Which yarns knot well

A rope table lists knots down one side and efficiencies across the top. The variable is the knot’s name.

The arithmetic says the variable is mostly the fibre.

The three inputs #

A knot’s efficiency is the fraction of the breaking strain that survives the bend:

(breaking strain − bending strain) / breaking strain

and the bending strain is half a fibre diameter over the bend radius.

So three quantities decide it. The bend radius, which is the knot’s. The fibre’s diameter relative to the yarn’s, which is the yarn’s construction. And the fibre’s breaking strain, which is the material’s.

Only the first is in any table.

The spread #

Across this collection’s own fibre table, at twenty tex and a bend radius of one yarn diameter:

Nylon keeps 83 per cent. Its breaking strain is twenty-one per cent, which is enormous, so the three and a half per cent the bend spends is a sixth of what it has.

Silk keeps 63 per cent. Breaking strain about nine and a half per cent, and a fine fibre.

Cotton keeps 46 per cent. Six and a half per cent breaking strain — the rule of thumb.

Polyester keeps 47 per cent, almost the same as cotton for entirely different reasons: a lower breaking strain offset by a finer fibre.

Viscose keeps 6 per cent. Its tenacity is 0.20 against cotton’s 0.35 at nearly the same modulus, so it breaks at under four per cent of strain and the bend has spent nearly all of it.

That is a range of fourteen, at the same knot.

What the range means #

It means the rule of thumb is a statement about one fibre, and the fibre it is a statement about is a natural cellulosic at ordinary fineness.

That is not a criticism of the rule. Rope has been made of hemp, manila, sisal and cotton for most of history, and all of those have breaking strains in the same band as cotton’s. A rule of thumb built on them is a good rule of thumb for them.

It stops being one the moment the rope is nylon, and it stops being one in the opposite direction the moment it is viscose or aramid.

Why nylon knots well #

Nylon’s breaking strain is twenty-one per cent, which is three times cotton’s and about five times aramid’s. A fibre that stretches a fifth of its own length before breaking has a great deal of strain to spend, and a bend that spends three and a half per cent of it is barely noticed.

That matches practice completely. Nylon rope is the traditional choice where knots are unavoidable and shock loads are expected — climbing, mooring, towing — and its knot efficiencies are the highest in any table.

The usual explanation is that nylon is “forgiving” or “elastic”, which is right and is not a mechanism. The mechanism is that the criterion is a strain criterion, and a material with a large breaking strain has more room in it.

Why aramid knots badly #

Aramid’s breaking strain is two and a half per cent, because its tenacity is very high and its modulus is very high, and a strain is the first divided by the second.

Three and a half per cent of bending strain against two and a half per cent of breaking strain leaves nothing. The arithmetic says an aramid yarn cannot be knotted at a bend radius of one diameter at all.

That is very nearly what is observed. Aramid ropes lose half or more of their strength to a knot, they are spliced wherever possible, and the manufacturers’ literature warns against knots in terms that no cotton rope’s ever did.

The same argument covers the whole high-performance class. High-modulus polyethylene, carbon, glass: all of them buy tenacity with modulus, and a strain criterion punishes the second as hard as it rewards the first.

A stiff fibre is a fibre with nothing to spend on a bend.

The fineness, which no table has #

The second input is the one nobody varies, and it makes a prediction.

The bending strain is half a fibre diameter over the bend radius, and the bend radius is quoted in yarn diameters. So the strain depends on the ratio of the two diameters, which is the square root of the packing factor over the fibre count.

A finer-fibred yarn of the same count has more fibres, each thinner, so the ratio is smaller and the knot costs less.

For a factor of two in fibre fineness — one and a half decitex against three, which is an ordinary difference — the bending strain falls by root two, and a cotton’s efficiency rises from forty-six per cent to about sixty-two.

Sixteen points of efficiency from a variable no rope table has a column for.

That is a testable prediction and it needs two yarns of the same count and fibre type spun from different staples, which any spinner can supply.

And the count, which has one #

A coarser yarn has more fibres of the same size, so the ratio of fibre diameter to yarn diameter falls as one over the square root of the count — and the efficiency rises.

At sixty tex rather than twenty, a cotton’s efficiency at one diameter of radius rises from forty-six to about sixty-nine per cent.

