Which knots a random loop ties

Almost every long loop is knotted closed random walks into loops and found that the share that can be proved knotted climbs steadily with length, and that the unknotted share falls exponentially, by a factor of $e$ every few hundred sticks. Its test was a single number, the determinant, which certifies a knot whenever it is not 1 but does not say which knot. The essay ended on the question the determinant could not reach. Simulations since the 1990s have suggested that every knot type’s probability eventually decays at the same exponential rate as the unknot’s, with only a power of the length in front that depends on the knot. That is a statement about knot types, one at a time, and testing it means identifying them.

This essay identifies them. Each closed random polygon is sorted by two values of its Alexander polynomial, computed from its crossings exactly and fast enough to classify fourteen thousand polygons, the longest with several hundred crossings. What comes out is a census: which knots random loops tie, how each one’s share moves with length, and a direct test of the picture behind the conjecture — that a long knotted loop is an unknotted loop carrying a few small tangles, placed independently along it.

Naming a knot from two numbers #

The polygons are the same as before. Each of $n$ steps is a random three-dimensional Gaussian displacement; the average step is subtracted from every step so that the walk returns to its start; the loop is projected onto a plane and its crossings are read, with which strand passes over. The hero figure draws three of forty sticks. The first has twelve crossings and is unknotted; the second has eighteen and is a trefoil; the third has sixteen and is a figure-eight knot. Nothing about the pictures says which is which.

A polynomial behind the colourings built the Alexander polynomial $\Delta(t)$ from a diagram’s crossings: a square matrix with one row per crossing and entries $1 - t$, $t$ and $-1$, any of whose minors is $\Delta(t)$ up to a factor $\pm t^k$. Its value at $t = -1$ is the determinant that the colourings count. Computing the whole polynomial symbolically is expensive for a diagram with hundreds of crossings, and the census does not need it. Two values are enough to separate the small knots that random loops actually tie.

The table lists the values. The trefoil has determinant 3 and $\Delta(2) = 3$; the figure-eight has determinant 5 and $\Delta(2) = 1$; the knot $5_1$ also has determinant 5, which is why the colourings cannot tell it from the figure-eight, but $\Delta(2) = 11$. The knot $6_1$ and the trefoil tied twice, $3_1 \# 3_1$, both have determinant 9; the first has a polynomial that vanishes at $t = 2$ and the second has $\Delta(2) = 9$. Each value is computed as a determinant of the Alexander matrix with $t$ set to $-1$ or $2$, by elimination modulo the prime 67,108,859 — small enough that every product fits exactly in ordinary double-precision arithmetic — so that a polygon of four hundred sticks and several hundred crossings takes a few milliseconds. Because the minor is only $\pm t^k \Delta(t)$, the computed value at $t = 2$ is compared with each candidate times every power of two that could arise.

What two values cannot do is also clear. Mirror images share every value of the Alexander polynomial, so a left-handed and a right-handed trefoil are counted together, as a polynomial that tells left from right explained they must be by any invariant of this kind. The granny knot and the square knot — two trefoils of the same or opposite handedness tied in succession — share both values and are counted together as $3_1 \# 3_1$. A knot whose polynomial is 1, though knotted, is counted with the unknotted loops. And a polygon whose two values match none of the table’s small knots is counted as unidentified: a larger prime knot, or a composite of three or more pieces. The identification was checked on every small diagram it was given against the full polynomial, computed symbolically, and agreed on all 262.

The trefoil leads at every length #

With identification cheap, the census can be large: two thousand polygons at each length from 25 to 200 sticks, and a thousand at 300 and at 400.

The share of loops with polynomial 1 falls from 97% at 25 sticks to 26% at 400. Every knot type’s share rises from zero, and the trefoil’s rises fastest: 2.5% at 25 sticks, 13.9% at 100, 25.0% at 400. The figure-eight is a distant second, between a sixth and a quarter of the trefoil at every length, and the knots $5_2$ and $5_1$ — the simplest five-crossing knots — come next, $5_2$ the more common of the two. The order matches the knots’ complexity in the sense six sticks tie a trefoil, and five cannot measured: the trefoil needs six sticks, the figure-eight seven, and the five-crossing knots eight, and a knot that needs fewer sticks is easier for a random loop to tie by accident.

