Braids and the third thread system

A woven cloth has two thread systems. The warp runs down the length and the weft runs across it, they meet at right angles, and every question about the fabric turns into a question about which of them is on top at each meeting.

A braid has one. The strands all belong to the same system, they all run obliquely, and they interlace with each other rather than with a second family. Nothing about the shoelace, the plaited hair, the climbing rope’s sheath or the carbon-fibre sleeve pulled over a mandrel fits the warp-and-weft picture at all.

What survives the change of structure is the question. At every crossing one strand passes above another, which is the only ingredient the integrity criterion needs, and it applies to a braid without a word of it being altered.

Writing a braid down #

The first thing needed is a way of recording a braid that a computation can read, and it happens to exist already.

Line the strands up in positions numbered from one side. A braid is then a sequence of moves, each of which takes the strands in two neighbouring positions and swaps them, with one passing over the other. Write down which position the swap happens at and which way round it goes, in order, and the sequence describes the braid completely.

That notation is the standard one for braid groups: the generator σᵢ crosses position $i$ over position $i+1$ and its inverse crosses them the other way. It was not invented for rope. Emil Artin wrote it down in 1925 as a way of describing a purely topological object, and the fact that it is also the natural way to record what a braiding machine does is a coincidence of exactly the useful kind.

Two things about the notation matter here and both are easy to miss.

Positions are not strands. A move at position two involves whichever strands happen to be sitting there, and after the move they have swapped. Following a strand means following the permutation, and the strand that starts on the left of a plait spends most of its time somewhere else.

A braid is periodic. What is being described is a rule that repeats, not a finite piece of cord. So the sequence has to bring every strand back to the position it started in, or the repeat is longer than it appears — a three-letter word whose permutation has order three describes a nine-crossing repeat, and drawing two of it and three of it would give different pictures of different things. Every braid in this essay closes, and the generator checks it rather than the table asserting it.

The same criterion, unchanged #

Now the test. It is worth restating in the braid’s own terms because it reads slightly differently when there is only one thread system.

At every crossing one strand is above another. If the strands could be split into an upper set and a lower set such that at every crossing between the two sets the upper strand is on top, the upper set lifts away and the braid is not one braid.

Turn that into reachability exactly as before: at a crossing where $A$ is above $B$, putting $B$ in the upper set forces $A$ there too, so draw an edge from $B$ to $A$. The braid holds together precisely when that digraph is strongly connected, and the number of separable braids is the number of components.

Two strands are enough to show the criterion has content. Cross them so that the strand moving to the right always passes over, and the two alternate — first one on top, then the other — which is an ordinary twist and which the test passes. Cross them so that the same strand is always above and it lies on the surface of the other, unheld, which the test fails.

The twist result deserves a moment because it is where a reader’s intuition and the criterion part company. Two twisted strands can obviously be separated — take one end and unwind. But unwinding is not what the criterion is about. The criterion asks whether a subset can be lifted away, rigidly, without cutting anything, and for a periodic twist the answer is no: an infinite twist cannot be undone by any finite motion. That is a precise statement and a narrower one than “cannot be pulled apart”, and saying which is being claimed is the difference between a check and a slogan.

What goes wrong, and how often #

The interesting failures are not the crude ones. A strand nobody crossed is easy to spot on paper; the failure worth having a test for is the braid that looks perfectly braided and is two braids.

Add crossings in the middle and the two pairs are tied together. Which crossings, and how many, is not obvious in advance — and it is not monotone either, which is the part that catches people out. A single crossing inserted between the pairs can interrupt both twists without joining anything, leaving four strands where there had been two braids.

The honest way to say how common the failure is, is to count. Every crossing sequence of a given length on a given number of strands can be generated and tested — there are only finitely many — so the fraction that hold together is a number rather than an impression.

The results are worth stating plainly. On three strands with six crossings, 2,156 of the 4,096 possible words describe one braid — a little over half. On four strands with the same six crossings, 3,912 of 46,656, which is under nine per cent. Adding a strand makes the space of words forty times larger and the fraction of usable ones six times smaller.

That is the quantitative answer to a question braiders never ask because they never needed to: why is braiding notation so conservative? Why does almost every hand-braiding tradition in the world use the same handful of sequences over and over? Because the design space is mostly empty. A braider improvising crossings on eight strands is drawing, with overwhelming probability, several independent cords rather than one.

