Many crossings and a small linking number

Two loops and one number defined the linking number by a recipe: draw two closed curves from some direction, give each place where one passes over the other a sign of $+1$ or $-1$ by the handedness of the crossing, add the signs and halve. Every later essay on this shelf used particular loops, chosen to make a point — the Hopf link, the Whitehead link, the Borromean rings, the periodic orbits of the Lorenz equations, the fibres of a Seifert fibration — and computed their linking numbers exactly.

This essay asks what the number is for loops nobody chose. Take two closed random walks, each of $n$ steps in three-dimensional space, both centred on the same point, so that they are as tangled together as two random objects can be. How linked are they? The question has the flavour of almost every long loop is knotted, which asked whether one random loop is knotted, but here there is a number to measure rather than a yes or no.

The figure shows the recipe at a small size. The two loops cross 56 times in this projection, 25 crossings positive and 31 negative, and the linking number is $(25 - 31)/2 = -3$. Viewed along another axis the picture is entirely different — different crossings, in different places, with a different total — and the linking number is again $-3$. Every count on this page is checked that way, in two projections at once, because the invariance proved in two loops and one number is the one property the count cannot be allowed to lose.

The figure already shows the theme. Fifty-six crossings yielded three. Most crossings cancelled in pairs, and the rest of this essay is about how much cancellation there is, why, and what happens when it is prevented.

Crossings come in pairs #

Order the crossings along one of the two loops and keep a running total of their signs. If the signs were independent coin tosses, the total would wander like a random walk, straying about the square root of the number of crossings from nought.

It does not. The total for 616 crossings stays between $-4$ and $8$, where a fair coin tossed 616 times would typically stray about 25. The ticks show why. The signs alternate far more often than chance would make them: 406 changes of sign between neighbouring crossings, against about 308 for independent signs. A crossing of one sign is usually followed, a step or two later, by a crossing of the other.

That pairing has a plain geometric source. When one loop passes over a strand of the other and then, a little further along, comes back over the same strand the other way, it makes two crossings of opposite sign. In space nothing has happened: the first loop has dipped near the second and left again without going round it, and a small push would remove both crossings together — a Reidemeister move of the second kind, the one that two loops and one number showed changes nothing. A random walk does this constantly. Its path is so crumpled that any strand of the other loop it passes over, it is likely to pass over again, nearby.

The strip of ticks has another feature worth reading: long gaps. Between steps 80 and 110, and again between 210 and 260, the first loop crosses the second not once. Those are stretches where the first walk has wandered out to its own edge, beyond anything the second loop covers in this projection, and the running total sits perfectly still across them. A random walk spends a good part of its length in excursions of that kind, and an excursion away from the other loop is a stretch that can contribute nothing to the linking.

The running total moves for good only where one loop genuinely goes round the other. Those places are rarer than crossings by a wide margin, and they are the subject of the next figure.

How the two counts grow #

Two loops of $n$ steps each, measured over many seeded pairs at each size from 16 to 1,024 steps, give two numbers: the average count of crossings in a projection, and the mean square linking number, which measures its typical size.

Take the walks first. A random walk of $n$ steps has a size of about $\sqrt n$, so its projection fills an area of about $n$; two such projections overlapping in that area, each a path of length $n$, cross each other about $n$ times, and the measured slope is $0.98$. If the signs of those $n$ crossings were independent, the mean square of half their sum would grow like $n$ as well. It grows with slope $0.56$ instead — barely faster than $\sqrt n$. At 1,024 steps the crossings number nearly a thousand and the typical linking number is about 3.4.

A comparison sharpens how small that is. Every loop of the butterfly links every other found the periodic orbits of the Lorenz equations linking each other by about an eighth of the product of their periods: two orbits of length $n$ link about $n^2/8$ times, because a deterministic flow winds every orbit round the same two holes in the same sense, and nothing cancels. Random loops are the opposite extreme. Their windings point every which way, and a winding one way is almost always undone by a winding the other way somewhere nearby.

