hundred. A drawing at that scale would show the ties and not the components; the components are found by a search over the whole width and reported as a count, and no picture on this page is evidence for the count rather than an illustration of what is being counted.