A knit has no hole to lose

Everything in this ladder has been planar. Threads at a spacing, a rectangle between four of them, a channel down the rectangle, a distribution of channels over a repeat. It has produced a rating, a permeability, an openness to the sky and a head of water, and every one of them starts from the hole between four threads.

Ask it about a jersey and it returns nothing at all.

A loop’s occupancy is its length times its diameter over the cell it sits in. Munden’s constants make that cell ℓ²/(k_c·k_w), where k_c and k_w are the course and wale constants, so the loop length cancels once and what is left is

$$\text{occupancy} = \frac{d\,k_c k_w}{\ell} = \frac{d\,k_s}{\ell}$$

where k_s is Munden’s own stitch-density constant, published separately and checked against the product. There is no gauge in that, no count, and no fabric dimension. Whether a knit has a hole between its loops depends on d/ℓ and on nothing else.

For a 20 tex cotton at a 3.5 mm loop, the occupancy is 0.955 dry-relaxed, 1.037 wet-relaxed and 1.129 fully relaxed. Past one there is no hole: the loops overlap in projection, and the planar model has been handed a fabric it cannot describe.

The claim #

A knit at any ordinary tightness has no planar opening, the threshold is a pure number in the yarn’s diameter over its loop length, and the crossing is inside the range the trade knits at rather than far outside it.

The second half is what makes the first interesting. If ordinary knits sat at an occupancy of three, the planar model’s refusal would be a triviality — of course a dense fabric has no holes. They sit between 0.95 and 1.13, which is at the threshold, and the difference between one side of it and the other is a relaxation state rather than a construction.

So a jersey knitted with holes closes them in the wash, and this collection has the arithmetic for that: Munden’s three states are two steps on one path, and the path goes towards more courses and more wales per centimetre, which is a smaller cell for the same loop.

The argument #

The closed form is where the content is, and it is worth deriving in one line because the cancellation is the whole of it.

A loop of length ℓ occupies a cell one wale wide by one course deep. Munden’s constants give courses per unit length as k_c/ℓ and wales as k_w/ℓ, so the two spacings are ℓ/k_c and ℓ/k_w and the cell area is ℓ²/(k_c k_w). The yarn’s projected area inside that cell is its length times its diameter, ℓd. Divide:

$$\frac{\ell d}{\ell^2/(k_c k_w)} = \frac{d k_c k_w}{\ell}$$

One power of ℓ cancels and the other does not. So a knit made with a longer loop of the same yarn is more open — not because the loop is longer but because the cell grows as the square while the yarn in it grows as the first power.

That is the same ratio-of-two-lengths structure this collection found when it computed the tightness factor, and the identity between them is asserted to machine precision: occupancy is the knitter’s own √tex/ℓ index times one constant. The threshold at occupancy one is therefore a threshold in the tightness factor, and it lands at K between 11.3 and 13.4 depending on the state.

The trade knits cotton jerseys at K between 13 and 17. The whole commercial range is above the threshold for the fully-relaxed state and most of it is above for all three.

What follows for what a knit passes #

A jersey is not impermeable, so the air is going somewhere, and the planar refusal says where.

It goes through the threads and around them in the third dimension. The first of those is exactly the floor a woven cloth has — the Kozeny–Carman path through a bed of fibres at the yarn’s own packing — and for a knit it is not a floor but the main term.

The second is not computed here and cannot be by any planar model. A loop is a three-dimensional object: the yarn passes over and under, and the space between two interlocked loops is a curved channel that has no projection onto the plane at all. That is a genuinely different geometry from a woven cloth’s, and this collection has no machinery for it.

So the honest statement about a knit’s permeability is that its structure is closer to a nonwoven’s than to a woven’s — a bed with tortuous paths through it — and that the arithmetic that works for a woven cloth is not merely inaccurate for a knit but is asking a question the fabric does not have an answer to.

The one place the planar model does apply #

There is a class of knitted fabric with occupancy well under one: a mesh, a net, a powernet, a knitted screen — anything at a tightness a knitter would call slack. For those the planar model works, and it produces a result with no free parameter in it.

An open knit extended along its courses opens before it closes, and the turning point is √k_r − 1.

