Warp knitting, which is a different thing entirely

The knitting most people have seen is weft knitting: one thread travelling across the fabric, bent into a course of loops, each drawn through the loop below. It is what hand knitting is, what a jumper is, and what the loop describes.

Warp knitting is not a variation on it. Every wale has its own thread, running down the length of the fabric rather than across it, and a whole course is formed at once by hundreds of needles acting together. The machine is different, the fabric is different, and the failure modes are different.

The structural question this raises has exactly the shape of the one the site began with, and the answer has exactly the shape of the satin theorem. Both of those are worth working out, because neither is obvious from the machine.

The question a warp knit raises #

Consider the simplest thing a warp-knitting thread can do: form a loop, then form another loop on the same needle, and keep going. That is a chain, also called a pillar stitch, and it is a perfectly good structure — a chain of loops, each through the one below.

It is also not a fabric. Every wale is a chain, no chain touches any other, and what comes off the machine is a few hundred separate cords lying side by side.

So the thread has to move sideways. Between one course and the next the guide carrying the thread shogs — swings across some number of needle spaces — and the length of thread lying between the two loops is the underlap. Whether the fabric holds together is a question about the pattern of shogging, and it is the same question a weave answers.

The criterion again, on a different graph #

Set it up in the site’s usual terms. Build a graph whose nodes are the wales and whose edges are the ties made by threads passing between them. A thread that occupies wale $j$ on one course and wale $k$ on the next has tied $j$ to $k$; every wale carries a thread, so the ties repeat all the way across.

The fabric holds together exactly when that graph is connected, and the number of independent fabrics is the number of components. It is the same criterion that decides whether a draft describes one cloth, applied to a different object — and the chain lapping is a fabric that would have been rejected without anyone needing to look at it.

The tricot lapping ties each wale to its neighbour, so the graph is a path across the fabric and connected. The chain lapping ties nothing, so the graph is a set of isolated nodes.

The coprimality condition #

Now the part that would not have been guessed, and which the enumeration produces rather than confirms.

A lapping with a longer underlap — moving two needle spaces rather than one — ties each wale to the wale two along. That looks like a stronger tie: more thread lying across the fabric, and a firmer cloth. Run the criterion and it is not.

Moving two spaces ties wale zero to wale two, two to four, four to six — and never to any odd-numbered wale. On a fabric with an even number of wales that is two completely independent fabrics, occupying the same space, interlocked nowhere.

The general statement is a small piece of arithmetic. If the lapping’s shogs generate the subgroup of the integers modulo the wale count that they generate, then the wales split into the cosets of that subgroup, and the number of them is

$$\gcd(\text{shog}, \text{wales})$$

so a single-bar lapping joins every wale exactly when its shog is coprime with the width. On an odd width a two-space underlap works; on an even one it does not. On any width a one-space underlap works, because one is coprime with everything.

That is the same argument, in the same words, as there is no satin on six ends — where a satin’s move must be coprime with its order or the interlacings fail to visit every end. Two structures with nothing mechanically in common, made on different machines in different centuries, running into the same number-theoretic wall.

The figures state the count from the connectivity search and check it against the greatest common divisor, so the two arrive independently and have to agree.

What machines actually do about it #

The condition is not a theoretical curiosity; it is the reason warp-knitting machines have more than one guide bar.

A machine with a single guide bar can only make lappings this argument permits. A machine with two bars runs two threads to every wale, each with its own shogging pattern, and the fabric is held together if the union of the two ties is connected. That is a much weaker requirement, and it is what makes the long-underlap constructions possible.

The commonest fabric on earth made this way is tricot, which is a two-bar fabric: one bar making a short lap in one direction and the other a short lap in the other, so the two sets of underlaps cross and the fabric is stable. The lingerie, the shirt lining, the mosquito net and most of the synthetic fabric in a car interior are two-bar tricot.

Atlas is the other classic, and it shows how to get a long traverse without breaking the condition: instead of shogging two spaces at once, shog one space at a time in the same direction for several courses, then come back. The thread ends up far from where it started and every individual tie is a one-space tie, so the greatest common divisor is one and the fabric holds.

That is a real design principle falling out of the arithmetic. Traverse far, but one space at a time.

Reading the notation #

The trade writes a lapping as a pair of numbers per course, and the notation is worth decoding because it says more than the diagrams here do.

Needle spaces are numbered across the machine. A movement is written as the space the guide is at before the needle passes through and the space it is at after — so 1-0 means the guide swings from space one to space zero across the needle, forming a loop. The movement between courses, the underlap, is the shog from the end of one pair to the start of the next.