Now the box, the second model. Its loops are made by choosing $n$ points uniformly at random in a unit cube and joining them in order. Edges are long — two-thirds of the cube’s width on average — and every edge of one loop has a fixed chance of crossing every edge of the other, so the crossings grow like $n^2$, with measured slope $2.00$. The mean square linking number grows with slope $2.03$: it keeps pace with the crossings, a fixed fraction of them, and the typical linking number grows in proportion to $n$. Javier Arsuaga and colleagues proved in 2007 that this is the true rate for uniform random polygons in a confined space.

The two models are made of the same ingredients and behave entirely differently. The box’s loops cross more and cancel less. The walks’ loops cross plenty and cancel almost everything.

Why the walks cancel so much #

The heuristic behind the slow growth is a matter of counting close approaches in space. Gauss’s integral expresses the linking number as a double integral over the two loops, and its integrand falls off like the inverse square of the distance between the two points. Contributions from pairs of points far apart are small, and they come with signs that average out as the loops wander; what remains is dominated by the pairs of points that come close.

How many such pairs are there? The answer rests on the fact that a walk in space does not come home: in three dimensions a random walk is transient, so it does not keep revisiting a neighbourhood, and its steps are spread thin. A walk of $n$ steps spreads its $n$ steps through a region of volume about $n^{3/2}$, so near any given point the density of steps is about $n^{-1/2}$. Each of the other loop’s $n$ steps therefore has about $n^{-1/2}$ steps of the first loop within a unit distance, and there are about $n \cdot n^{-1/2} = \sqrt n$ close approaches in all. If each close approach contributes a winding of either sign, at random, the linking number is a sum of about $\sqrt n$ independent terms, and its mean square grows like $\sqrt n$ — the arithmetic of the average settles and the wobble does not, applied to a count of terms that is itself only $\sqrt n$.

The same counting says why the crossings are so much more numerous. A crossing in projection needs only that two strands overlap when flattened, whatever their depths; two strands a whole radius apart in depth still cross in the picture. A close approach needs the strands near each other in all three directions at once. The crossings are counted in an area of $n$, the close approaches in a volume of $n^{3/2}$, and the ratio between the two counts, about $\sqrt n$, is the factor by which the projection exaggerates the entanglement.

The measured slope is a little above a half, at $0.56$. Approaches at larger scales — whole sections of the two walks winding round each other — add contributions of their own, which the crude count above leaves out, and at these sizes they may push the slope up; or the true exponent may be slightly larger than a half. The measurements here do not separate the two, and the heuristic is not a proof. What they do settle is the order of magnitude: the linking number of two random walks grows roughly like the fourth root of their length, far below the square root of their crossings that independent signs would give.

The same counting explains the box. There the loops are not crumpled: a uniform random polygon’s edges run across the whole cube, two edges that cross in projection are typically far apart in depth, and an edge that passes over another rarely passes back. Every pair of edges is a close approach on the scale of the cube, and there are $n^2$ of them.

What the distribution looks like #

The mean square is one number. The whole distribution of the linking number for walks of 256 steps is drawn below, from 2,000 seeded pairs.

The histogram is symmetric, as it must be: reflecting both loops in a mirror reverses every crossing’s sign and so the linking number’s, and a random walk is as likely as its mirror image. It is close to a bell, centred on nought, with spread about 2.2. But it is not quite a bell. The middle is taller, the tails are longer, and the kurtosis is $3.7$ against the bell’s $3$; pairs three spreads or more from nought turn up about three times as often as the bell would allow, 0.75 per cent against 0.27.

The heavy tails have the same source as everything else here. The linking number is a sum of contributions from close approaches, but the number of close approaches is itself random — two walks that happen to pass through each other’s densest regions have many more of them than two that merely overlap at their edges. A sum of a random number of random signs is a mixture of bells of different widths, and mixtures of bells always have heavier tails than a single one. The central limit theorem’s bell would need the number of terms to be fixed.

About one pair in five is unlinked by this measure, with linking number exactly nought, and zero can mean two different things is a warning against reading that as “not linked”. Two loops with linking number nought can be as inseparable as the Whitehead link. The linking number underestimates how often random loops are tangled, just as a single knot polynomial underestimated how often one random loop is knotted.