The reason is that a loop is a length of yarn and the yarn does not stretch, so an extension one way is very nearly a contraction the other and the cell’s area is conserved. Occupancy is yarn over cell area, so the fabric’s open fraction does not change at all — exactly, and asserted. But what passes is decided by the hole, and the hole’s own area is (wale − d)(course − d), which is the cell less d times the sum of the two spacings. The sum of two numbers with a fixed product is smallest when they are equal, so the hole is largest when the cell is square.

Munden’s relaxed jersey has a cell about a quarter longer one way than the other — that ratio is k_r, published as 1.30 — so the strain that squares it up is √k_r − 1, which is 13.1 per cent for a fully relaxed jersey and about fourteen for the published constant. It is the same number for every jersey there is, because it is made of two relaxation constants and nothing else: not the yarn, not the loop length, not how hard it is pulled.

And the effect is negligible, which is also a result. The flow rises by 0.3 per cent to the turning point and falls to 95 per cent of its peak by eighty per cent strain. A knit’s cell is not far from square to begin with, so it never reaches the aspect ratios at which the slot penalty does any work. Stretching a knit does not change what it passes, and the reason it does not is the same conservation that makes its open fraction constant.

What the woven arithmetic would have said, and why it is wrong #

It is worth running the planar machinery on a jersey anyway, once, to see what it would have produced if the refusal had not been there.

Take the fully-relaxed 20 tex jersey at a 3.5 mm loop. Its wale spacing is 0.814 mm and its course spacing is 0.636. Subtract a diameter from each and the “hole” is 0.647 by 0.469 mm — larger than any hole in this collection’s woven table except a cheesecloth’s. Feed that into the duct arithmetic and the fabric passes several thousand millimetres per second.

Every step of that is arithmetically correct and the answer is nonsense, because the subtraction is wrong: the yarn crosses its own cell four times and the arithmetic has subtracted it once. A woven cell has exactly one thread along each edge, so spacing-less-diameter is the clear gap. A knit’s cell has a loop wandering through it, and the same subtraction describes a rectangle that is not there.

That is why the occupancy formulation is the right planar question and the gap formulation is not. Occupancy counts the yarn’s whole projected area against the cell’s, so it counts all four crossings; the gap counts one. The two agree for a woven cloth and diverge by a factor of four for a knit, which is exactly the sort of quiet disagreement a model inherits when it is carried across a structure boundary without being re-derived.

What was counted, and how #

The occupancy is computed two ways and the two are required to agree to twelve decimal places: from the spacings, which is what a figure draws, and from the closed form, which is what the argument uses. A change to the diameter route or to the constants that reached one and not the other would leave them disagreeing with nothing to notice.

The refusal is a function rather than a branch: asking for a knit’s open area at an over-full tightness raises an error naming the loop length that would be needed, rather than returning zero. A zero is a number somebody can print.

The turning point is asserted twice over. The closed form √k_r − 1 is computed, the hole area is sampled either side of it, and the flow’s own peak is required to land at that strain exactly rather than near it — which would catch a factor of two in the area-conservation step, since a wrong exponent would move the peak without changing the shape.

The threshold hardly moves with the fibre, and moves with the packing #

The critical tightness factor is quoted above as 11.3 to 13.4 depending on the relaxation state, for a cotton yarn. It is worth asking what happens to it in another fibre, because the answer decides whether the finding is about knitting or about cotton.

Occupancy is the diameter times Munden’s stitch-density constant over the loop length, and the tightness factor is the square root of the tex over the loop length. Divide one by the other and the loop length goes:

occupancy = k_s × (d ÷ √tex) × K.

So the critical K is fixed by the yarn’s diameter per root tex, which is where the fibre enters — and it enters through one group only. A yarn’s diameter comes from its mass, its fibre’s density and its packing factor, so d/√tex is proportional to one over the square root of the density times the packing, and nothing else about the fibre survives.

That group barely moves across the textile fibres. Against cotton at 1.52 g/cm³, polyester at 1.38 shifts the critical tightness factor down by 4.7 per cent and wool at 1.31 by 7.7. Nylon and viscose sit inside that span. The whole range of common fibres moves the threshold by under a tenth, which is a good deal less than the spread between the three relaxation states — so the finding is about the geometry of a loop and not about what it is made of.

