A whole number split into two that are not

Two closed curves have a linking number: count their signed crossings in any projection, halve, and the result is an integer that no deformation can change unless one curve passes through the other. The count deliberately throws away a curve’s crossings with itself, because a kink can be added anywhere and each one changes that self-count by one.

There is a situation where the thrown-away crossings come back and are worth something. Take a ribbon — a thin strip whose centre line is a closed curve in space — and look at its two edges. They are two closed curves running side by side, and they have a linking number like any other pair. But the ribbon has a single shape, and that shape can be described by two separate numbers: how much the strip turns about its own centre line, and how much the centre line coils in space. Neither is an integer, neither is invariant, and their sum is the linking number of the edges.

$$\mathrm{Lk} = \mathrm{Tw} + \mathrm{Wr}.$$

This is Călugăreanu’s theorem, proved between 1959 and 1961 and put in its general form by James White in 1969. The essay measures it.

A ribbon on a flat circle #

The simplest ribbon has a flat circle for its centre line.

Take a band of paper, give one end two full turns, and tape the ends together. Its edges then wind round each other twice, and their linking number is 2 — counted, in the figure, from the signed crossings of the drawing itself. The twist, measured as how far the strip rotates about its centre line in one circuit, is exactly two turns.

The writhe here is zero. It measures the coiling of the centre line in space, and a flat circle does not coil: seen from any direction, it never crosses itself. So on a flat circle $\mathrm{Lk} = \mathrm{Tw}$, and the theorem says nothing surprising. Everything interesting happens when the centre line is allowed to leave the plane.

A word on what the pictures draw. Every ribbon is shown as its core line and one edge, joined by short cross-ties so that the strip between them can be seen turning. Both lines are closed curves in space, projected onto the page, and the linking number is counted from where they cross.

The same number on a curve that is not flat #

Now put the ribbon on a figure eight that has been lifted out of the plane, so that one loop passes over the other where a flat figure eight would cross itself.

The ribbon was built to twist as little as possible: its edge was carried along the core by parallel transport, the rule that turns the strip only as much as the bending of the core forces it to. And yet the edges link once. Where did the linking come from?

From the shape of the core. The figure eight’s two loops pass one over the other, and the core has writhe: $0.533$. The strip, forced to close up after following a curve that coils, picks up a twist of $0.467$ of a turn on the way round. Those two real numbers add to $1.000$, the integer counted from the crossings, to three decimal places. The three numbers were computed by three unrelated methods — a crossing count, an integral of the strip’s rotation, and a double integral over the core — and they satisfy the identity to the accuracy of the computation.

What the three numbers are, one at a time #

Linking number. The edges of the ribbon are two disjoint closed curves, and their linking number is the signed crossing count of a projection, halved. It is an integer, and it cannot change unless the ribbon is cut or passed through itself — the same reason the winding of a loop round a hole cannot change by a little: a quantity that moves continuously and takes only whole values cannot move at all.

Twist. At each point of the core the ribbon has a direction across it — a unit vector at right angles to the core, pointing to the edge. As the point moves along the core, that vector rotates about the core’s direction. The twist is the total of that rotation in one circuit, measured in full turns. It is a real number, and it depends continuously on the shape: bend the core and the twist changes smoothly.

Writhe. This is a property of the core alone, and it is where the self-crossings return. Gauss’s double integral, applied to one curve against itself instead of to two curves, gives

$$\mathrm{Wr} = \frac{1}{4\pi} \oint\!\!\oint \frac{(\mathbf{r}(s) - \mathbf{r}(t)) \cdot (\dot{\mathbf{r}}(s) \times \dot{\mathbf{r}}(t))}{|\mathbf{r}(s) - \mathbf{r}(t)|^3} \, ds \, dt.$$

For two separate curves this integral is the linking number and always an integer. For one curve it is a real number that measures how much the curve coils. It is zero for any flat curve, since the three vectors in the integrand then lie in a plane and their triple product vanishes.

An average of integers that is not one #

The writhe has a second description that shows exactly how the thrown-away self-crossings come back.

Project the core along some direction and count its self-crossings with signs. That count — the directional writhe — is an integer, and it depends on the direction: from most directions the figure eight’s loops cross once, positively; from some they do not cross at all; from a few the crossing appears the other way round. The writhe is the average of the directional writhe over all directions, each direction weighted equally on the sphere.

