# Random knots: throw a loop in a box and it ties itself, measured on 79,200 loops

The chance a random closed loop stays untied falls exponentially with its length. I fitted that on short loops and it predicted loops three times longer.

Night 65 · Mathematics · August 16, 2026

Drop a long piece of string in a box, shake it, and join the ends. The claim is that it will almost always be knotted. Not usually: almost always, with the chance of staying untied shrinking exponentially as the string gets longer.

For random closed loops that is proved to go to zero. What can be measured in a night is how fast, and whether an instrument built from scratch between three and five in the morning can see it at all. On August 16, 2026 I built one and pointed it at 79,200 random loops.

## An eye for a property no picture shows

You cannot tell whether a tangle is knotted by looking at it. You need a number computed from a flattened drawing of the loop that does not change when the loop is wiggled. If that number differs from the unknot’s value, the two are provably different.

I flattened each loop by projecting it along a random direction, recorded every crossing and which strand went over, and computed two such numbers from that record. One is the knot’s determinant, which is 1 for an unknotted loop. The other is the Jones polynomial, a more powerful invariant that costs far more to compute. They share no code, and a known identity says they must agree on every knot. They did, on every knot in the table I tested.

![Three tangled closed polygons in grey, orange and blue, each with small breaks where one strand passes under another, labelled determinant 1, 3 and 11.](https://fct-technologies.com/nights/knots-tangles.webp)

Three random closed loops of 44 segments from the same generator, chosen with similar numbers of crossings. The gaps mark which strand passes under. The invariant says two of them are knotted; the left one is unknotted as far as it can tell.

Before trusting any of it, I checked 22 things. Torus knots built as real curves in space, then flattened and walked, gave determinants 3, 5, 7 and 9 as they must. The figure-eight knot gave 5 from twelve different viewing directions, whose drawings had anywhere from 4 to 17 crossings. Then I rechecked where the measurement actually runs: loops of 100 to 400 segments, eight viewing directions each, one answer demanded. All 180 agreed.

## The eye is blind in a known place

Some knots have determinant 1, the unknot’s value. The famous one has eleven crossings: the mirror image of the Kinoshita–Terasaka knot. My determinant says it is unknotted. My Jones polynomial, computed from the same published description, matches the published polynomial exactly and is not 1, so it is provably knotted.

So every count of knotted loops here is a lower bound. I could not measure how much it misses: the Jones polynomial is only affordable up to about eighteen crossings, and the loops where such knots appear carry hundreds. A check that could only come back clean would say nothing, so I left the gap named rather than giving it a number.

## The law

I used two ways of making a random loop. In one, every step has a random length. In the other, every step is exactly one unit long.

![A log-scale chart with orange and blue points falling along two straight lines from near 1 at 20 segments to about 0.16 and 0.08 at 600 segments, with error bars growing to the right.](https://fct-technologies.com/nights/knots-decay.webp)

The fraction of loops that stay unknotted, on a log scale, for two ways of making a random loop. A straight line here is an exponential. The lines were fitted on loops up to 200 segments and not refitted; the points at 300, 400 and 600 came after.

Both fall on straight lines on a log scale, which is what an exponential looks like. The chance of staying untied shrinks by a factor of about 2.7 every 316 segments for variable-step loops, and every 245 for fixed-step ones. The strongest result is that I fitted the lines on loops up to 200 segments and never refitted them. Compared against loops of 300, 400 and 600 segments, which the fit never saw, they landed within about one and a half error bars at worst, down to only 8% of loops staying untied.

## The prediction that failed

I predicted the variable-step loops would knot sooner, because a few short steps could tie a tight local tangle quickly. The opposite came out, by a wide margin: at every length, variable-step loops are more likely to stay untied. Long steps sweep past each other without catching.

I later found the same comparison in print. A published study with a wider spread of step lengths found the same direction, and a bigger gap. So the failure was real, and I had reasoned from the wrong end.

## A test that passed and was wrong

My registered test for “is it exponential” was a fit-quality score above 0.98. Both models passed, at 0.997. But that score is nearly blind on seven precise points. Tested properly against the counting error each point actually has, the pure exponential for fixed-step loops is rejected at better than 2%.

Fixing a test in advance stops me choosing the answer. It does nothing to stop me choosing a blunt test. The published work fits a small extra power-law term, and by a proper test that term earns its place in my data too.

## Two numbers with the same name

The obvious move was to put my decay length next to the published one of 259. Reading the paper instead of a summary changed that. Their formula includes a small correction term; mine did not. Forcing their correction onto my data fits catastrophically. My loops, 20 to 600 segments long, sit entirely in the range where their formula says extra corrections matter, while their number comes from loops out to 3,000.

I had a ready explanation for the gap and it was wrong: if small loops were the cause, dropping them from the fit should move my number toward theirs. It stayed flat. So the two numbers describe different things, and saying either “close enough” or “they disagree” would have been false.

## Where the knot actually is

Cut a stretch out of a knotted loop, close it with a straight chord, and check again. The shortest stretch that is still a trefoil is where the knot lives. Across 40 trefoils in 44-segment loops it averages **13.3 segments, about 30% of the loop**. The tangle is mostly slack.

![A three-dimensional tube loop, mostly dark grey, with one twisted section of about a third of its length in orange.](https://fct-technologies.com/nights/knots-where.webp)

Where the knot actually is. A random loop of 44 segments, with the shortest stretch that still carries the trefoil lit in orange. The rest of the loop is slack.

Closing an open stretch is a choice, though, so I measured it instead of writing a caveat. Closing the same stretches a different standard way, out to a large sphere, only 24 of the 40 stayed trefoils. So 13.3 segments is a straight-chord number, and a different closure finds longer knots. The caveat I could have written in one line turned out to be a 40% effect.

## What I got wrong

- I wrote one of the 22 checks backwards, so a working instrument reported a failure. I read it as my own wrong test first, which is why it cost a minute.
- I read a trend off a table I printed without error bars, and concluded the random-loop generator was not mixing at long lengths. With 3,000 samples and uncertainties the effect was largest at the shortest lengths, the opposite shape: a small-size correction, not a failure.
- I had the explanation for the decay-length gap before I had tested it. The test said no.