Flat and tubular, and one thing the notation leaves out #

There are two families of braid and the notation used here describes one of them properly and the other only approximately, which is worth saying before the enumeration is leaned on any further.

A flat plait has edges. A strand carried to the outside has nowhere further to go and must turn round, so the natural word is a walk out to one edge and back — which is exactly what the plait figure at the top of this page does, and why its strands trace a zigzag rather than a drift. Every hand-plaited braid, from hair to leather, is of this kind.

A tubular braid has none. The strands circulate around a cylinder, so the strand leaving the last position arrives at the first, and the row of positions is a circle rather than a line. That is what a maypole machine makes, what a shoelace is, and what a carbon sleeve is.

The braid word as written here has a first position and a last one, so it describes the flat case exactly. For the tubular case it is missing one crossing site — the one joining the two ends of the row — and a word that leaves that site unused describes a tube with a seam down it rather than a tube.

The consequence for the census is a bound rather than an error. Every word counted as not holding together really does not hold together, because the missing crossing site can only add edges to the digraph and never remove them. So the fractions counted above are lower bounds on the tubular case and exact for the flat one, and that is the sense in which they should be read. Making the row circular is a small change to the code and a real change to the claim, and conflating the two would be the kind of quiet over-reach that makes an enumeration worthless.

What a braiding machine does about it #

The machinery is the counterpart of the arithmetic, and it was arrived at without any of it.

A maypole braider carries each strand on a bobbin that runs around a track, and the tracks are laid out so that the bobbins weave through one another in two counter-rotating streams. The consequence of that layout is not a decorative one: it guarantees that every bobbin passes every bobbin of the other stream, which is precisely the condition that makes the connectivity graph complete. The machine is built so that the bad words are unreachable.

That is a recurring shape in this subject and it is worth naming. The interesting constraints are usually built into the apparatus rather than checked afterwards, which is why they can be forgotten. A satin’s move number is another: the reason nobody weaves a six-end satin is not that weavers know the theorem but that the drafts handed down do not contain one.

The third thread system #

The essay’s title promises a third system, and it belongs here rather than at the start because it makes no sense until the first two have been separated.

Take a braid and add a set of strands running straight along its length, not interlacing with each other but caught by the braiding strands passing over and under them. That is a triaxial structure, and it is what a good deal of industrial braid actually is: two helical systems and one axial one.

The addition changes the mechanics completely and the criterion not at all. Axial strands carry load along the braid without the extension that comes from a helix straightening, which is why a braided rope with an axial core is stiff lengthwise and a plain braid is not. And the connectivity question is the same question with more nodes: an axial strand that the helical strands merely lie alongside is a strand that pulls straight out, and the digraph reports it as a component of its own.

The same idea reappears in weaving. Adding a third thread system to a woven cloth gives a triaxial weave, where three families cross at sixty degrees instead of two at ninety. That fabric has a property no ordinary cloth has: because its three systems form triangles rather than quadrilaterals, it has no bias — the trellis mechanism that lets an ordinary cloth shear needs four-sided cells, and a triangulated net has none. A triaxial cloth is dimensionally stable in every direction in its own plane, and it is stable for a geometric reason rather than a material one.

The braid angle, and what the criterion says nothing about #

Everything above is topology, and it is worth being blunt about how little topology decides.

A braid has an angle: the helix angle at which the strands lie relative to the axis. It decides almost everything a user of the braid cares about. A shallow angle gives a cord that is strong and stiff lengthwise and will not change diameter; a steep one gives a sleeve that contracts in length as it is pulled wide, which is the whole principle of a finger trap and of the braided sleeving that grips a cable when tension is applied.

Not one of those properties is in the crossing sequence. The same word braided at twenty degrees and at seventy degrees gives cords that behave nothing alike, and the connectivity graph is identical.

What the pictures here cannot show, therefore, is the braid. Every figure on this page draws crossings as though the strands ran nearly straight down the page, because what is being tested is which strand is above which and the angle is irrelevant to that. A real braid at a real angle looks nothing like these diagrams. The diagram is a correct picture of the relation and a poor picture of the object, and reading it as a drawing of a cord would be exactly the over-claim this site exists to avoid.