That is the precise sense in which the self-crossings of the linking-number recipe were not worthless. From any one direction the self-count is arbitrary, since kinks can be added at will. Averaged over every direction it becomes a definite quantity, depending smoothly on the curve’s shape — and not an integer, because an average of integers need not be one. Four hundred random directions give $0.512$; the exact average is the integral’s $0.533$, and the gap is the sampling error of four hundred draws.

The same number has a life inside knot diagrams, where it is used in the opposite spirit. There the writhe of a diagram — its signed self-crossings from the one direction the diagram is drawn from — is exactly the quantity that the Kauffman bracket must be corrected by before it becomes an invariant, because the first of the three moves changes it by one. In a diagram the writhe is a nuisance to be cancelled; averaged over all directions of a curve in space it is a geometric quantity with a meaning. The two are the same count, taken once and taken on average.

Twist and writhe trade #

Hold the linking number fixed and deform the core. Nothing is cut and nothing passes through anything, so the linking number cannot change. The twist and the writhe can, and must, in opposite directions.

As the lift shrinks, the figure eight’s two loops come closer to crossing, and from almost every direction the projection shows one positive self-crossing: the writhe climbs towards 1. The twist gives way by exactly the same amount. In the limit the core would pass through itself — which is not allowed, and at which point the writhe would jump by 2 and the linking number would be free to change.

This trade is visible in ordinary objects. A telephone cord that has been twisted too far about its own length will, if its ends are brought together, throw itself into coils: the twist in the cord converts to writhe in the cord’s centre line, because a coiled shape stores the same linking number with less elastic energy in twisting. A rubber band twisted and relaxed does the same. And a closed ribbon cannot get rid of its linking number by any amount of wriggling; it can only redistribute it between the two forms.

A coiled loop #

The coil is the extreme case, where a core wound round a tube carries a large writhe.

Here the core has writhe close to $-2.8$, and the ribbon built with one deliberate extra turn has twist $+0.77$: a linking number of $-2$, with most of it stored in coiling. The sign is the coil’s handedness: wind the core round its tube the other way and every number in the figure changes sign, as a mirror image must.

This is the shape a supercoiled ribbon settles into, and it explains a counter-intuitive fact about coiled springs and cables. A coil of wire looks as though it must be twisted — it is visibly helical — and yet the wire along a coil can carry little twist or none, because the helix of the centre line is writhe, not twist. Conversely a perfectly straight-looking cable can be carrying a great deal of twist that will turn into coils the moment its ends are let go. Which quantity a ribbon holds cannot be read from how helical it looks; only their sum is visible to the edges. The figure is also a check on accuracy. The coil’s writhe is the hardest of the three numbers to compute, because the core passes close to itself on every winding, and the identity holds to $0.003$, which is the quadrature’s error rather than any failure of the theorem.

Seven ribbons, one identity #

The rows repay reading in pairs. The two figure eights with lift 0.2 have the same core and so the same writhe, $0.722$; one ribbon was built with no extra turn and links once, the other with one extra turn and links twice, and their twists differ by exactly one. So writhe belongs to the core and twist to the strip, and adding a full turn to the strip changes twist and linking number together, leaving writhe alone. The coiled loops show the other thing: a large writhe with a small twist, the linking number mostly carried by the shape.

The figure eight with lift 0.8 is the quiet row, and perhaps the most instructive. Its linking number is 0, but its twist and writhe are not: $-0.118$ and $+0.118$. A ribbon can be unlinked at the level of its edges while its strip and its core are both doing something, in equal and opposite amounts. It is the same moral as zero meaning two different things for the linking number itself: an integer summary, however exact, can conceal a structure that only the parts reveal.

A half turn, and the ribbon that has one edge #

The theorem asks for a ribbon with two edges, and that excludes one famous strip. Give a band a half turn before joining its ends and the result is a Möbius band: its “two” edges are one closed curve that runs round twice, and there is no second curve for it to link with. The identity has nothing to say about it, because its left-hand side does not exist.

Seen from the identity, the Möbius band is the ribbon whose twist would have to be a half-integer on a flat core — and a two-edged ribbon on a flat circle can only close up after a whole number of turns, since its edge must come back to where it started, not to the opposite side. So the integrality of the linking number and the two-sidedness of the strip are the same condition. A strip that closes after half a turn has traded its second edge for the extra half, and the bookkeeping of twist and writhe starts again only when two edges are available to be linked.

Double the Möbius band — cut it down the middle — and a two-edged ribbon comes out, famously in one piece, with two full turns of twist and edges that link. Every step of the paper-and-scissors trick is an instance of the arithmetic above.

