The Lorenz Flow

In 1963 Edward Lorenz cut a weather model down to three equations:

$$\dot x = σ(y - x), \qquad \dot y = x(ρ - z) - y, \qquad \dot z = xy - β z,$$

with $σ = 10$, $ρ = 28$, $β = 8/3$. Trajectories never settle and never repeat, winding around two lobes and swapping between them unpredictably — the first strange attractor, and the origin of the phrase butterfly effect.

#A single trajectory, seen the way Lorenz drew it

Drag to turn the attractor over — from the side it is a pair of wings, from above a pair of spirals. LorenzCurve(n) integrates n steps and evaluates to the points.

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Buried in that tangle are the trajectories that do close up. Each one is a loop in space — and a loop in space is a knot.

The template turns a flow into a word

Birman and Williams collapsed the attractor onto a branched surface: two ribbons joined along a branch line, each ribbon being one lobe. Squashing the flow onto it loses nothing that matters topologically, and what remains is a doubling map, $x \mapsto 2x \bmod 1$.

An orbit is then just the record of which ribbon it took each time round — a word in $L$ and $R$. Periodic orbits are periodic words, read up to rotation, because there is no distinguished starting point on a loop.

That is the whole reduction: a knot becomes a cyclic word, and the braid comes out of the word by the recipe on the previous page — order the rotations lexicographically, see where the flow sends each, take the positive permutation braid of that permutation.

#A closed orbit, as a knot

The shortest word whose orbit is knotted: its closure is the trefoil.

In$\mathrm{LorenzBraid}(\text{LLRLR})$Out$\mathrm{Braid}(5, \bigl\lbrack2, 1, 3, 2, 4, 3\bigr\rbrack)$In$\mathrm{AlexanderPolynomial}(\text{LLRLR})$Out$t^2-t+1$In$\mathrm{SeifertGenus}(\text{LLRLR})$Out$1$

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<notatio-cell value='LorenzBraid("LLRLR")' data-allow-mismatch="children" /> <notatio-cell value='AlexanderPolynomial("LLRLR")' data-allow-mismatch="children" /> <notatio-cell value='SeifertGenus("LLRLR")' data-allow-mismatch="children" />

Not every knot turns up this way. Among the 250 prime knots with ten crossings or fewer, only eight are Lorenz knots — and the figure-eight, the second-simplest knot there is, is not one of them. Lorenz knots are all positive braid closures, which is a strong condition: it forces the genus, and it rules the figure-eight out immediately, since that knot is amphichiral and a positive braid closure never is.

The same knots, from lattices

Now the surprise. Take the space of lattices in the plane with unit-area fundamental domain. Each lattice $L$ has two classical invariants,

$$g_2(L) = 60!!\sum_{ω ∈ L ∖ 0}!! ω^{-4}, \qquad g_3(L) = 140\!\!\sum_{ω ∈ L ∖ 0}\!\! ω^{-6},$$

and the discriminant $Δ = g_2^3 - 27 g_3^2$ vanishes exactly where the lattice degenerates. The set where it vanishes, inside the three-sphere of normalised lattices, is a trefoil — so the space of genuine lattices is a trefoil complement, which is to say the unit tangent bundle of the modular surface.

Let the flow be $\varphi_t(L) = e^t L$, stretching one direction and squeezing the other. Its closed orbits correspond to conjugacy classes of hyperbolic matrices in $\mathrm{PSL}(2,\mathbb{Z})$ — which, as the modular page shows, are exactly cyclic words in two letters.

Ghys's theorem. Those modular knots are the Lorenz knots. Every periodic orbit of the lattice flow can be deformed to a periodic orbit of the Lorenz equations, and the word in $L$ and $R$ is the same word either way.

#One word, both readings

LLRLR is L²R¹L¹R¹, so its run lengths are 2, 1, 1, 1 — and those are exactly the partial quotients of a continued fraction. Here it is as a matrix in PSL(2,ℤ), and as the braid whose closure is its knot.

In$\mathrm{ModularFromSTWord}(\bigl\lbrack2, 1, 1, 1\bigr\rbrack)$Out$\mathrm{ModularMatrix}(-1, -2, 0, -1)$In$\mathrm{LorenzBraid}(\text{LLRLR})$Out$\mathrm{Braid}(5, \bigl\lbrack2, 1, 3, 2, 4, 3\bigr\rbrack)$In$\mathrm{TripNumber}(\text{LLRLR})$Out$2$

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<notatio-cell value="ModularFromSTWord([2, 1, 1, 1])" data-allow-mismatch="children" /> <notatio-cell value='LorenzBraid("LLRLR")' data-allow-mismatch="children" /> <notatio-cell value='TripNumber("LLRLR")' data-allow-mismatch="children" />

Linking with the trefoil counts the letters

The trefoil is still sitting there, as the discriminant locus. Every modular knot links it some number of times, and that number has a name in a much older subject: the Rademacher invariant, from the theory of the Dedekind eta function.

It is computed by pushing the knot through a Seifert surface for the trefoil and counting signed punctures — and the answer is simply

$$\operatorname{lk}(k_A, \text{trefoil}) = \#R - \#L,$$

the letters of the word, counted with sign. One ribbon of the template links the trefoil once positively, the other once negatively, so the linking number can only be the difference. A number theorist's invariant of a modular transformation, and a topologist's count of how a curve winds — the same integer.

#Rademacher, as a letter count

Dedekind sums are the arithmetic side of the same invariant; the modular page follows them further.

In$\mathrm{DedekindSum}(1, 5)$Out$\frac{1}{5}$In$\mathrm{DedekindSum}(2, 7)$Out$\frac{1}{14}$

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<notatio-cell value="DedekindSum(1, 5)" data-allow-mismatch="children" /> <notatio-cell value="DedekindSum(2, 7)" data-allow-mismatch="children" />

Things worth knowing

• Every torus knot is a Lorenz knot. $(σ_1 \cdots σ_{p-1})^q$ is positive, and the template realises every positive braid of that shape — see torus knots.
• Lorenz knots are fibred, with genus fixed by the braid, because a positive braid closure always is.
• Trip number is braid index. The number of $LR$ corners in the cyclic word is the minimal number of strands, so trip number $1$ means the unknot however long the word.
• The doubling map is why the words are all words: any cyclic binary word occurs, so there are as many Lorenz knots as there are necklaces.

What this connects to

The modular group supplies the words and the arithmetic; knots and braids supplies the braid machinery. This page is the claim that those are one subject, which is Ghys's theorem, and the trefoil is where all three meet: the discriminant locus, the first torus knot, and the first Lorenz knot. $$