Does a double jersey hang together

This site exists to ask one question of a drawing that the drawing cannot answer: does it describe one cloth. A weave falls into two when its above-and-below relation is not strongly connected, and the failure is invisible — a two-layer draft looks exactly like a one-layer draft, and the only way to see the difference is to compute it.

Weft knitting has looked immune. A knit is one continuous thread travelling across the needles, so its fabric is connected by construction and the criterion has nothing to work on; the knitted failure is a needle that never casts off, which is a different theorem in different clothes.

That is true of one bed. It stops being true the moment there are two.

Two fabrics made at once #

Set a rib machine to knit the front bed on odd courses and the back bed on even ones. Every needle in work knits; every course knits somewhere; no needle holds anything. The array passes every condition the previous rung states, and a machine will make it without complaint.

What comes off is a tube. The front bed’s loops make one fabric and the back bed’s make another, and there is nothing between them anywhere across the width. On a flat machine the two are joined at the two edges, where the yarn turns; on a circular one they are joined by the yarn spiralling from one to the other once per revolution. In the body of the cloth they are two.

That is precisely the draft that falls into two cloths — a fabric that looks like one object, passes every local check, and comes apart when it is asked to be one thing. Hosiery has been made this way on purpose since the rib frame existed, which is the tell: an effect nobody would design by accident and everybody would sell as a single fabric if they did.

So the question transfers, and the useful part is that the answer transfers too. It is a connectivity question, it is decidable from the array, and it is not a matter of looking at a picture.

The graph is over wales, and its edges come from the yarn #

Building it takes one observation and no machinery.

A course is one traverse of one yarn. The carrier passes across the width once and every needle selected to take yarn — knitting or tucking — takes it from that single continuous length. So everything one course touches is threaded onto one thread, and any two wales a course feeds are joined.

That gives a node per wale and, per course, a chain through the wales that course feeds. Whether the fabric is one fabric is whether the graph is connected.

The relation is symmetric, and that is the one place the knitted question differs in form from the woven one. A weave’s relation is above and below, which is asymmetric — the site’s criterion is strong connectivity of a directed graph, and the direction is what encodes which layer could be lifted off. A knit’s relation is the same length of yarn, which points both ways. So the answer here is components rather than strong components, computed by the same implementation on a symmetric adjacency, because the site has one connectivity routine and three would drift apart.

The criterion anybody would state, and the census that refutes it #

Ask a knitter when two beds make one fabric and the answer is immediate: some course has to knit on both beds. It is the right shape of answer, it is what the machine setting looks like, and it is what this essay expected to confirm.

It is wrong, and the enumeration says so on sixty-two arrays.

One fabric exactly when the wales cannot be split into two groups with no course feeding both. That is the criterion, and the difference from the plausible one is that the split need not follow the beds. A course that feeds both beds joins whatever it touches; it does nothing for a wale it does not touch.

The smallest counterexample is four cells long. Let the front bed knit needle one on course one and needle two on course two, and let the back bed knit both needles on course two. Course two feeds both beds — the plausible criterion is satisfied — and course one feeds one wale and one wale only, so that wale is joined to nothing at all. The graph has two components and one of them is a single wale, which is a cord lying in a fabric rather than a layer of one.

The failure runs the other way too, and more often. A structure with one bed entirely out of work has no course feeding both beds, so the plausible criterion calls it two fabrics; it is single jersey and it is obviously one.

What was counted, and how #

Three symbols on two beds over two needles and two courses is 3⁸ = 6,561 arrays. Every one is built, tested and counted.

1,135 are fabrics — 17.30 per cent. The rest fail on the machine conditions: 5,265 have a wale that takes yarn and never knits, and 161 have a course that knits nowhere.

Of the 1,135:

Three or more is not a gap in the counting; it is arithmetic. Each course chains everything it feeds into one component, so two courses can leave at most two components, and a two-course repeat cannot produce three. Running the same enumeration over three courses does produce them — 60 arrays of 117,951 fall into three — which is the check that the zero above is a fact about the repeat rather than a bug.

The interesting half is how the fifty split. Twenty-two of them come apart along the beds, front from back, which is the tube and the cases like it. Twenty-eight come apart across the beds — each component draws wales from both — so neither half is a layer and neither could be described as the front or the back of anything. Those are the ones no amount of looking at a machine would find.

