A knot must turn twice, and a little more

Walk once round a closed curve and keep track of which way the walker faces. By the time the walk is back where it started, its direction has turned — and the total amount it turned, added up without regard to which way, is the curve’s total curvature. A circle turns through exactly one full turn, $2\pi$. A square turns through four right angles, again $2\pi$. Any closed curve at all has to turn at least that much, because it has to come back.

Werner Fenchel proved that in 1929, and added that only flat convex curves manage exactly $2\pi$. Twenty years later István Fáry and, independently, John Milnor — then an undergraduate — proved something much stranger: a knotted curve turns through more than $4\pi$. Whatever knot it is, however it is drawn, the tangent must turn more than twice round. Being knotted is a matter of topology, which ignores bending entirely — the three Reidemeister moves let a diagram be bent and stretched at will; and yet it forces a definite amount of bending.

The stick knots earlier in this collection asked how few straight pieces a knot needs. This is the smooth version of the same economy, and the answer is not a whole number of pieces but an angle.

A trefoil, and how far it turns #

The simplest knotted curve is the trefoil, and the tidiest way to draw one is on a doughnut: wind twice round the hole and three times round the tube.

This trefoil turns through $5.365\pi$. Computing that takes nothing more than adding angles: the curve is drawn as a polygon of three thousand sides, and the total curvature of a polygon is the sum of the angles by which each side turns away from the one before. As the polygon is refined, that sum converges to the integral of the smooth curve’s curvature. It is the same accounting as the gaps at the corners of a solid, one dimension down: there a surface’s angle deficits always added to $4\pi$, fixed by its shape; here a curve’s turning angles add up to whatever the drawing makes them, and topology fixes only a floor.

Nothing about $5.365\pi$ is special. Draw the trefoil on a fatter tube and the number rises; on a thinner one it falls. The theorem says it can never fall to $4\pi$. To see why, the turning has to be counted a different way.

Counting hilltops #

Pick a direction — call it up — and look at the height of each point of the curve in that direction. Going round the curve, the height rises and falls, and has some number of local maxima: hilltops. A circle tilted any way has one hilltop and one valley. A trefoil, it turns out, never has fewer than two.

The map covers 4,050 directions spread over the sphere, and none of them shows the trefoil with a single hilltop. Two is the least, and three is the commonest. The average, weighting each direction by its share of the sphere, is $2.6807$.

Now multiply by $2\pi$: $5.3615\pi$. The total curvature was $5.3652\pi$. The two agree to within a part in a thousand, the error of the finite grid of directions. That is Milnor’s identity, and it is the whole engine of the proof:

$$\text{total curvature} = 2\pi \times (\text{average number of local maxima over all directions}).$$

The reason is local. At a point where the curve turns through a small angle $\alpha$, the set of directions in which that point is a local maximum is a thin band on the sphere whose area is proportional to $\alpha$ — a sharp bend is a hilltop from many directions, a gentle one from few. Add over the whole curve and the total turning is the total area of those bands, which is the average number of hilltops times the sphere’s area, scaled. Integral geometry of this kind goes back to Crofton’s formula of 1868, which measures a curve’s length by counting how often random lines cross it.

Fenchel’s floor, from the same count #

The identity proves Fenchel’s theorem in one line, and seeing that first makes the knotted case clearer. Every closed curve has a highest point in every direction — a continuous height on a closed loop must reach a maximum somewhere. So the number of local maxima is at least one in every direction, the average is at least one, and the total curvature is at least $2\pi$.

When is it exactly $2\pi$? Only when almost every direction gives exactly one hilltop. Working out which curves achieve that takes more care than the inequality, and the answer is the expected one: a loop that is not flat, or a flat loop with a dent in it, shows a second hilltop from a whole range of directions, enough to lift the average above one. Only a flat convex loop escapes. One hilltop in every direction is the defining property of a flat convex loop.

The knotted case is the same count one level up. Fenchel’s floor comes from the fact that a loop has to have a top; Fáry and Milnor’s comes from the fact that a knotted loop cannot get away with having only one. Neither proof ever computes a curvature directly. Both count how a curve looks from outside, averaged over every point of view.

