Torus knots that fill the three-sphere

Every loop of the butterfly links every other found that the periodic orbits of the Lorenz equations are all linked with one another, and that the linking number grows with the lengths of the two orbits. It ended on Étienne Ghys’s discovery that Lorenz knots are the same as the “modular knots” of the space of lattices, and on the question of what the linking numbers can say about the flow. This essay looks at the simplest flow on a three-dimensional space in which every orbit is closed and every orbit is a knot, and in which the linking numbers can be found exactly: a flow on the three-sphere whose orbits are torus knots.

The starting point is the fibration Heinz Hopf found in 1931, which the circles that fill a three-sphere drew. Write the three-sphere as the pairs of complex numbers $(z_1, z_2)$ with $|z_1|^2 + |z_2|^2 = 1$, and move every point by $(z_1, z_2) \mapsto (e^{it} z_1, e^{it} z_2)$. Each point traces a great circle, the circles fill the sphere, no two meet, and every two are linked exactly once. Now change one thing. Let the two coordinates turn at different rates:

$$(z_1, z_2) \mapsto \left(e^{ipt} z_1,\ e^{iqt} z_2\right),$$

with $p$ and $q$ whole numbers with no common factor.

Every orbit a torus knot #

The motion keeps $|z_1|$ and $|z_2|$ fixed, so a point with both coordinates non-zero stays on the torus where $|z_1|$ has its value, and on that torus it turns $p$ times round one way while it turns $q$ times round the other before closing up. A curve on a torus that winds $p$ times one way and $q$ times the other is the $(p, q)$ torus knot; with $p = 2$ and $q = 3$ it is the trefoil. So every orbit through a point with both coordinates non-zero is a trefoil, on its own torus, and these orbits fill everything except two circles.

The two exceptions are where one coordinate vanishes. On the circle $z_2 = 0$ the point $(z_1, 0)$ simply turns, and on the circle $z_1 = 0$ likewise. These two orbits are plain unknotted circles, and they are special in another way: going once round $z_2 = 0$ takes only $1/p$ of the time the regular orbits take, and going round $z_1 = 0$ takes $1/q$ of it. Herbert Seifert classified such decompositions in 1933, and spaces filled by circles in this way, with finitely many exceptional circles that close up early, are called Seifert fibred spaces; the three-sphere with this action is the simplest of them after Hopf’s.

The orbits can be described without the motion as well. The quantity $z_1^q / z_2^p$ is unchanged by the motion, since $z_1^q$ picks up $e^{ipqt}$ and so does $z_2^p$, and each value of it picks out exactly one orbit. The orbits are the level sets of a single complex function, as Hopf’s circles are the level sets of $z_1/z_2$.

The space of fibres #

Every orbit is a circle, so the set of orbits is itself a space, and it is worth knowing what it looks like. Each orbit is picked out by the value of $z_1^q / z_2^p$, which can be any complex number or infinity, so the orbits correspond to the points of a sphere — the Riemann sphere, which stereographic projection identifies with the plane plus one point. Two of its points are special. The value 0 belongs to the exceptional circle $z_1 = 0$ and the value $\infty$ to $z_2 = 0$, and near those points the sphere of orbits is not smooth but has a cone, of angle $2\pi/q$ at one and $2\pi/p$ at the other, because nearby regular orbits wind round the exceptional one several times before closing.

A sphere with two cone points has an Euler characteristic that counts each cone point fractionally: $2 - (1 - 1/p) - (1 - 1/q) = 1/p + 1/q$. Seven hundred and twenty degrees of gap found that a surface’s angle deficits add to $2\pi$ times its Euler characteristic, and the same holds here with the cones’ missing angles $2\pi(1 - 1/p)$ and $2\pi(1 - 1/q)$ counted in. The fractional Euler characteristic is positive, which places the three-sphere, fibred this way, among the Seifert fibred spaces whose geometry is that of the round sphere. With three cone points of orders 2, 3 and 5 the count is $2 - \tfrac12 - \tfrac23 - \tfrac45 = \tfrac1{30}$, still positive, and one of the spaces fibred over that orbit space is Poincaré’s dodecahedral space, the famous three-dimensional space with the same homology as the sphere and a different shape.

Linked pq times #

Two regular orbits are disjoint closed curves, so they have a linking number, and it is the same for every pair: $pq$.

