Two loops and one number

Two closed curves in space either come apart or they do not, and the question of which is surprisingly hard to make precise. Come apart means there is a way of moving them, without either passing through itself or through the other, that ends with them in separate boxes — a statement about infinitely many possible motions, and no drawing settles it.

What settles it, at least in one direction, is a number that can be read off any picture with arithmetic a child could do — and which turns out not to depend on the picture at all.

The recipe #

Orient each curve: choose a direction to travel round it, and draw an arrow. Now look at the places where the two curves cross in the picture, ignoring any crossings of a curve with itself. At each such place one strand passes over the other, and the two directions of travel form a frame. Give the crossing $+1$ if turning the under-strand’s direction anticlockwise by less than half a turn brings it onto the over-strand’s direction, and $-1$ if the turn goes the other way.

Add the signs and halve. That is the linking number.

Halving is not a fudge. Two closed curves cross each other an even number of times in any projection — going into the region enclosed by the other and coming out again — so the signed sum is even, and the halved value is a whole number. The figures check both facts on every link they draw rather than asserting them.

Three things about the recipe deserve saying at once. It needs orientations: reversing one curve’s arrow flips every sign and negates the number. It ignores self-crossings entirely, so it says nothing about whether either curve is knotted. And it is arithmetic on a drawing — which is exactly why it is startling that the answer is a property of the curves.

Why the drawing does not matter #

Every planar picture of a pair of curves in space is one projection among many, and pushing the curves about changes the picture. The claim is that the halved signed sum survives all of it.

The standard proof is a case analysis over the three ways a diagram can change without the curves passing through each other — the Reidemeister moves — and it is short because two of the three cases are trivial.

The first move adds or removes a kink in one strand, which is a self-crossing, and self-crossings are not counted. The third move slides a strand across a crossing of the other two, which changes where the crossings are but not which strands cross or with what signs. The second move pushes one strand across another, creating or destroying two crossings at once — and those two always have opposite signs, because the strand comes back the way it went. So the signed sum changes by $+1-1$, which is nothing.

That is the whole argument, and it is the model for how every invariant in the subject is proved to be one: exhibit a quantity, check it against the three moves, and it is then a property of the curves rather than of the paper. The figure above measures the second move happening — the crossing count going from two to four while the signed sum stays put — rather than only describing it.

The other definition, which never looks at a picture #

There is a completely different way to compute the same number, and it belongs to the eighteenth century rather than the twentieth.

Gauss, working on terrestrial magnetism, wrote down a double integral over the two curves:

$$\operatorname{lk} = \frac{1}{4\pi}\oint\!\!\oint \frac{(\mathbf{r}_1 - \mathbf{r}_2)\cdot(\mathrm{d}\mathbf{r}_1 \times \mathrm{d}\mathbf{r}_2)}{|\mathbf{r}_1 - \mathbf{r}_2|^3}.$$

Nothing in it mentions a projection, a crossing or a sign; it is an integral over pairs of points, one on each curve, and it converges because the numerator vanishes to the right order when the points approach each other. Its value is always a whole number, which is not obvious from looking at it.

Every figure in this essay computes both quantities and requires them to agree to two decimal places. That agreement is the evidence that the drawing is of the link it claims to be — a diagram is only as good as the three-dimensional data behind it, and here the data is a parametrisation and the depths come from it rather than from a convention imposed on the picture.

The integral has an interpretation that makes the whole number visible. Think of one curve as carrying a current; then the integral is the circulation of the magnetic field it produces around the other curve, and Ampère’s law says that circulation counts how many times the second curve encircles the first. Which is the same statement as the crossing count, arrived at from physics. It is also, viewed from the plane, the winding number with one dimension added: how many times one loop goes round another, when round has to be defined in space.

What it is good for #

The number’s usefulness is entirely in one direction. If two curves have linking number other than zero, they cannot be separated — because separated curves can be projected with no crossings at all between them, giving a signed sum of zero, and the number is unchanged by any motion that gets them there.

That gives an honest proof of inseparability, which is the sort of thing that is hard to come by. The alternative — trying every motion — is not a proof of anything, and the eye is a poor judge: a pair of curves can look hopelessly tangled and come apart in one move.

The number also distinguishes links from each other, coarsely. A pair with linking number $2$ is not the same link as a pair with linking number $1$, whatever either looks like. And because the number changes sign when one component is reversed, it detects something about orientation as well: a link and its mirror image have linking numbers of opposite sign, so a link with non-zero linking number is not the same as its own mirror image as an oriented link.

What it cannot see #

The limitation is sharp, and there is a famous example of it.

Every pair of the Borromean rings has linking number zero, and the three cannot be separated. That is not a failure of arithmetic; the pairwise numbers are genuinely zero, because any two of the rings really can be pulled apart once the third is removed. What holds the three together is a relation among all three at once, and a quantity computed two at a time cannot see it.

The same limitation appears with only two components. The Whitehead link has two components clasped through each other, its diagram has four crossings between them, and its linking number is zero — the crossings cancel in pairs. It cannot be separated, and proving that needs an invariant of a different kind altogether: a Milnor invariant, or the Alexander polynomial, or an argument about the fundamental group of the complement.

So the honest summary is that the linking number is a lower bound on tangling. Non-zero proves inseparable; zero proves nothing. That is the standing situation with topological invariants, and it is why the subject accumulated so many of them: each is easy to compute and blind to something, and the working method is to try several.