Why one hilltop means no knot #

The identity converts the theorem into a statement about hilltops: a knotted curve has at least two in every direction. Suppose instead some direction gave a closed curve only one maximum. Then it also has only one minimum, and between them the curve consists of exactly two arcs, each rising steadily from the bottom to the top. Slice the space by horizontal planes: each plane between the bottom and the top meets the curve in exactly two points, one on each arc. Join those two points by a straight segment in each plane, and the segments sweep out a disc whose edge is the curve. A curve that bounds an embedded disc is unknotted — the disc is the simplest of the surfaces a knot can bound, and only the unknot bounds one.

So a knotted curve has at least two maxima in every direction, the average is at least two, and the total curvature is at least $4\pi$. Strictness needs a finer look, which Milnor supplied: for a knotted curve the directions with three or more hilltops cannot be avoided — they fill a region of the sphere of positive area — so the average is strictly above two, and the inequality is strict.

The argument never asks which knot the curve is, and never needs an invariant that tells knots apart. It asks only whether a curve can be swept out by horizontal segments, and a knotted curve cannot. That is why the same bound, $4\pi$, holds for the trefoil, the figure-eight, and knots with a thousand crossings — and why, for knots far more complicated than the trefoil, $4\pi$ is far from the truth.

Milnor’s identity on eight curves #

The identity holds for every closed curve, knotted or not, and checking it on several is a good test of both the computation and the claim.

The circle and the tilted ellipse sit at exactly $2\pi$, as Fenchel’s theorem says flat convex curves must. The five knotted curves are all above $4\pi$, the thinnest-tubed trefoil only just, at $4.075\pi$. The figure-eight knot in a standard drawing reaches $6.61\pi$, the $(2, 5)$ torus knot $8.58\pi$.

The flower is the warning. Its counterpart is the theorem that seven points joined in every way always contain a knotted cycle, which by this theorem must then turn more than twice round however the points are placed; the flower is the opposite case. It is a plane curve with five petals, as unknotted as a curve can be, and it turns through $9.39\pi$ — more than any of the knots in the table. Bending a great deal does not make a curve knotted; the theorem goes in one direction only. Long random loops are almost always knotted and turn through enormous angles, but the curvature is the consequence of being crumpled, not the cause of being knotted.

How close to 4π a trefoil can get #

The theorem gives a floor of $4\pi$. Can a trefoil come close to it? The torus drawing answers that, and it answers more interestingly than expected.

Shrink the tube and the trefoil drawn twice round the hole hugs its core circle more and more tightly. Its tangent then mostly follows the core, which it traverses twice — two full turns, $4\pi$ — plus small wiggles that shrink with the tube. At radius $0.02$ the total is $4.001\pi$. The limit is exactly $4\pi$, but it is approached and never reached: at radius nought the curve collapses onto the doubled circle and is no longer a knot at all. In the language of hilltops the squeezed trefoil is easy to read: from almost every direction it shows exactly the two highest points of the doubled core circle, one on each pass, and the third hilltop appears only in a thin band of directions close to the axis, where the wiggles round the tube are what the eye sees. As the tube thins, that band shrinks to nothing and the average falls to two.

The second curve is the same knot. The trefoil is also the torus knot that winds three times round the hole and twice round the tube, and drawn that way it tightens onto a circle traversed three times — so its curvature falls towards $6\pi$ and stays there. The bound depends on the drawing; the infimum depends on the knot. And the $(2, 5)$ knot, a genuinely different knot with five crossings, can also be squeezed to just above $4\pi$ — so the Fáry–Milnor bound does not distinguish the trefoil from the cinquefoil. Both have the property that matters: they can be drawn with two hilltops in nearly every direction.

Where the tangent points #

There is a second picture of the same quantity, and it is the one Fáry used.

As the curve is traced, its unit tangent moves over the sphere, and the distance the tangent travels is exactly the angle it turns through. So the total curvature is the length of this path on the sphere. A circle’s tangent runs once round a great circle. The trefoil’s tangent wanders over the whole sphere, and its path is longer than $4\pi$ — longer than two great circles laid end to end.

In this picture the hilltop count becomes Crofton’s formula on the sphere: a point is a local maximum of the height in direction $u$ exactly when the tangent there is perpendicular to $u$, which is when the indicatrix crosses the great circle perpendicular to $u$. Counting crossings with random great circles measures the length of a spherical curve, just as counting crossings with random lines measures a plane one. Fáry’s proof showed that the indicatrix of a knotted curve must meet every great circle at least four times — which makes its length more than $4\pi$.