Everything is computed twice. The graph gives a component count by walking edges; a second routine walks every partition of the wales and asks whether any course straddles it. The two must agree on every one of the 1,135, and the agreement is asserted rather than spot-checked — which is the discipline the cloth census uses for the same reason, because a canonical form that misses a case and a count that assumes one both produce numbers in the right range.

Interlock comes out as two, and that is not a mistake #

The census throws up one result that took some believing, and it is worth the space because the honest answer is more interesting than the tidy one.

Interlock is two components. Its two courses feed {front 1, back 2} and {back 1, front 2}, so the graph has two parts and each part is a one-by-one rib. The criterion says: two fabrics, interpenetrating, neither of them a layer.

The trade says the same thing in words and does not notice it is saying it. Every knitting textbook describes interlock as two one-by-one rib fabrics knitted one inside the other, and then says they are locked together. The lock is the point. Rib A’s yarn crosses the gap from a front wale at one position to a back wale at the next; rib B’s crosses from a back wale at the first to a front wale at the second. The two runs cross each other in the thickness of the cloth — and crossing is not linking. Nothing passes through anything.

So an interlock is held together by its sinker loops crossing, which is a geometric fact about where the yarn is, and not by any thread passing through any loop, which is the only fact the graph carries. The criterion is exactly right and it is answering a question about topology while the fabric is being held by geometry.

That is not a new embarrassment. The site has already found a cloth the criterion cannot see, where friction holds a leno’s crossed ends and the connectivity says nothing about it. This is the knitted version and it is the same lesson: the criterion is exact about what it measures and silent about everything else, and knowing which is which is the whole of using it.

What the criterion cannot see, and it is the usual complaint #

Three limits, and the first is the one that decides whether any of this is useful.

It is topological, so it has no strength in it. A single tuck every twenty courses joins the two layers, by this criterion, exactly as firmly as a rib joins them every course. Both are one fabric and the answer is the same word. One of them delaminates in the wash and the other does not, and nothing here distinguishes them — the criterion counts whether a path exists, not how many there are or what each carries.

That is a real limitation and it is also the criterion’s whole value, in the same way the woven one’s is. A binary answer that is exact is worth having precisely because it is not trying to be a strength model: it tells a designer that a structure cannot hold, which no amount of sampling will, and it says nothing about whether a structure that can hold, holds well.

It is blind to the crossing. Interlock is the standing example and it is not the only one. Any construction where two systems are jammed rather than linked comes out separable and behaves as one thing, and the site now has three of them — the leno’s friction, the milled cloth’s felted fibre, and this.

And it says nothing about the selvedges. Every count here is taken on the repeat, which is toroidal across the width as well as down the courses, so a tube’s two layers are two. A real tube is a closed surface joined by the yarn turning at the edge of a flat piece or spiralling on a circular machine, and the join is real: a stocking does not fall into two sheets. The criterion measures the body of the fabric and the edges are outside it — which is exactly the same convention every weave count on this site uses, and worth restating because a tube is where the convention shows.

The array is a hypergraph, and that bounds the damage #

The construction above builds a graph by chaining each course through the wales it feeds, and the chaining is doing something worth naming: a course is not an edge between two wales, it is a single object touching many of them at once. That is a hypergraph — wales as vertices, courses as hyperedges — and the question is hypergraph connectivity.

Saying so is not a change of vocabulary for its own sake. Two useful facts fall straight out of it, and neither is visible when the courses have been flattened into chains of ordinary edges.

The number of fabrics can never exceed the number of courses. Every course merges everything it feeds into one piece, and every wale is fed by at least one course, so m courses can leave at most m pieces. The essay uses this once, to explain why a two-course repeat cannot give three fabrics; as a general bound it says something a designer can act on. A one-course repeat can never separate. A four-course repeat can fall into at most four, and no structure of any size on a two-course repeat can be worse than a tube.

And any repeat containing one course that feeds every wale is one fabric, whatever every other course does. That course is a hyperedge covering the whole vertex set, so it connects everything by itself. Half-Milano is exactly that case — a rib course plus a front-bed course — and it is why the census puts it in the one-fabric column without the second course contributing anything. A rib course is a guarantee, and it is the only single-course structure that is one.