Two curves, one surface #

A second reading of the number explains where its whole-numberedness comes from, and it is the reading that generalises.

Take one of the two curves and find a surface whose boundary it is — a Seifert surface, which always exists for any closed curve in space. Now count how many times the other curve passes through that surface, with a sign for the direction of passage. That count is the linking number.

The equivalence with the crossing recipe is not hard to see: pushing the second curve about changes where it punctures the surface, but a new puncture can only appear together with one of the opposite sign, since the curve is closed and must come back. And the count is manifestly a whole number, because punctures are counted rather than measured.

This is the version that says what the number means. It is an intersection count between a curve and a surface, and intersection counts modulo the freedom to deform are what algebraic topology is built out of. The same construction, one dimension up or down, gives the winding number of a loop around a point and the degree of a map between spheres; and the sum of the indices of a vector field’s zeros on a surface is the same kind of accounting, with the surface’s own topology as the thing being counted against.

The number as a degree #

There is a third definition, and it is the one that says why the answer is a whole number without any counting at all.

Given two disjoint closed curves, consider the map that takes a pair of points — one on each curve — to the unit vector pointing from the first to the second. The pairs form a torus, since each point runs round its own circle; the unit vectors form a sphere. So the configuration is a continuous map from a torus to a sphere, and every such map has a degree: the number of times, counted with sign, that it covers the sphere.

That degree is the linking number, and Gauss’s integral is the formula for it — the integrand is the area element pulled back from the sphere, and dividing by $4\pi$ turns total area into a count of coverings. The whole-numberedness is now free: a degree counts preimages of a generic point, and a count of points is an integer.

This reading also explains why the number is unchanged by deformation. Moving either curve deforms the map continuously, and the degree of a continuous map does not change under continuous deformation — the same fact that makes the winding number of a loop an integer immune to wobbling, and the same fact that forces a fixed point when a degree cannot be zero. Three definitions, three subjects: a diagram count from knot theory, an integral from electromagnetism, a degree from homotopy theory. That they agree is the reason the object is worth a name.

Where it came from #

Gauss wrote the integral in an 1833 notebook, in a page on electrodynamics, without a proof that it is a whole number and without an account of what it measures. Listing, his student, gave the subject its name — topology — and studied linking; Maxwell knew the integral and its magnetic reading. But the systematic treatment waited for the twentieth century, when Reidemeister’s moves turned diagram arguments into proofs and it became possible to say what an invariant is.

The physical reading has not gone away. Linking numbers count the entanglement of magnetic field lines in plasmas, the supercoiling of closed loops of DNA, and the crossings enzymes must undo to separate two daughter loops after replication; in each case the number is a conserved quantity and the physics is the process that changes it, which it can only do by cutting. That the same integer governs a nineteenth-century magnetic circulation and a twentieth-century enzyme assay is the sort of thing that makes an invariant worth its name.

Where it fails, and what it costs #

Orientation is not optional. Without arrows there is no sign and no number. For an unoriented link only the absolute value survives, and even that requires care about which component is which.

It is blind to knotting. Both components can be knotted in the worst way imaginable, and the linking number is computed the same way and reports the same value, because self-crossings are never counted.

Zero is uninformative. As above: the Whitehead link and the Borromean rings are both linked and both report zero, and no amount of care with the arithmetic changes that.

Computing it from a real curve is not free. For a smooth curve given by data rather than a formula, finding the crossings needs the projection to be generic — no tangencies, no triple points — and near-degenerate crossings are exactly where a numerical computation goes wrong. The Gauss integral avoids the projection but converges slowly when the curves come close.

What the pictures cannot show #

A diagram of a link is a projection, and the third dimension appears only as breaks in the strands. That is enough information to reconstruct the link — the breaks say which strand is nearer — but it is not enough to see it, and a reader looking at the Borromean rings cannot see why removing one ring frees the other two. The convincing demonstration is a physical model, and no static picture substitutes for it.

Nor can a figure show inseparability. What the pictures show is a number, and the number’s invariance under the drawn deformations; the step from this number is the same in these four pictures to this number is the same in every picture is the Reidemeister argument, which is a case analysis in prose.

And the failure case is the hardest thing to draw honestly. The Whitehead link is not drawn here at all, because a static picture of it looks exactly like a picture of two loops that could plainly be pulled apart — which is the point being made, and a picture that makes a true statement look false is not a good figure.

The ladder from here #

Below: the three moves, which are what makes any of this an invariant, and the winding number, which is the same accounting one dimension down. Sideways: tricolourability, an invariant of a knot rather than of a pair, computed by a completely different mechanism and blind to different things. Above: the invariants that see what this one misses — polynomial invariants, Milnor’s higher linking numbers for the Borromean case, and the fundamental group of the complement, which sees nearly everything and is nearly impossible to compare.

What is worth carrying away #

The construction here is the template for the whole of algebraic topology, and it has three steps that recur without change. Attach a number to a presentation of the object — here, to a diagram. Show that the number does not change under the moves that relate two presentations of the same object. Conclude that the number is a property of the object, and use it to tell objects apart.

Every step is doing work. Without the first, there is nothing to compute; without the second, there is a number that describes a drawing; and the third is the only one that gives a theorem, in the form of an implication that goes one way. Different numbers, same shape — and the discipline of the method is that the third step never runs backwards. Two objects with the same invariant have not been shown to be the same, and forgetting that is how a knot table gets a duplicate entry.