The computation behind the table makes no use of the formula it confirms. It does have to cope with the projection. Stereographic projection sends the three-sphere to ordinary space by sending one point to infinity, and an orbit that passes near that point is drawn as a curve that swings far out and comes back — the long arcs in the hero figure are fibres passing close to the point at infinity. Linking numbers do not care: projection is a continuous one-to-one map away from the single point, and two curves that avoid it link in space exactly as they linked on the sphere. Each orbit was sampled at 720 points, the three-sphere was turned by a fixed rotation of four-dimensional space so that no orbit runs through the point from which it is projected, and the points were projected stereographically into ordinary space, where the orbits become closed curves — some of them very long, swinging far out and back. The linking number of each pair of resulting polygons was then computed by an exact formula for polygons, described below. For Hopf’s fibration, $p = q = 1$, the answer is the familiar 1. For the trefoils it is 6, for the $(3, 4)$ torus knots 12, for the $(4, 5)$ knots 20. Every pair taken, on any tori, at any angles, gives the same number.

Windings round the two circles #

The reason is a pair of windings. Two loops and one number defined the linking number, and one of its equivalent forms is a winding number: the linking number of a curve with the circle $z_2 = 0$ is the number of times the coordinate $z_2$ winds round nought as the curve is traversed, because that circle is exactly where $z_2$ vanishes.

A regular orbit has $z_2 = b\,e^{i(\beta + qt)}$, which winds $q$ times as $t$ goes once round, so the orbit links the circle $z_2 = 0$ exactly $q$ times; by the same argument it links $z_1 = 0$ exactly $p$ times. The two exceptional circles link each other once, for every $p$ and $q$ — they are a Hopf link. And two regular orbits link $pq$ times. One way to see it: push one regular orbit outward through the tori until it reaches the exceptional circle $z_2 = 0$, never crossing the other orbit, which stays on its own torus. The linking number does not change during the push. At the end the pushed orbit has become the circle $z_2 = 0$ traversed $p$ times, since a regular orbit near that circle winds round it $p$ times before closing; and the other orbit links that circle $q$ times. So the linking number is $p \cdot q$.

The number has a second name. The map that sends each point of the three-sphere to its orbit, a point of the Riemann sphere, is a map from a three-sphere to a two-sphere, and Hopf attached to every such map a whole number, its Hopf invariant: the linking number of the curves that sit over two different points. For Hopf’s own fibration it is 1, and that was the first proof that a map from a three-sphere to a two-sphere can be essential — impossible to shrink to a point — even though the three-sphere has no holes a two-sphere could wrap round. For the map here, sending $(z_1, z_2)$ to the orbit labelled $z_1^q / z_2^p$, the Hopf invariant is $pq$, and every whole number arises as a Hopf invariant of some map, $pq$ among them. The table above is a measurement of Hopf invariants, ten of them, made from drawn curves.

Counting crossings in one drawing #

The linking number can also be read from a single flat drawing, by counting crossings.

Flatten the two projected curves onto a plane and look at the places where one passes over the other. At each such crossing the directions of the over-strand and the under-strand give a sign, and half the sum of the signs is the linking number. In the drawing above the two trefoils cross twelve times and all twelve crossings have the same sign — two curves that turn the same way round, as all orbits of one motion do, cross one another always in the same sense — so the linking number is 6, agreeing with the exact computation. Counting crossings is the definition two loops and one number began with; the agreement of the two methods is the theorem that the count does not depend on the drawing.

Six trefoils, one number #

The figure below takes six fibres, from one hugging the exceptional circle $z_1 = 0$ to one hugging the other.

Every one of the fifteen pairs links six times. Two neighbours on adjacent tori and two fibres at opposite extremes behave identically. That is exactly what the pushing argument predicts — any regular fibre can be moved to any other through regular fibres — and it is a strong statement about the structure. The sphere is filled with knots so uniformly that the linking of any two is a property of the whole decomposition, not of where the pair sits. In Hopf’s fibration the same uniformity gives the number 1, and a rotation of four-space takes two of them explained it by symmetry: the rotations of four-space that carry Hopf’s circles to each other act transitively. Here the circle action is still there, but the uniformity of $pq$ comes from the windings, not from a symmetry carrying every fibre to every other.