How many joining courses a wide structure needs #

The bound runs the other way too, and it is the more useful direction because it prices the connection rather than merely permitting it.

Connecting n wales needs the hyperedges to span them, and a hyperedge touching s wales can reduce the number of pieces by at most s − 1. So a structure of n wales needs at least

(n − 1) ÷ (s − 1)

courses that feed more than one wale, where s is the largest number of wales any single course reaches.

For a rib, s is n and one course does it. For a two-bed jacquard, where each course carries one colour and feeds only the wales that colour appears in, s can be small — and then the requirement bites. A design in which each course feeds four wales across a repeat of thirty-three needs at least eleven multi-wale courses in its repeat before it can be one fabric at all, however the colours are arranged.

That is a floor no arrangement can get under, and it is the first thing in this ladder that puts a number on the trade’s rule about tie-in courses. The rule is usually stated as a frequency — one binding course in so many — and the arithmetic says the frequency that matters is not per unit of length but per wale: the wider the repeat and the fewer wales a course touches, the more joining courses are needed, in exact proportion.

It also explains a familiar jacquard failure. A two-bed jacquard whose design has a large area of one colour has, across that area, courses that feed almost every wale — plenty of connection. Where the design is finely broken up, each course feeds few wales, s collapses, and the requirement rises sharply in the very region where the designer was thinking about the picture rather than the structure. Delamination in a jacquard double jersey concentrates in the detailed parts of the design, and the bound above says why without any mechanics in it.

The caution is the standing one. This is a floor on the number of courses, not a guarantee that a structure meeting it holds: the criterion still counts whether a path exists rather than what it carries, and eleven courses arranged badly connect nothing.

The generalisation, which is the woven one turned over #

Set the two criteria side by side and the difference is instructive rather than cosmetic.

A weave separates when a set of threads can be lifted off the rest — the failure is that no crossing forces the two sets together, and it is measured on a relation between threads. A knit separates when a set of wales is fed only by courses that feed nothing else — the failure is that no traverse of yarn visits both sides, and it is measured on a relation between wales and courses.

The shapes are different and the consequence is the same, and there is one sentence that covers both: a fabric is one fabric when its parts are connected by the thing that makes it a fabric. In a weave that thing is the interlacing; in a weft knit it is the traverse of the yarn; in a braid it is the oblique crossing of the strand systems; in a nonwoven it is the percolation of the fibre network. Four constructions, four relations, one question and one algorithm — and the site now runs the same connectivity routine on all four, which is not a tidiness argument but a correctness one, because four implementations would have four sets of edge cases.

The place it stops being one question is where the connection is not a link at all. Friction, felting and jamming hold cloth together and are not relations between threads in any graph, and every one of them is invisible to all four.

Who found it, and when #

The tube is ancient and the observation is not.

Knitting a tube on two beds is as old as having two beds, which on the hand flat is the eighteenth century and on the circular machine the nineteenth, and the technique is described everywhere as tubular knitting — a name for the shape rather than for the structure. No source this essay draws on describes it as a fabric that fails to be one fabric, because there is no reason to: a tube is what was wanted, and a thing that does what was wanted is not a failure.

What the knitting literature does have is the vocabulary of layer connection, which is the same idea approached from the other end. A double-jersey designer speaks of tie-in courses, of stitches that bind the two faces, of how often a two-bed jacquard must knit across the beds — all of which are statements about how much connection there is, made by people who take for granted that there has to be some. The condition is understood and it is understood as a rule of practice rather than as a property of the array.

The gap between those is where this rung sits, and the gap is the same one the weaving side had. Weavers have always known that a backed cloth needs binding points, and it took a matrix to say exactly which drafts have enough. Here it took an array to say exactly which two-bed structures have enough, and the answer — that the plausible criterion is wrong on 62 arrays in 1,135, in both directions — is not something practice would ever surface, because practice does not knit arrays it has no use for.

Where the ladder goes next #

The array now says which two-bed structures exist and which of them are one fabric, and it says exactly how much yarn each of them takes. What it cannot say is how large the fabric will be, and therefore what it weighs — which is the shortfall an earlier essay here recorded and the next essay measures the size of. The yarn count composes exactly. The dimensional constants do not compose at all, and settling that for the 1,085 structures counted here would take a little under fifty thousand fabrics.