A chain is 1-0/0-1: over the needle one way, back the other, never leaving the space. A tricot is 1-0/1-2: over the needle at one place, then a shog of one space, then over the needle at the next. A cord is 1-0/2-3, a shog of two. Atlas is a run of one-space shogs in the same direction and then back.

The notation also encodes something the graph does not: whether the two halves of the movement go the same way round the needle or opposite ways. Same way gives a closed lap, whose loop is crossed at the base; opposite ways gives an open lap, whose loop is not. Closed laps make a firmer, less extensible fabric; open laps a softer one. The connectivity criterion is blind to the difference, which is a good example of it being necessary and a long way from sufficient.

Two machines, and what each is for #

Warp knitting divides into two machine families and the division is about needle spacing rather than about structure.

Tricot machines are finely gauged, run at very high speed, and use compound or bearded needles. They make the light, smooth, closely knitted fabrics the word tricot suggests: lining, lingerie, swimwear, automotive headliner. Two guide bars is the norm and the lappings are short.

Raschel machines are coarser, slower, and use latch needles, which lets them handle heavy or irregular yarn and hold long underlaps. They make lace, netting, sacking, geotextile, and the elastic fabrics with laid-in yarns. A raschel can carry dozens of guide bars.

Structurally the difference matters for one reason: with many bars, the connectivity condition of this essay stops binding. A raschel lace is held together by whichever combination of bars happens to tie the wales, and the designer’s problem is no longer whether the fabric coheres but where it deliberately does not — the holes in a lace are places where the graph is intentionally sparse, and the skill is in leaving them exactly where they are wanted.

That is a pleasant inversion of the site’s usual position. Everywhere else, a fabric falling into pieces is the failure the check exists to catch. In a lace it is the product, and the check becomes a way of specifying the design rather than of validating it.

Why a warp knit does not ladder #

The most useful practical difference between the two knittings is what happens when a thread breaks, and it follows directly from the structure rather than from the yarn.

In a weft knit, one thread makes a whole course. Break it and the loop above has nothing holding it, so it frees the loop above that, and the failure climbs the wale — a ladder, or in the trade a run. That is the topological argument the knits ladder is built on.

In a warp knit, the thread that made a loop in wale $j$ made the next loop in wale $k$. Break it and the loops it made are freed — but they are scattered across several wales rather than stacked in one, and each of them is also held by the underlaps of the neighbouring threads passing over it. The failure spreads sideways and dies out, rather than climbing.

Which is why tricot does not ladder, and why it is used for anything that must not fail conspicuously: seat covers, medical textiles, and the fabric of a parachute’s sliders. It is not a stronger fabric. It has a better failure mode, for a reason that is entirely structural.

The fabric this makes #

Three properties of a warp knit follow from the structure and are worth stating because they are not shared with either weaving or weft knitting.

It is stable. The underlaps run diagonally between wales, and in a two-bar fabric they run diagonally in both directions, which triangulates the structure. A tricot fabric therefore has far less extension than a weft knit and far less bias give than a woven — the trellis mechanism needs four-sided cells, and the crossing underlaps do not leave many.

It is fast. Every wale is knitted at once rather than in sequence, so a warp-knitting machine produces fabric at a rate a weft-knitting machine cannot approach. That is the commercial reason it exists.

It needs a warp. Every wale must be supplied with its own thread, so the machine has to be dressed with a beam exactly as a loom does — which is why it is called warp knitting, and why it is not a hand technique. There is no hand equivalent of warp knitting, and that absence is why the structure is unfamiliar despite being everywhere.

Where it came from #

Warp knitting is a machine invention with no handcraft ancestor, which is unusual in this subject and explains why it is so little known outside the trade.

The first warp-knitting machine was built by Josiah Crane in 1775, and the tricot machine as recognised today dates from the early nineteenth century. Every other structure on this site — weaving, weft knitting, braiding, felting — was made by hand for thousands of years before anyone built a machine for it. Warp knitting was designed.

The consequence is visible in the vocabulary. There are no folk names for warp-knit structures because no folk made them; the terms are machine terms, and the fabrics are named after the machines rather than the other way round. Tricot is the exception and it is a borrowing: it is simply French for knitting, applied to a machine-made fabric that hand knitters would not recognise.