The packing factor is the larger lever and it is the one that is not a material constant at all. Going from a spun yarn’s 0.6 to a filament yarn’s 0.75 raises the critical K by twelve per cent, because a more tightly packed yarn is thinner for its mass and fills less of the cell.

That is the structural reading and it matches the trade. An open knitted mesh — a net, a powernet, a marquisette — is made from filament yarn, and the usual reasons given are strength and abrasion resistance in a fabric with few threads. This adds a geometric one: a filament yarn’s higher packing puts the threshold further up the tightness scale, so a filament mesh can be knitted tighter than a spun one and still have holes in it. A spun yarn at the same tightness factor has already closed.

And it says which way the uncertainty runs. The packing factor of a spun yarn is not a measured constant here but a stated one, and every threshold on this page moves as its square root. A yarn at 0.55 rather than 0.6 has a critical K four per cent lower, which pushes even more of the trade’s range onto the closed side. The direction of the finding is robust against the one number in it that is least well known, which is the check worth having when a conclusion turns on a threshold falling inside a range rather than outside it.

Where the model stops #

Munden’s constants are measured and are a plain jersey’s. They come from regression on relaxed cotton jerseys, this collection has already found that they do not compose the way a naive reading suggests, and everything above inherits whatever they inherit. A rib, an interlock or a purl has its own constants and this arithmetic has not been run on them.

Occupancy is a projection and a loop is not planar. The yarn crossing its own cell four times is being counted as though it lay flat, which double-counts nothing and under-counts the third dimension entirely. An occupancy of 1.13 does not mean the cell is 113 per cent covered; it means the projection would need that much area and the yarn has gone somewhere else, which is out of plane.

Area conservation under extension is an approximation. A loop robbed from a neighbouring course is a real mechanism and it breaks the conservation; so does yarn extension at large strains. The turning point is exact for the idealisation and approximate for a fabric.

And nothing here is wet. A knit’s swelling is not what its change of state is about, which this collection established separately, but a swollen yarn is a larger d and moves the threshold directly — so a fabric on the open side of it dry can be on the closed side wet.

The generalisation #

A model that returns nothing for a whole class of objects has found a boundary rather than failed, and the boundary is worth more than another number inside it.

The useful move is to compute where the model stops rather than to note that it does. Here the boundary is a threshold in one ratio, the threshold is a pure number, and the commercially interesting fabrics sit on it — so the refusal is informative in a way that a general caution would not have been. A model that says this does not apply to knits teaches nothing; one that says this applies below d/ℓ = 1/k_s and the trade works just above it says where to look.

The diagnostic is to solve for the argument at which the answer degenerates. Any model with a positivity condition in it has such a value; finding it costs one inversion and it is almost always more informative than the model’s output at a typical input, because it is the one place the model’s assumptions are visible.

And the second lesson is about refusing rather than returning. An over-full knit has an open area of zero under a max-with-zero, and zero is a perfectly usable number that would propagate silently into a permeability, a rating and a transmission. Raising an error instead costs nothing and is the difference between a model that knows its own domain and one that does not.

Who found it, and when #

Munden’s constants are from 1959 and 1962 and are the foundation of knitted-fabric geometry; the tightness factor √tex/ℓ is the knitter’s own index and long predates the constants.

That a commercial jersey’s loop is over-full in projection is not new here either — this collection reached it from the other direction when it asked whether a knit has spare area to swell into and found that it does not.

What appears to belong here is the consequence for transport: that the threshold expressed as d k_s/ℓ has no fabric dimension in it, that the trade’s tightness range straddles it, and that a knit’s air path is therefore the same mechanism as a woven cloth’s floor rather than the same mechanism as a woven cloth’s holes. The two fabrics are usually compared on permeability as though they were two settings of one structure, and they are two structures.

Where the ladder goes next #

This is where the ladder built on the hole between four threads runs out, and it runs out at a fabric rather than at a question. What is left undone is named plainly: a knit’s air path is a three-dimensional channel between interlocked loops and nothing in this collection can price it. The nonwoven has the same difficulty and has a percolation argument instead of a geometric one, which may be the shape the answer takes.

Sideways, the same over-fullness that removes a knit’s holes is what makes its own dimensions a state rather than a construction, which is where a knit relaxes for as long as it is allowed to picks the argument up.