That is not the same as saying a thicker rope knots better in absolute terms, because a knot in a thicker rope takes a bigger bend radius too and the two effects partly cancel. What it says is that at the same relative bend radius, the coarser yarn keeps more.

Rope tables do record a count dependence, usually as a note that thin lines lose more to a knot — the same kind of note a yarn count system invites and does not settle. The arithmetic gives it a magnitude and a mechanism.

What a specification should carry #

Putting the three inputs together, a knot efficiency figure is only meaningful with three things attached, and rope tables carry one.

The bend radius, in yarn or rope diameters, which is what the knot’s name is standing in for. Naming the knot is a poor proxy: two knots with the same first-curve radius have the same efficiency and two dressings of one knot do not.

The fibre, which supplies the breaking strain and is the largest of the three effects.

And the fibre fineness, which supplies the rest and is absent everywhere.

A table with those three columns would be much shorter than the tables that exist, because it would have one row per rope rather than one per knot, and it would predict rather than record, which is what a specification quoting the wrong quantity never can.

The two fibres that break the pattern #

Two rows of the table do not behave the way a reader would guess and both are instructive.

Wool. Its breaking strain is high — about five per cent — but its fibre is very coarse, five decitex against cotton’s one point seven. The two effects fight and wool comes out in the middle, keeping about half.

That is worth knowing because wool’s reputation is for being weak, and its knot efficiency is unremarkable rather than poor. What makes a woollen rope weak is the yarn’s own strength rather than what a knot does to it.

Glass. Its breaking strain is a little over one per cent, the lowest in the table, and a bend of one diameter spends three times it. Glass yarn cannot be knotted at all, and everybody who has worked with glass reinforcement knows it: the fibre breaks when it is bent, in the hand, before any load is applied.

That is a prediction the arithmetic makes flatly and the material confirms flatly, which is the most useful kind of agreement — not a number matching a number, but an impossibility matching an impossibility.

Why extensibility and strength pull against each other #

There is a structural reason the ordering above looks the way it does, and it is worth surfacing because it constrains what a fibre can be.

Breaking strain is tenacity over specific modulus. Making a fibre stronger raises the first; making it stiffer raises the second. And the two are made by the same process: orienting the molecules along the axis raises tenacity and modulus together, and usually raises the modulus more.

So the fibres at the strong end of the table are also at the stiff end, and their breaking strains are low. Aramid is five times cotton’s tenacity and thirty times its stiffness relative to weight.

That means a fibre cannot be optimised for knotting and for strength at once by the usual route. The way to a knottable strong fibre is to raise the tenacity without raising the modulus, which is what nylon does and is why nylon occupies the position it does in rope.

Which is the same trade-off this collection found on its torsion ladder, where orientation lowers the torsional stiffness ratio exactly as it raises the tensile modulus. One process, three consequences, all of them running the same way.

What was counted, and how #

The efficiency is computed from three quantities, all of them this collection’s own.

The fibre diameter comes from the fibre’s fineness by the site’s own volume arithmetic, at a packing factor of one because a fibre is not a bundle. The yarn diameter comes from the count, the fibre density and a packing factor of 0.6.

The breaking strain is a tenacity in newtons per tex over a specific modulus in the same units, where the specific modulus is the bulk modulus over the density. That conversion is written once and used everywhere.

The bend radius is an input and is quoted in yarn diameters throughout, so the numbers are comparable across counts.

The check is the same one the previous rung carries: the coherent bound leaves nothing at a bend of one diameter, and the free bound leaves between a third and seven tenths for a cotton and a polyester. Viscose falls outside that band and is excluded from the check by name rather than quietly — the model is not failing on viscose, viscose is simply a poor thing to knot.

Where the model stops #

The friction does not enter the efficiency at all. It decides whether the knot holds, not what it costs, because the capstan relation has no radius in it and the bending term has no friction in it. That separation is exact and is unusual — most things in this subject couple.

The bend radius a real knot takes is not computed, and it varies by fibre: a stiff fibre resists being pulled into a tight bend, so an aramid knot may take a larger radius than a cotton one and partly rescue itself.

That is a real and unmodelled interaction and it runs in the direction of compressing the range above. How much is a contact problem this collection cannot solve.

And the compression is ignored, as it is throughout this ladder — the same omission the fabric’s own contacts carry.