The curves also show the beginning of the turn the conjecture predicts. The trefoil’s share is still rising at 400 sticks, but the shares of $5_2$ and $5_1$ have levelled off, and the figure-eight’s is barely rising. At great lengths every individual knot type becomes rare, because the loop is then almost always a composite of several knots rather than any single one; the smaller shares reach that turn first. And the composite $3_1 \# 3_1$, the trefoil tied twice, is climbing faster than any prime knot. At 50 sticks it is rarer than all four prime knots shown; by 300 sticks it has passed all of them except the trefoil itself.

What makes a knot easy to tie at random #

The order of the census — trefoil, figure-eight, then the five-crossing knots — is the order of every measure of a knot’s complexity, and it is worth seeing why a random loop should respect it. A knot of a given type can only be tied by a stretch of the loop that does enough work. A knot must turn twice, and a little more showed that any knotted closed curve turns through more than two full circles in total, and the stick numbers say the same thing in a discrete form: a trefoil cannot be made with fewer than six straight pieces, a figure-eight with fewer than seven, and the five-crossing knots with fewer than eight. A random loop ties a knot when some short stretch of it happens to wander into one of those configurations and then closes off, and the fewer sticks a configuration needs, the more ways a random stretch has of hitting it.

The diagrams carry the same ordering in another form. Three moves, and what they cannot undo reduced the question of whether two diagrams show the same knot to Reidemeister’s three moves, and a random polygon’s diagram typically has dozens or hundreds of crossings, almost all of which those moves would remove. What survives is a handful of essential crossings — three for a trefoil, four for a figure-eight — and a random tangle with only a few essential crossings is far more likely to arise than one with many. Colours that count more than three found that the trefoil is the only knot of three crossings and the figure-eight the only one of four; there are two of five, three of six and seven of seven, and the census shows the corresponding knots appearing in that order and at shares falling with their complexity.

None of this says how much more common the trefoil should be than the figure-eight. The ratio the census measures — between four and seven trefoils for each figure-eight — is a property of the Gaussian polygon model, and other models of random loops — polygons on a lattice, or with steps of equal length — need not give the same ratio. What is common to the models is the structure: small knots first, composites later, and the shares of the individual types governed by the number of ways a short stretch of the loop can tie each of them.

The identification method matters to this ordering too. A census that used the determinant alone would have counted the figure-eight and $5_1$ together, since both have determinant 5, and would have counted $6_1$ with the double trefoil, since both have determinant 9; the second value of the polynomial is what separates them, and it is what lets the census see that the double trefoil, not $6_1$, is the knot of determinant 9 that long loops tie. At 300 sticks the census finds 66 double trefoils and 10 copies of $6_1$ per thousand polygons: a composite of two trefoils is far easier to tie at random than the prime knot of the same determinant, because it is two easy tangles rather than one hard one.

Shares divided by the unknotted share #

The conjecture concerns the rate at which each type’s probability eventually falls. A cleaner quantity to look at directly is each type’s share divided by the share with polynomial 1, which removes the common exponential factor if there is one.

On logarithmic axes the relative shares are nearly straight lines, which means each grows like a power of the length, and the powers separate into two groups. The prime knots — the trefoil, the figure-eight, $5_2$ and $5_1$ — grow with slopes between $1.13$ and $1.37$. The composite $3_1 \# 3_1$ grows with slope $2.47$, about twice the trefoil’s. The picture behind the universality conjecture predicts exactly this pattern. If a long loop is an unknotted loop carrying small knotted tangles, each tangle confined to a short stretch of the loop and able to sit anywhere along it, then the number of places a single tangle can sit grows like $n$, and the number of ways to place two grows like $n^2$. Each prime knot’s relative share should grow like $n$, and each knot with two prime pieces like $n^2$, with any common exponential decay cancelling in the ratio.

The slopes are steeper than the predicted one and two, by about a quarter for the trefoil and more for the rarer knots. That is expected at lengths where the loop is still short compared with the size of a typical tangle: a tangle of a trefoil occupies dozens of sticks of a Gaussian polygon, and a loop of a hundred sticks has room for it in far fewer than a hundred places. The predicted powers are a statement about the limit, and these loops are not yet there. What the figure establishes is the ratio. The composite’s power is about twice the prime’s, at every length the census covers.