The squeezed trefoils of the previous figure look simple in this picture. As the tube thins, the trefoil’s tangent mostly follows the tangent of the core circle, which lies along the equator of the sphere; going twice round the core, the indicatrix runs twice round the equator, length $4\pi$, with small loops added wherever the curve swings round the tube. The loops shrink with the tube and never vanish. The same picture shows why a knotted curve cannot do with one pass round a great circle plus small loops: a tangent that ran once round the equator and otherwise stayed close to it would belong to a curve with a single hilltop from nearly every direction, and that curve, by the argument of the disc swept out by horizontal segments, would be unknotted.

The floor for each knot #

Milnor went further than the bound. He showed that the least total curvature any curve of a given knot type can approach is $2\pi$ times a whole number attached to the knot: the fewest hilltops a drawing of it can have in its best direction. Horst Schubert, a few years later, studied that number under the name bridge number: the trefoil, the figure-eight and every two-bridge knot have bridge number two, so their floor is exactly $4\pi$; more complicated knots have higher floors.

For torus knots the numbers are explicit. The $(3, 4)$ knot drawn three times round the hole turns through $6.002\pi$; drawn four times round, $8.000\pi$. Schubert proved that the bridge number of the $(p, q)$ torus knot is the smaller of $p$ and $q$, and the table shows the drawings approaching exactly $2\pi$ times it. For the $(2, 7)$ knot the thinnest drawing is still at $4.126\pi$, because a curve that winds seven times round a thin tube has to bend sharply to do it; the limit is still $4\pi$, approached more slowly.

So the Fáry–Milnor theorem is the first case of a sharper statement. Knotted means bridge number at least two, which means total curvature above $4\pi$; and each further strand of the bridge costs another $2\pi$.

Polygons, a grid of directions, and an inequality no drawing reaches #

Every curve here is a polygon, and every number a finite sum. The total curvature of a polygon is exactly the sum of its turning angles, so those numbers are exact for the polygons drawn and converge to the smooth curves’ values as the polygons are refined; at three thousand sides they agree with the smooth integrals to the precision printed. The hilltop map is a grid of 4,050 directions, and the claim that no direction gives a single maximum is a statement about all directions, which the grid supports and the theorem proves. Milnor’s identity is checked to about one per cent, which is the accuracy a grid of directions gives; the identity itself is exact.

The squeezing figure shows trefoils approaching $4\pi$; that no trefoil reaches it is the strict inequality, which no sequence of drawings can display. And the bridge numbers in the last table are Schubert’s theorem: the figure shows drawings that achieve $2\pi$ times the smaller index, and cannot show that nothing does better.

Still open: how tightly a knot can be tied #

Total curvature measures bending and ignores thickness: a curve can approach $4\pi$ only by collapsing towards a doubled circle, which a real rope could not do. Put a thickness on the curve — require a tube of radius one around it to stay embedded — and ask for the shortest length of rope that ties a given knot. This is the ropelength, and it is the physical sharpening of the same question.

For the trefoil the shortest ropelength found by computer, after much refinement of simulated tightening, is about $32.74$ times the rope’s radius. The best lower bound proved, by Elizabeth Denne, Yuanan Diao and John Sullivan in 2006, is about $31.32$, and their proof uses a strengthening of the Fáry–Milnor argument: every knotted curve has an alternating quadrisecant, a straight line meeting it in four points in a particular order, which Denne proved in 2004. The gap between $31.32$ and $32.74$ is not closed, and for no knot at all is the exact ropelength known. The tight trefoil is not even known to be unique. Even the simpler-looking count of crossing changes needed to undo a knot is unknown for many small knots; how much rope a knot needs is a question about geometry layered on top of that.

A floor that topology sets and geometry pays #

The theorem is a bridge between two kinds of question that usually have nothing to say to each other. Whether a curve is knotted is invariant under every bending; how much a curve bends is changed by every bending. Milnor’s identity connects them through counting: bending is averaged hilltops, and topology bounds the hilltops from below. Fenchel’s theorem says a closed curve must turn once. Knotting makes it turn twice — and since $4\pi$ is approached only by curves collapsing onto a doubled circle, it must always turn a little more.

That pattern — a geometric quantity bounded by a topological one, through an average over directions — is the same one the corners of a solid showed with equality instead of inequality, and the same one that makes every loop of the Lorenz flow carry a linking number its geometry cannot change. Here the topology cannot fix the amount, only its least value. The rest is up to how the knot is drawn.