Every loop of the butterfly links every other

The linking number of two closed curves in space is a whole number that survives every deformation in which the curves do not pass through each other. So far in this collection the curves have been drawn by hand, chosen to make a point: two rings, three Borromean rings, the triangles in a graph forced to hook together however the graph is placed. Here the curves are not chosen at all. They are produced by a differential equation, and the question is what that equation does to their linking.

The equation is Lorenz’s, from 1963:

$$\dot x = 10(y - x), \qquad \dot y = x(28 - z) - y, \qquad \dot z = xy - \tfrac83 z.$$

Its solutions wander forever over a two-lobed set without settling or repeating. But among the solutions that never settle there are infinitely many that do repeat — periodic orbits, closed loops in three-dimensional space, unstable, each one invisible in a simulation because the slightest error throws the solution off it. Each closed loop is a knot, and any two of them form a link. In 1983 Joan Birman and Robert Williams showed what those links must be, and the answer has a property no hand-drawn example in this collection had: every pair of these loops is linked, and always the same way round.

Naming a loop by the lobes it visits #

A trajectory of the Lorenz equations winds round one of two unstable centres — call them left and right — a few times, then flips to the other, winds there, and flips back. A periodic orbit does this in a fixed pattern and then repeats, so it can be named by the sequence of lobes it visits on one circuit. The simplest goes once round each lobe: LR. Next come LLR and LRR, then LLLR, LLRR, LRRR, and at period five the six words LLLLR, LLLRR, LLRLR, LLRRR, LRLRR and LRRRR.

Finding these loops in the equations is a computation in its own right. Each loop crosses the plane $z = 27$ on its way down once per circuit of a lobe, so following a solution from that plane back to it defines a return map, and a periodic orbit of period $n$ is a point that this map brings back after $n$ returns. A long chaotic trajectory passes near every such point eventually; taking its near-returns as starting guesses and refining them by Newton’s method finds the loops to ten decimal places. Ten of the twelve words of length up to five gave loops that close up and visit the lobes in exactly the order their names say. The other two, LLLLR and its mirror LRRRR, did not converge to an orbit with that name from the starting guesses available. The periods grow with the length of the word, about three-quarters of a time unit per lobe: 1.5587 for LR, 2.3059 for LLR and LRR, between 3.02 and 3.08 for the three of length four, and 3.82 to 3.87 for those of length five. Mirror-image words have identical periods to the last decimal computed, as they must, since turning the whole picture half a turn about the vertical axis carries the equations to themselves and swaps the two lobes.

The drawing is a projection, and in it the loops cross many times. Those crossings are where the linking number is counted — but in this view the crossings come with both signs, and the count is not visible by eye. It needs either an integral or a better picture.

The integral, done by computer #

The first way is Gauss’s. The linking number of two closed curves $A$ and $B$ is

$$\mathrm{lk}(A, B) = \frac{1}{4\pi} \oint_A \oint_B \frac{(\mathbf a - \mathbf b) \cdot (d\mathbf a \times d\mathbf b)}{|\mathbf a - \mathbf b|^3},$$

a double integral that Gauss wrote down in 1833 to count how often an orbit winds through another. Nothing in it is discrete; it is a continuous function of the two curves integrated over every pair of points. And its value is always a whole number, because it measures the degree of the map that sends each pair of points to the direction between them.

For two computed Lorenz loops, each sampled at a few thousand points along one period, the integral can be summed segment by segment. It knows nothing about lobes, words or templates; it is just geometry.

The forty-five integrals land on whole numbers — the farthest of them is $0.024$ away, and that is the closest pair of loops, where a finite sampling of the integrand is least accurate. The integers run from 1 to 5. None is nought and none is negative. That last fact is the theorem, and the horizontal axis says where it comes from: each integer is also predicted, before any integration, from the two names alone.

The template, and the braid it makes #

The second way to count is the better picture. Most of the Lorenz attractor is very nearly a surface: the flow squeezes volumes so hard that the attractor is almost two-dimensional, and if the thin direction is collapsed what remains is a branched surface — two bands, one round each lobe, joined along a single line where they meet and fold over one another. This is the Lorenz template. Birman and Williams proved that every periodic orbit of the flow can be pushed onto the template without passing through any other, so the knots and links of the orbits are those of loops drawn on that surface.