What this predicts about sewing thread #

The arithmetic has a consequence for a product that is knotted constantly and is never described in these terms.

A sewing thread is knotted at the start and end of every seam, and it is bent round the needle eye, the hook and the tension discs thousands of times a minute at radii of a few of its own diameters. Every one of those is the same bending-strain problem.

So the fibre that makes the best sewing thread should be the one with the largest breaking strain, and the ordering the arithmetic gives is nylon, then silk, then polyester and cotton together.

That is very nearly the historical ordering of sewing threads by quality: silk was the premium thread before synthetics, nylon and polyester displaced cotton, and cotton survives for its heat resistance rather than its performance.

It also explains a puzzle. Polyester is stronger than cotton by a good margin and its threads are not proportionately better in use, and the arithmetic says why: polyester’s breaking strain is almost the same as cotton’s — a higher tenacity offset by a higher modulus — so it has no more to spend on a bend.

The improvement polyester brings to sewing thread is durability and moisture behaviour rather than a better tolerance of the geometry it is put through.

Where the argument would break #

A fair test of an account is what would falsify it, and this one has a clean failure mode.

If knot efficiency were measured across fibres at a controlled bend radius — not by naming a knot, but by bending each yarn round a mandrel of a stated diameter and pulling — the ordering should follow the breaking strains, in the order this rung gives, with the magnitudes the arithmetic predicts.

If instead the ordering followed the tenacity, the criterion would be a stress criterion rather than a strain one and the whole account would be wrong.

Those two orderings are very different. By tenacity, aramid and carbon are at the top and viscose at the bottom. By breaking strain, nylon is at the top and aramid near the bottom.

Nobody has run that experiment in a form that separates them, because knot tests are run on knots and a knot’s radius is not controlled. A mandrel test would settle it in a morning and would be a better measurement of knots than any knot test.

The generalisation #

The rung is an instance of a pattern this collection has met repeatedly and it is worth stating in its general form.

When a table’s rows are named after the wrong variable, the table cannot predict.

A rope table’s rows are knots. The arithmetic says the strong variable is the fibre and the second-strongest is the fibre fineness, and the knot enters only through one number — a radius — which the knot’s name is a poor proxy for.

So the table records what happened when somebody tested that knot in that rope, and it cannot say what will happen in a different rope. Every user of such a table has to interpolate on a variable the table does not have.

That is the same complaint this collection has made about thread count as a measure of quality: a real correlation, recorded against a variable that is not the mechanism, so the number travels badly.

The remedy in both cases is the same. Find the quantity the mechanism actually depends on, and record that instead — even when it is harder to measure, because a harder measurement of the right thing beats an easy measurement of a proxy.

Why viscose is worth keeping in the table #

Viscose keeps six per cent and it would be tidier to drop it as an outlier. It is worth keeping for two reasons.

The first is that it is correct. Viscose is a poor thing to knot, it is not used for cordage, and its low tenacity at a cellulosic modulus is exactly why. The number is not a model failure; it is the model saying something true and unflattering about a real fibre.

The second is that it calibrates the range. A table whose members all agree is a table that has not been tested at its edges, and viscose and nylon are the two edges. Between them the account has to span a factor of fourteen in efficiency using only two inputs, and it does.

That is the same reason the shear-modulus table on this collection’s other ladder keeps glass in it. A row nobody will use, whose answer is known independently, is worth more than another row of the same kind — and the rows that make a table testable are usually the ones that look least useful.

Who found it, and when #

Knot efficiency has been measured since the nineteenth century and the tables are extensive. The dependence on the rope material is universally noted and rarely explained; the dependence on fibre fineness is, as far as this collection can find, not noted at all.

The strain criterion is standard and is the usual explanation given for the material dependence, generally in the form “more extensible fibres knot better” without the arithmetic that makes it quantitative.

What is this collection’s own is computing all three inputs from its own tables, finding a range of fourteen across the fibres, and noticing that the second-largest effect is a variable nobody records.

Where the ladder goes next #

The knot group closes here. The contact ladder returns to fabric, and the first question it left open is whether the extension ceiling — the model’s most conspicuous quantitative failure, three times any measured jersey — is a contact problem at all.

It is not. Giving the yarn a thickness lowers the ceiling by seven per cent against a gap of two thirds, so contact is not why a jersey stops, and the candidate this ladder was written to test is ruled out.