Almost every long loop is knotted

Six sticks tie a trefoil, and five cannot asked for the fewest sticks that can make a knot, and found that random corners rarely make one: one hexagon in two hundred is a trefoil, one heptagon in 1,700 a figure-eight. That was the view from the smallest end. From the other end the picture reverses. A loop of many sticks with random corners is knotted more often than not, and the longer it is, the harder it becomes to find one that is not.

The question was first asked about molecules. A long polymer in solution wriggles at random, and if its two ends join to form a ring, the ring is knotted or not, permanently, since the chain cannot pass through itself. Harry Frisch and Edel Wasserman, in 1961, and Max Delbrück, in 1962, independently conjectured that the probability of a knot tends to one as the ring grows. That is now a theorem for several models of random loops, and the figures below measure how quickly it happens for one of them.

The measurement uses nothing beyond tools already built for knots. Each loop is projected to a picture, its crossings are found, and the colouring determinant of colours that count more than three is computed exactly from them. A determinant other than 1 proves the loop knotted. A determinant of 1 proves nothing — some knots have it — so every knotted share reported here is a lower bound.

A random walk, closed #

The loops are built from random walks. Each step is a displacement in three dimensions whose three coordinates are independent draws from a normal distribution, so the step has no preferred direction and a typical length a little over one and a half units. Forty such steps wander away from the start and end somewhere else, as a walk that comes home described for walks on a lattice: in three dimensions a walk does not reliably return.

To close it, subtract the average of the forty steps from each one. The steps then add to exactly zero and the path returns to its start. The correction is small — under a tenth of a unit per step in the figure, against steps averaging 1.6 — and it leaves the walk’s shape almost untouched while making the knot type a well-defined question. Loops built this way are called Gaussian random polygons, and they are the standard model in the physics of polymer rings: the steps stand for segments of chain long enough that their directions are independent.

A random polygon does not avoid itself in any special way; it simply almost never passes exactly through itself, since that would need two sticks to meet at a point, an event of probability zero. So every polygon here is a genuine knot of some type, and the question is which.

One loop of a hundred sticks #

Seen from above, a random polygon of a hundred sticks is a scribble with 58 crossings. No one could say by eye whether it is knotted, and no sequence of the simplifying moves of three moves, and what they cannot undo is obvious: most of the 58 crossings can be removed, but finding the order in which to remove them is a search that grows quickly with the number of crossings. The determinant settles it in a fraction of a millisecond: it is 5, the determinant of the figure-eight knot and of the cinquefoil, and since the unknot’s determinant is 1, the loop is knotted.

The computation behind that number is the one used for the six-stick trefoil, scaled up. Every pair of sticks that cross in the picture gives a crossing; the crossings cut the loop into 58 arcs; each crossing gives a row of the colouring matrix; and the determinant of a 57-by-57 minor, computed exactly in whole numbers, is 5. The answer does not depend on the direction of projection, although the picture and the matrix do, and that independence is checked on the small polygons of the previous essay, where it is cheap.

That speed is the point of using the determinant. Deciding outright whether a diagram shows the unknot is a much harder problem. Wolfgang Haken showed in 1961 that it can be decided at all; Joel Hass, Jeffrey Lagarias and Nicholas Pippenger proved in 1999 that an unknotted diagram always has a proof of the fact that can be checked in polynomial time; and Marc Lackenby announced in 2021 an algorithm running in quasi-polynomial time; whether it can be done in polynomial time is open. The determinant is a polynomial-time computation that answers a weaker question — it can prove knottedness, never unknottedness — and for a census of thousands of loops that weaker question is the one that can be afforded.

What the determinant cannot say is which knot this is. Several knots have determinant 5, and a knot with dozens of crossings in its picture might be a figure-eight hidden under many removable crossings or something far more complicated. For the question asked here — knotted or not — the lower bound is enough.