On the template, an orbit crosses the branch line once per circuit of a lobe. The order of those crossing points along the line is decided by the words: list every rotation of every orbit’s name, read each as an infinite repeating word, and sort them in dictionary order with L before R. One trip round a lobe moves a point to the place of the same word shifted one letter, so the return to the branch line is a braid: each point is joined to its image.

Two things make this braid special. The return map on each band preserves order — two points on the left band arrive in the order they left — so two strands from the same side never cross. Every crossing is a strand from the left band passing a strand from the right band, and the template’s fold fixes which goes over. So every crossing in a Lorenz braid has the same sign. A braid with all crossings of one sign is called positive, and the closure of such a braid has linking numbers that are simply half the number of crossings between the two components.

For LR and LLR the five points sort as LLR, LRL, LR on the left, RLL and RL on the right. Two strands cross between the two orbits, so their linking number is one — the same one the Gauss integral found from the computed loops.

Why dictionary order #

The sorting rule looks arbitrary and is not; it is the one place where the dynamics enters the combinatorics, and it is worth checking on the example.

A point on the branch line is followed forward, and the word read from it records which band it takes at each return. Two points that start close together take the same band for a while, because the return map is continuous on each band; they first separate at the letter where one goes left and the other right. The return map on each band is also increasing: the band is stretched and laid back over the branch line without being turned over. So if two points take the same first band, their order after one return is the same as their order before. Repeat that, and the order of two points on the line is decided by the first letter where their words differ — exactly the rule for ordering words in a dictionary, with the left band’s letter first.

For LR and LLR the five words to be sorted are the rotations LR and RL of the first orbit, and LLR, LRL and RLL of the second, each read as an infinite repetition. Three start with L. Among them LLR comes first, because its second letter is L. LRL and LR agree for three letters — LRL·LRL… against LR·LR·LR… — and part at the fourth, where LRL repeats with L and LR continues with R; so LRL comes before LR. The two words starting with R sort the same way: RLL before RL. Shifting each word by one letter sends LLR to LRL, LRL to RLL, LR to RL and so on, and those are the strands of the braid in the figure above.

A flow whose bands were turned over on the way round would reverse the order on that band, and the dictionary would have to be read with a twist on every letter from that band. That is the kind of template other flows have, and on such templates crossings of both signs can occur. The Lorenz flow is untwisted on both bands, which is the whole reason its braids are positive.

Forty-five pairs, all positive #

The same count, done for every pair among the ten orbits found, gives a table.

The table has structure that can be read off the words. LR, the shortest loop, links every other loop once or twice. Mirror pairs — LLR and LRR, LLLR and LRRR — link each third orbit in the mirrored way, because the equations are symmetric under turning the picture half round the vertical axis. The largest entries, five, come from pairs of period-five orbits such as LLLRR with LLRLR, whose words interleave so that many left-strands pass many right-strands. And nowhere is there a nought.

That absence is the striking thing, and it is worth seeing why it could have been otherwise. Linking number nought is what two unlinked loops have, and it is also what many linked configurations have when their crossings cancel. A flow could easily produce loops that pass on either side of each other with crossings of both signs. The Lorenz flow cannot, because the template has only one kind of crossing. Cancellation is impossible, so any two orbits that cross at all on the template are linked — and every orbit visits both lobes, which gives every pair the chance to cross; in every pair counted below, they do.

Every pair, up to length eight #

The template makes the computation free, so it can be run far past the orbits that are easy to find numerically.

All 2,346 pairs link, at least once. The numbers grow with the lengths of both words, roughly in proportion to their product, and on the longer pairs the ratio settles near one eighth. There is a heuristic for the eighth. A strand of one orbit and a strand of the other cross only if one leaves the left band and the other the right — a chance of about a half — and if their images come back inverted, about another half if the order on the branch line were random. Counting crossings in both directions and halving to get the linking number leaves $n_1 n_2 / 8$. The scatter around that line is the template’s order not being random: words that interleave their L’s and R’s finely cross more than words in long blocks.