The absence of a handcraft ancestor also explains why the coprimality condition of this essay was never folklore. A weaver who tried to weave a six-end satin found out that it did not work; a hand knitter who dropped a stitch watched it run. Nobody ever hand-made a chain-lapped fabric and discovered it fell into cords, because the only way to make one is to build the machine first — and a machine builder who made that mistake would have fixed it in the design and never written it down.

What a second bar buys, exactly #

The essay says a second guide bar makes the condition “much weaker” and leaves it there. The weakening has a closed form, and it is the same arithmetic one step up.

Two bars shogging s₁ and s₂ tie each wale to the wale s₁ along and to the wale s₂ along, so the ties generate the subgroup of the integers mod w generated by both. That subgroup’s index is the greatest common divisor of all three numbers, so

a two-bar fabric is one fabric exactly when gcd(s₁, s₂, w) = 1.

Which is a genuinely different condition, not merely a laxer one, and the difference is best seen in a case where it changes the answer. On six wales, a two-space shog gives two fabrics and a three-space shog gives three. Put both bars on and gcd(2, 3, 6) = 1: one fabric, from two lappings each of which fails alone.

That is the whole commercial argument for a second bar stated in one line, and it generalises without effort. Three bars need gcd(s₁, s₂, s₃, w) = 1, and adding bars can only ever help, since adding a generator cannot enlarge a greatest common divisor.

How much of the design space a single bar loses #

The condition also admits a count, and the count is worse than the essay’s even-width example suggests, because a machine’s wale count is never a prime.

A single-bar shog works exactly when it is coprime with w, so the fraction of shogs that give one fabric is Euler’s totient of w over w:

On a 720-needle machine, barely a quarter of the possible single-bar lappings produce a fabric at all, and which ones they are depends on a number — the needle count — that a designer has no reason to be thinking about. A lapping that worked on one machine can fail on the next one along, with nothing about the fabric changed.

In practice the shogs used are one, two and three, so the exposure is narrow and specific: a shog of one is always safe, a shog of two fails on every even machine, and a shog of three fails on every machine whose needle count is a multiple of three. Machine widths are chosen as round numbers, and round numbers are divisible by two and often by three, so both of the failing cases are the normal case rather than the exception.

Why atlas beats cord on both counts #

The design principle the essay extracts — traverse far, one space at a time — looks like a compromise, and it is not one. Atlas beats the long-shog alternative on yarn as well as on connectivity, which is unusual enough to be worth the arithmetic.

An underlap’s length is roughly its shog times the wale spacing. A cord lapping traverses two spaces in one course and lays two units of underlap to do it. An atlas traverses the same two spaces over two courses and lays one unit in each — two units again, but spread over twice as many courses.

So per course of fabric, atlas lays half the underlap that cord does, reaches the same distance, and connects every wale where cord connects half of them.

The honest counterweight is that the underlap is not only a cost. It lies on the back of the fabric and covers it, so a cord lapping’s heavier underlap per course is buying back-cover, weight and a firmer hand — which is what such lappings are actually specified for. Cord buys cover per course; atlas buys traverse without it, and the two are not competing for the same job.

What is not a trade-off is the connection. Nothing about a cord lapping’s extra yarn ties the odd wales to the even ones, and no amount of it ever will.

What the figures cannot show #

The lapping diagrams here draw threads as smooth curves between needle positions, which records the topology — which wale a thread is in at which course — and nothing else.

A real warp knit is nothing like that picture. The loops are pulled tight, the underlaps lie flat against the back of the fabric, and the whole structure is compressed to a fraction of the height the diagram uses. What the diagram is a correct picture of is the graph; what it is a poor picture of is the fabric.

And the connectivity graph says nothing about the amount of thread. A one-space underlap and a four-space underlap can both give a connected fabric, and they give fabrics of quite different weight, stiffness and cost, because the underlap length is where a good deal of the yarn goes. The criterion tells a designer which lappings are possible; it says nothing at all about which is wanted.

The figures also draw six wales, which is enough to see the pattern and small enough to make the even-width splitting visible. A machine has hundreds, and on hundreds the two-space lapping still splits into two — the arithmetic does not care about scale.

Where the ladder goes next #

This is the base of its own ladder: warp knitting is a structure with its own questions rather than a rung on the weft-knitting one. The rung beside it is the loop, which is the weft-knit base, and the two ladders meet at ravel, fray and run, where the failure modes of the three families are compared directly.

The nearest companions elsewhere are rib and interlock, which is what a weft knit does to gain the stability a warp knit has by construction, and the satin theorem, which is the same coprimality argument met in a woven cloth.