The knotted share, by length #

At ten sticks, fewer than one loop in a hundred is provably knotted. At forty it is one in sixteen, at a hundred more than one in five, and at two hundred and fifty more than half. The curve has not flattened by the end of the range; its apparent levelling between 160 and 250 sticks is within the sampling error of three hundred loops. That error is the standard deviation of a proportion, $\sqrt{p(1-p)/300}$, about three percentage points at $p = \tfrac12$, and how far from the average a thing can be gives the guarantee that goes with it: a share measured this way is off by more than three standard deviations at most one time in nine, whatever the distribution.

The grey curve is the average number of crossings in the picture, which grows faster than the number of sticks — 67 crossings at a hundred sticks, 205 at two hundred and fifty. More crossings mean more opportunities for the strands to be entangled, and the knotted share rises with them. But the relation is not simple: most crossings in a long random loop are removable, and the knotting comes from a few places where the strands are genuinely tangled.

The unknotted share falls exponentially #

On a logarithmic scale the share with determinant 1 falls roughly along a straight line, and a straight line on that scale is exponential decay. The fitted rate is a factor of $e$ every 290 sticks or so: each further 290 sticks multiplies the share by about $0.37$, each further thousand by about $0.03$. The unknotted share is smaller still, since it is part of the determinant-1 share.

That exponential decay is what was proved. De Witt Sumners and Stuart Whittington showed in 1988 that for self-avoiding polygons on the cubic lattice the probability of being unknotted falls exponentially with length, and Nicholas Pippenger proved the corresponding statement for lattice random walks in 1989; Yuanan Diao and others extended it to random polygons in space in the 1990s, including Gaussian ones and those with sticks of equal length. The measured rate here is an estimate from a few thousand loops, with an uncertainty of perhaps twenty percent. The theorem guarantees the decay; it does not say how fast.

Knots the determinant cannot see #

The test is one-sided, and it is worth knowing how badly it can miss. A knot’s determinant is the absolute value of its Alexander polynomial at $t = -1$, as a polynomial behind the colourings showed, and some knots have determinant 1. The torus knot that winds three times one way and five the other, with ten crossings, is one: its polynomial evaluated at $-1$ gives exactly 1. The Kinoshita–Terasaka knot and the Conway knot, with eleven crossings each, are worse — their whole Alexander polynomial is 1, the same as the unknot’s.

Such knots are invisible to every test used here. How often a random loop is one of them is not known exactly, but it can be bounded from the data. At a hundred sticks the knots that are seen are overwhelmingly the small ones — trefoils and figure-eights — and knots with ten or eleven crossings in their best diagrams need far more winding than a typical loop of that length does. So the undercount at these lengths is probably small, and certainly smaller than the share that is detected. At much greater lengths the loops are composites of many small knots, and a composite has determinant 1 only if every piece does, which becomes steadily less likely as pieces are added.

Why long loops must knot #

The proof is an argument about patterns, and its idea can be stated without its technical apparatus. Fix a small knotted tangle: a few sticks that tie a trefoil inside a small ball, with two loose ends leaving it. A random walk produces that tangle somewhere along its length with some small probability per step — tiny, but not zero and not dependent on how long the walk is. A walk of $n$ steps has about $n$ places where the tangle could start, and the chance that it appears at none of them falls like $(1 - p)^n$, exponentially in $n$.

A loop containing such a tangle is knotted, whatever the rest of it does: the rest can be as complicated as it likes, but it cannot undo a trefoil tied inside a ball it never enters. The reason is the genus of the surface a knot bounds. Tying one knot after another along the same string adds their genera — Horst Schubert proved it in 1949 — and the trefoil has genus one, so any loop that contains it as a separate tangle has genus at least one, while the unknot has genus nought. That last step needs care — the rest of the loop might pass through the ball — and the careful version is Harry Kesten’s pattern theorem of 1963 for self-avoiding walks, which Sumners and Whittington used. It says that any pattern that can occur at all occurs, in almost every long walk, many times.

The argument predicts more than knotting. It predicts that long loops are usually knotted by several separate tangles, each small, rather than by one large knot. The last figure looks for that.

What the determinants show #

Among 600 loops of a hundred sticks, 468 have determinant 1, 77 have 3 and 24 have 5; the rest are spread thinly over larger odd numbers. Every determinant is odd, as a knot’s must be — a check on the computation that could have failed and did not.