Where the identity does real work: circular DNA #

The identity would be a curiosity if it were not the governing equation of a molecule. A double-stranded DNA molecule is, geometrically, a ribbon: two strands wound round a common axis. In bacteria, in mitochondria and in many viruses the molecule is closed into a loop, and then the two strands are two closed curves with a linking number — an integer, fixed as long as neither strand is broken.

Relaxed DNA winds about once every 10.5 base pairs, so a closed loop of 5,250 base pairs would have a natural linking number of about 500 if it lay flat with no coiling. Most closed DNA in cells is underwound: its linking number is a few per cent below that. Since the twist of the double helix is set largely by chemistry, the deficit shows up as writhe, and the loop coils on itself into a supercoil — a shape seen directly in electron micrographs as a loop wound into a twisted rope.

Cells control this with enzymes called topoisomerases, and the identity is exactly how their actions are described. A type I topoisomerase cuts one strand, lets the other pass through the gap, and reseals: the linking number changes by exactly one. A type II topoisomerase cuts both strands, passes another segment of the double helix through, and reseals: the linking number changes by exactly two. Between those events the molecule can writhe and twist as it likes; the linking number is conserved. Measuring it — by gel electrophoresis, which separates loops by how compact their coiling makes them — was one of the first places where a topological invariant was read off a laboratory experiment.

What the figures compute and what they assume #

Every number is numerical except one. The linking number is exact, counted from crossings in a projection where no two strands cross tangentially. The twist and the writhe are integrals approximated by sums over hundreds of points on the curve, and they carry errors of the order shown in the tables’ third decimal place. The identity is checked to within $0.03$, which is a statement about the quadrature and not a proof of the theorem.

The frame is a choice. A ribbon on a given core is determined by which way its edge points at every place, and the figures build it by parallel transport plus a chosen number of extra turns. Other ribbons on the same core, with the edge pointing elsewhere, have different twists and different linking numbers, and the same writhe.

The writhe histogram is a sample. Four hundred directions give an average within a few hundredths of the integral. The equality of the average over all directions with Gauss’s integral is a theorem, due to F. Brock Fuller; the figure illustrates it with a finite sample.

Nothing here shows a ribbon’s elastic energy. Why a real telephone cord or DNA loop chooses to store linking as writhe rather than twist depends on stiffness to bending and to twisting, which the figures do not model. The identity says what trades are possible, not which one physics picks.

Still open: which shapes a ribbon settles into #

The identity is exact and purely geometric. The questions left are about which configuration a physical ribbon adopts, and they are hard. For an elastic rod closed into a loop with a given linking number, the shapes of least energy are known for small linking numbers and computed for larger ones, but the full picture — which supercoiled shapes are stable, when a loop snaps from one to another, and how self-contact changes the answer — is worked out case by case rather than by a theory.

For DNA the questions are sharper still, because the molecule is not a uniform rod: its stiffness varies with its sequence, it carries electric charge, and it is surrounded by proteins that bend it. How the linking deficit of a real chromosome is distributed between twist and writhe, and how that distribution regulates which genes can be read, is an active experimental question. The identity is the one fixed point in it: whatever else happens, twist plus writhe is an integer that only an enzyme can change.

There are purely mathematical questions too. For three or more ribbons tied together — a braided cable, a strand of rope — the pairwise linking numbers do not capture everything, just as three rings can be linked with no two linked, and the corresponding decompositions into twist-like and writhe-like quantities for multi-component ribbons are not settled in a form anyone uses.

One integer, two ways to hold it #

The linking number of two curves was an integer because it could not change by a little. The ribbon shows where that integer can live when the two curves are bound together as the edges of one strip: partly in how the strip turns, partly in how its centre line coils, in proportions that shift continuously with the shape. Neither part is a topological invariant. Their sum is, and the proof that it is — Călugăreanu’s, White’s, Fuller’s — is the reason a loop of DNA, twisted and released, coils rather than simply untwisting.

It is also a small change in what an invariant is for. The linking number was introduced as something that does not change, and so can tell two configurations apart. Here its constancy is put to work the other way round: because it cannot change, every change in one of its two halves forces an equal and opposite change in the other, and the invariant becomes a conservation law. A cell that wants to loosen its DNA’s twist in one place, to open the strands and read them, must accept extra writhe somewhere else — or call an enzyme that breaks the strands and changes the integer.