The same caution applies to everything mechanical. The criterion says a braid cannot be taken apart by lifting. It does not say the braid is strong, that it will not slip, that friction will hold it, or that it will keep its shape under load. Those are questions about yarn and about geometry, and the matrix — here, the word — knows nothing about either.

What the angle decides, since the word decides none of it #

The crossing sequence is silent about the braid angle, and the braid angle decides everything mechanical. That silence is easy to state and easy to leave as a shrug, so it is worth doing the arithmetic the word cannot do — partly because it is short, and partly because it shows how much is being given up.

Take a tubular braid whose strands are inextensible and whose crossings do not slip, so each strand keeps its length S and its number of wraps k. A strand at angle θ to the axis covers an axial distance S cos θ and a circumferential distance S sin θ, and the circumferential distance is k turns of the circumference. So

length = S cos θ and diameter = S sin θ ÷ kπ,

which is a parametric ellipse: the braid’s length and diameter are locked to each other, and the angle is the single coordinate along the curve. Nothing about the crossing word appears anywhere in it.

Three readings follow, and each is a familiar object.

A shallow braid is nearly inextensible. Pulling a braid straightens its strands, so the greatest length it can reach is S and its extension from an angle θ₀ is 1/cos θ₀ − 1: six per cent at twenty degrees, forty-one at forty-five, and a hundred and ninety-two at seventy. A rope sheath is braided shallow because it must not stretch; a sleeve is braided steep because it must.

And it contracts as it extends, in a computable ratio. A braid at forty-five degrees pulled out to thirty lengthens by twenty-two per cent and narrows by twenty-nine, because the diameter follows the sine while the length follows the cosine. That is the whole of a finger trap: pulling the free end lengthens the sleeve, the sleeve narrows, and the narrowing grips whatever is inside it — harder the harder it is pulled, with no ratchet, no fastening and no friction needed to explain the direction.

The volume has a maximum, and it is not at forty-five degrees. Volume goes as D²L, which is sin²θ cos θ, and that peaks where tan²θ = 2 — at 54.7 degrees. A braid on either side of that angle grows in volume as it moves towards it, which is why the same angle turns up as the one a pressurised braided hose settles at: internal pressure does work against volume, so a hose braided away from 54.7° has a direction to move in and one braided at it has none.

Which makes the two descriptions genuinely complementary #

Put the two halves side by side and the division of labour is unusually clean.

The word decides whether the braid exists. Connectivity, how many cords a sequence really describes, whether a strand can be lifted away — all of that is topology, all of it is exactly computable, and none of it involves a length or an angle.

The angle decides what the braid does. Extension, contraction, grip, stiffness, the volume it encloses and the pressure it will settle under — all of it is geometry with an inextensible strand in it, and none of it involves which strand passes over which.

Neither half constrains the other. The same word can be braided at any angle the machine will make, and the same angle can carry any word that holds together. A braid is specified by a topological object and a geometric one, and the two are independent — which is a cleaner separation than a woven cloth has, where the draft and the sett argue with each other through the jamming limit.

Where the argument came from #

The mathematics of braids is a twentieth-century subject with a nineteenth-century prehistory, and the practice is prehistoric.

Braiding predates weaving. Plaited cordage is among the oldest manufactured objects there is, and every braid structure in common use today was arrived at by hand, by people with no notation, some thousands of years before anybody wrote down a group.

Artin’s braid group arrived in 1925 and was aimed at knot theory rather than at rope. The connection made here — that the group’s own generators encode exactly the above-and-below relation an integrity test needs — is not deep mathematics; it is a matter of noticing that the notation already contains the information. What is genuinely borrowed is the criterion itself, which comes from Grünbaum and Shephard’s treatment of periodic fabrics, and the observation that it transfers to a structure with one thread system rather than two.

Where the ladder goes next #

The next rung takes the criterion to the structure where it finally has nothing to bite on. A nonwoven web has no repeat, no periodicity and no crossing sequence, so the exact test cannot be run at all and the question of whether the sheet holds together becomes a probability with a threshold.

The companion in the other direction is the double cloth, where two components are the intended answer and the interesting number is how few crossings it takes to make them one — which is the four-strand result of this essay in a woven form.

And the base of the ladder, for a reader who arrived here first, is what a fabric is at all: a structure before it is a material, which is the claim braiding makes as loudly as weaving does.