The loops are knotted too #

A single orbit’s own braid answers a different question: whether the loop is knotted. The closure of a positive braid with $n$ strands and $c$ crossings bounds a surface of genus $(c - n + 1)/2$ — the Seifert surface built from the braid has exactly that many handles, and for a positive braid no surface does better. Genus nought means unknotted.

No orbit of period four or less is knotted. At period five, LLRLR and LRLRR have genus one, and a positive braid of genus one closes to the trefoil — so the Lorenz equations contain trefoil-knotted periodic orbits, and both of them were among the ten found in the equations themselves. By period eight most orbits are knotted and the genus reaches four. Birman and Williams proved that every Lorenz knot is fibred and that every torus knot occurs among them, and Étienne Ghys later identified the Lorenz knots exactly with the modular knots, the closed geodesics on the space of lattices in the plane — an identification that connects a chaotic flow to the arithmetic of the modular group by a route nobody expected. The figure-eight knot, which is not a positive braid, never appears.

A model proved right, and ten loops found by computer #

The template is a model. Birman and Williams derived it assuming that the Lorenz flow behaves as its geometric idealisation does: that the attractor collapses along a stable direction onto a branched surface with an expanding return map. Whether the actual equations at these constants do this was an open problem for decades — it was the fourteenth of Stephen Smale’s problems for the century — and was settled by Warwick Tucker’s computer-assisted proof in 2002. With Tucker’s theorem, the template’s conclusions apply to the real flow, and in particular every pair of its periodic orbits has positive linking number.

What the computations here add is independent evidence for the individual numbers, not a proof. The ten orbits were found numerically, to ten decimal places on the return plane, and the Gauss integrals were summed over a few thousand segments each; agreement to within $0.024$ of an integer on all forty-five pairs is the evidence that the computed loops are the orbits the words name. The orbits whose Newton iterations did not converge, LLLLR and LRRRR, are not shown to be absent; the method found no loop with those names from the guesses it had. And the table of 2,346 pairs is computed on the template alone: it describes the equations only through the theorem that the template is right.

Still open: what the linking numbers can tell about the flow #

The template turns topology into combinatorics, and the combinatorics has questions that are not settled. One is quantitative: the linking number of two orbits grows like an eighth of the product of their periods on average, and the full distribution of linking numbers among orbits of given lengths — how it spreads about that line, and how the spread depends on the words — has no closed description. Ghys’s identification of Lorenz knots with modular knots points to where an answer might come from: he showed that the linking number of a modular knot with the trefoil — which sits naturally beside the space of lattices — is a classical arithmetic quantity, the Rademacher function of the matrix the geodesic comes from. William Duke, Özlem Imamoḡlu and Árpád Tóth later expressed the linking number of two modular knots through integrals of modular forms. That converts the statistical question into number theory without, so far, answering it.

The other question runs the opposite way. The linking of periodic orbits is a property of the flow that survives every deformation which keeps the orbits from colliding, so it constrains which flows can be deformed into which. For the Lorenz equations at other constants, where the attractor is no longer of Lorenz type, the periodic orbits still exist but the template changes, and which links persist across the change is known only in examples. A complete account of how the knotting and linking of periodic orbits changes as a parameter moves — which orbits are born together, which must die in pairs — is one of the directions of the subject that remains open beyond the cases worked out by hand.

A number fixed by the shape of the flow #

The linking number began in this collection as a count over a hand-drawn diagram, a number that survives deformation and can be nought for two different reasons. In the Lorenz flow it cannot be nought at all. The flow folds one band over the other in only one way, and that single fold fixes the sign of every crossing in every braid the flow makes, so the whole infinite family of periodic loops is linked together, each pair positively.

The two ways of computing it make the point from both ends. The Gauss integral is pure geometry, summed over thousands of segments of loops found by solving the equations; it knows nothing of words. The template count is pure combinatorics, a sort of words in dictionary order; it knows nothing of the equations. That they agree, pair after pair, on numbers like 3 and 5 is the theorem of Birman and Williams seen from outside — and a reminder that how fast nearby orbits part is not the only thing a chaotic flow fixes. It also fixes, exactly and for ever, how its repeating loops are woven together.