A fibre and its neighbour #

There is one more linking number hiding in the fibration: a fibre with a fibre right beside it. Push a regular fibre off itself a tiny amount along the fibration, to the neighbouring orbit, and the two link $pq$ times, as any two fibres do. That gives each trefoil a preferred way of being thickened into a ribbon — the ribbon whose two edges are the fibre and its neighbour — and the ribbon twists $pq = 6$ times relative to the ribbon a surface bounded by the knot would give. A whole number split into two that are not split such a ribbon’s linking number into writhe and twist; here the whole number is fixed at six by the fibration, and how it divides between the two depends only on how the trefoil is drawn. The difference between the fibration’s ribbon and the surface’s is exactly what makes these trefoils behave so uniformly: it is a single number for the whole structure.

An exact formula for polygons #

The linking numbers in the tables were computed from polygons with 720 vertices, and they came out as whole numbers to six decimal places. That is not an accident of fine sampling.

Gauss’s formula expresses the linking number of two smooth curves as a double integral, and the obvious way to compute it is to sample both curves and add up the integrand — the cool curve in the figure, whose error falls slowly, still about a hundredth at 384 points. But two closed polygons are themselves a pair of disjoint closed curves and have a linking number of their own, a whole number, which agrees with that of the smooth curves as soon as the polygons are fine enough not to pass through each other. Konstantin Klenin and Jörg Langowski gave in 2000 a formula that computes it exactly: for each pair of segments, one from each polygon, the four points span a tetrahedron, and the solid angle under which one segment sees the other is a sum of four arcsines. Adding the solid angles over all pairs and dividing by $4\pi$ gives the polygons’ linking number, a whole number up to rounding. In the figure it is exactly 12 from 24 points on.

Torus knots everywhere #

The torus knots that fill the sphere here are the same objects earlier essays met in other guises. Joan Birman and Robert Williams showed in 1983 that every torus knot occurs as a periodic orbit of the Lorenz equations, on the template the butterfly essay used, and Ghys’s identification of Lorenz knots with modular knots puts the trefoil in a special place there too: the space of lattices in the plane is the complement of a trefoil in the three-sphere, and the trefoil is what the Lorenz template’s missing orbit looks like. The fibration here is a much tamer flow than Lorenz’s — every orbit closed, every pair linked the same number of times — and it is the model against which the Lorenz linking numbers, which grow with the orbits’ lengths and vary from pair to pair, can be measured. In the Lorenz flow two orbits of lengths $m$ and $n$ link about $mn/8$ times, a product of lengths with a constant in front. In the fibration every regular orbit has the same period, and the linking number is again a product — of the two windings $p$ and $q$ — with no constant in front and no variation at all. It is the tidy case against which the statistics of the Lorenz linking numbers can be read: there, spread about a product; here, the product exactly, for every pair.

The fibration also connects to the oldest appearance of the trefoil in mathematics: as the place where a curve with a cusp meets a small sphere. The curve $z_1^3 = z_2^2$ in two complex dimensions has a singular point at the origin, and a small three-sphere around it meets the curve in a trefoil — one of the fibres of the $(2, 3)$ fibration, since the trefoil is exactly where $z_1^3 / z_2^2$ equals 1 on the sphere. John Milnor’s study of singularities starts from that picture.

Still open: linking numbers of flows in general #

For Seifert fibrations everything here is exact: the fibres, their knot types, their linking numbers. The open questions are about flows that are not so tidy. For a general flow on the three-sphere, the periodic orbits may be linked in arbitrarily complicated ways, and which patterns of knotting and linking can occur among the periodic orbits of a smooth flow with given properties is understood only for special classes — flows carried by templates like Lorenz’s, or flows that preserve a geometric structure. For flows with infinitely many periodic orbits, the asymptotic linking of long orbits — the average linking number per unit length, which Vladimir Arnold proposed as an invariant of a divergence-free field in 1974 — is known to equal the field’s helicity, but how the linking numbers of individual long orbits spread about that average is, as for Lorenz, a question with few answers.

A product of two windings #

Hopf’s fibration fills the three-sphere with circles linked once. Turning the two coordinates at different rates, $p$ and $q$, fills it instead with $(p, q)$ torus knots — trefoils for $(2, 3)$ — and leaves two exceptional circles that every knot winds round $q$ and $p$ times. The linking number of any two knots is the product, $pq$, the same for every pair: computed from drawn curves for ten fibrations, by an exact formula for polygons, it came out as $pq$ every time, and twelve crossings of one sign in a single drawing gave the trefoils’ six.