Two trefoils as often as two independent ones #

The independent-tangle picture makes a sharper prediction, and the census can test it with no fitting at all. If trefoil tangles occur along the loop independently, with a relative rate $\lambda$ for one, then two occur with relative rate $\lambda^2/2$ — the square for two independent placements, halved because the two tangles are the same type and placing them in either order gives the same loop. The trefoil’s relative share is measured, so the double trefoil’s is predicted.

From 100 to 300 sticks the agreement is close: $0.0135$ against a prediction of $0.0161$ at 100 sticks, $0.044$ against $0.041$ at 150, $0.062$ against $0.069$ at 200, and $0.168$ against $0.154$ at 300 — each within its counting error, which is large because a few dozen double trefoils are found at each length. Nothing was adjusted to make the two curves meet; the prediction is half the square of a different measured number. At 75 sticks the double trefoil is rarer than predicted, as it should be when the loop is too short to hold two tangles comfortably. At 400 sticks it falls short again, $0.280$ against $0.448$, by more than the counting error.

These samples do not say why. One possibility is that at 400 sticks the double trefoil is increasingly accompanied by a third knot, which moves the loop into the unidentified class; another is that the two polynomial values miss some composite of the trefoil with a knot whose own polynomial is 1. A third is that the independent-tangle rule itself is only approximate for long loops, where tangles can interfere with one another. The census establishes the agreement over a range of lengths and the departure beyond it; the explanation needs a larger census with finer identification.

The trefoil’s lead among knotted loops #

The last figure looks only at the knotted loops and asks what share of them each kind of knot accounts for.

Among the knotted loops of 25 sticks, more than three in four are trefoils and the rest are figure-eights or the two five-crossing knots; no composite appears. By 50 sticks the trefoil is two-thirds of all knots, and composites have appeared, 4%. By 400 sticks the trefoil is down to a third, composites the polynomial values identify are a fifth, and a third match nothing in the table — larger prime knots or composites of three or more pieces, which the two values cannot separate. The trefoil remains the single commonest knot at every length, but the knotted loops spread over more and more types, and an increasing share of them are several knots in a single loop. That is the mechanism by which every individual knot type’s share must eventually fall: the total knotted share approaches one, and it is divided among more and more composites.

The surface a knot bounds showed that the genus of a composite knot is the sum of the genera of its pieces, and how many changes undo a knot counted the crossing changes that undo a knot, a number whose behaviour when knots are tied in succession has proved far subtler than the genus’s. For random loops the composite structure dominates the long-length behaviour, and so the additivity of invariants under tying knots in succession is not a curiosity of the theory but the reason the census looks the way it does.

Still open: the common rate #

The universality conjecture — that every knot type’s probability decays at the same exponential rate as the unknot’s, with only a power of the length in front — remains unproved for any model of random loops. The census supports its mechanism: relative shares grow like powers of the length, the composite’s power is about twice the prime’s, and two trefoils occur as often as two independent ones over the range where the loops are long enough to hold them and short enough to be counted. None of that proves the conjecture, and the measured powers are not yet the predicted ones. A proof would have to show that a random loop really does decompose into an unknotted backbone and independent local tangles — that knotting is local — and the evidence for locality, in polygon models and in DNA, is strong and entirely numerical.

The census is also only one model. Gaussian polygons are the simplest random loops to generate, and they allow steps of any length, including very short ones that can slip between other parts of the loop; random loops on a lattice, or with steps of equal length and some thickness, model real polymers more closely and knot less readily. Whether the ratios found here — the trefoil’s lead, the composite’s doubled power — carry over to those models unchanged, or only in their pattern, is a question the conjecture’s universality is meant to answer and these figures cannot.

There is a narrower question the figures raise directly. The Alexander polynomial’s two values classify the knots random loops tie up to about the size where composites of three pieces begin to matter, and at 400 sticks a third of the knotted loops are beyond them. Identifying those requires stronger invariants — the full Alexander polynomial, the Jones polynomial, or hyperbolic volumes — computed for diagrams with hundreds of crossings, which is expensive but possible. Whether the independent-tangle rule holds for three tangles as it does for two, and whether its departure at 400 sticks is real, are questions such a census could settle.