The distribution is dominated by the smallest knots. Determinant 3 belongs to the trefoil, 5 to the figure-eight and the cinquefoil, and 7 to the knot $5_2$ and the seven-crossing torus knot, among others. The composites are there too. When two knots are tied one after the other along the same string, the determinant of the result is the product of their determinants, so 9 can be two trefoils and 15 a trefoil with a figure-eight. Nine loops have determinant 9 and three have 15 — fewer than the singles, as two tangles are rarer than one.

This is the pattern the proof predicts and not a proof of it: a determinant of 9 could also be a single knot, such as $6_1$. What the counts show is that knotting at this length is mostly a matter of small tangles, alone or in pairs.

Where this happens outside a computer #

The first test was on DNA. In the early 1990s two groups, one including Vsevolod Rybenkov, Nicholas Cozzarelli and Alexander Vologodskii and the other Stephen Shaw and James Wang, joined the ends of long DNA molecules in solution and measured how often the resulting rings were knotted, separating the knots by how fast they moved through a gel. The rates matched simulations of random polygons, once the stiffness and electric charge of DNA were put into the model — and they have been used since to measure those properties, by asking what they must be for the knotting rates to come out right.

Living cells face the same arithmetic. The DNA in a bacterium is a closed loop millions of base pairs long, packed into a tiny volume, and replication and repair keep cutting and rejoining it; left alone it would knot and tangle, and a knotted chromosome cannot be copied or divided. Cells carry enzymes, type II topoisomerases, whose job is to cut one double strand, pass another through the gap and reseal it — a crossing change of exactly the kind how many changes undo a knot counted — and they keep the knotting of DNA far below what a random loop of its length would show.

A household version was run in 2007 by Dorian Raymer and Douglas Smith, who tumbled strings of different lengths in a rotating box and counted knots. Short strings never knotted; the probability rose with length and then levelled off near one half, limited by the box and the tumbling time. The levelling is a feature of their apparatus. A random loop, left to wander without limit, knots with probability approaching one.

The Gaussian steps used here are the steps of the walk that becomes a curve, which converged, when rescaled, to Brownian motion. A random polygon of $n$ sticks is a closed Brownian path sampled at $n$ points and joined by straight lines, so taking more sticks does not only make the loop longer: it resolves finer and finer wiggles of the same kind of path, and each resolved wiggle is another chance to tie a small knot.

Still open: the same rate for every knot #

The probability that a random loop of $n$ sticks is a particular knot $K$ — a trefoil, say — rises at first, peaks, and then falls, because at great lengths the loop is almost always a composite of several knots rather than any single one. Simulations of lattice polygons, begun in the 1990s by Enzo Orlandini, Maria Carla Tesi, Esaias Janse van Rensburg and Stuart Whittington among others, found that the falling side of that curve decays exponentially at a rate that seems to be the same for every knot type, the unknot included, with only a power of $n$ in front that depends on the knot.

That universality is a conjecture. It is supported by extensive computation and by a heuristic — a knot of type $K$ is an unknotted loop with a few fixed tangles in it, and the tangles cost only a polynomial factor — but it has not been proved for any model of random loops, and neither the common rate nor the powers in front are known exactly. For the Gaussian polygons here, the characteristic length of about 290 sticks is a measurement, not a derived constant, and nobody knows a formula for it.

What the figures do not settle #

Every knot determination here is exact: crossings from exact geometry, determinants in whole-number arithmetic with no rounding. What is statistical is the sampling. Each point is a few hundred loops, so the shares carry errors of two to three percentage points, and the fitted decay length of 290 sticks could be off by a fifth. The determinant test undercounts knots, and so every knotted share is a lower bound and every determinant-1 share an upper bound on the unknotted share.

The theorems quoted — Sumners and Whittington’s, Pippenger’s, Diao’s — are about particular models and are stated, not proved, here; the figures agree with them over the range computed. The conclusion does not depend on the model’s details. Whatever builds the loop, a long one has room for a small knot, and room is all a knot needs.