Linked only while they overlap #

The loops above share a centre, which makes them as entangled as two random loops can be. Slide one of them sideways and the linking should fade.

On a common centre three pairs in four are linked. Half a radius apart nothing much changes; one radius apart, 62 per cent; one and a half, 32 per cent; two radii apart, one in ten; and three radii apart only two pairs in a thousand. The curve falls in the way the overlap of two clouds of points falls as they are separated: steeply once the dense centres no longer meet. The mean square linking number falls faster still, from $3.7$ on a common centre to $1.7$ one radius apart and $0.15$ two radii apart, as one would expect if the pairs that stay linked at a distance are the ones linked by a single winding near their edges, not the deeply interleaved ones that produce the large values.

That is the same lesson as the close approaches. Linking is decided where the loops come near each other, and two random walks a few radii apart have so few points near each other that the chance of a winding is tiny, however many times their projections might still overlap from some angles. The figure also gives a sense of how unusual the box’s behaviour is: there, every pair of edges is effectively close, because the box keeps everything within one cube’s width of everything else.

A box changes the answer #

The second model deserves a picture of its own, because its loops look nothing like the walks.

The loops are made of long straight edges running across the whole square, and their 140 crossings are scattered rather than gathered in nearby pairs. An edge that crosses another seldom crosses it back, because both are straight; there is nothing like the crumpled walk’s habit of dipping under a strand and returning. With the crossings unpaired, cancellation still happens — about a quarter of the variance that independent signs would give survives — but it removes a fixed share and no more, and the linking number grows in proportion to the number of edges.

There is a deterministic cousin of this behaviour on the shelf. Six points in space and a pair that must link placed six points anywhere and found that some pair of the triangles they span must link — straight edges between scattered points, exactly the box’s ingredients, forced to entangle by a parity. The box is the random version of the same situation: straight edges between points crowded into one region cannot avoid one another, and the more of them there are, the more they link.

Physically, the box is a model of long polymers crowded into a small space: DNA packed in a virus’s head, or chromosomes in a nucleus, where the chains are far longer than the container is wide and every part of each chain can reach every part of the others. The growth of the linking number like $n$ there, rather than like $n^{1/4}$, is one reason confinement makes entanglement so much harder to avoid, and why cells spend so much effort on enzymes that pass one strand through another to undo it.

Two projections, one count, and an exponent left open #

Every linking number on the page is counted from the signed crossings in one projection and recounted in a second, along a different axis, where the crossings are different; the two agree on every pair checked. The figures also check that the signed crossings along one loop add to twice the linking number, that independent runs agree on the mean square, and that the box’s crossings and linking grow at the rates the scaling argument predicts.

What they do not check is the walks’ exponent. The slope of $0.56$ is a measurement over sizes from 16 to 1,024 steps, from a few thousand pairs at each size, and the heuristic that predicts a half is an order-of-magnitude argument, not a theorem. Nor do they say anything about whether pairs with linking number nought are separable: that would need the invariants of linked, and no two of them are, which a random sample of this size could compute but these figures do not.

Still open: the exponent, and links of many loops #

For two random walks in unconfined space, the exact growth of the mean square linking number is not settled by these figures, and the honest summary is that a careful proof of the rate, with its constant, would be needed to decide between $\sqrt n$ and something slightly faster. The analogous question for a single loop concerns the writhe, which a whole number split into two that are not met as the share of a ribbon’s linking carried by the coiling of its core: the self-crossings of one loop, signed and averaged over every direction of view. For random walks its mean square grows with length too, and has been measured and estimated by several groups; whether the two growth rates are the same law in two guises — a loop’s coiling about itself and two loops’ winding about each other, both decided by close approaches — is a natural guess and part of the same open picture.

A wider question is about many loops at once. A melt of polymers is a tangle of thousands of random loops, and its physical properties — how it flows, how it resists being pulled — depend on how entangled the loops are with one another. The linking numbers of every pair are one crude measure; the Milnor invariants of every triple, which see the Borromean kind of linking, are another; and how these quantities scale with the length of the chains, the density of the melt and the size of the container is known only in a few cases. The box and the open walk here are the two extremes, and almost everything physically interesting lies between them.