Seven points and a knot they cannot avoid

Six points in space and a pair that must link placed six points anywhere, joined every pair, and found that among the ten ways of splitting the six into two triangles, at least one pair of triangles is always linked. The proof was a parity: the ten linking numbers add to an odd number, and an odd number is never nought.

John Conway and Cameron Gordon proved that theorem in 1983, and in the same paper they proved its sequel. Add a seventh point. Now there are closed paths through all seven — each visiting every point once and returning to the start — and the question is not whether two paths link but whether one path knots. The answer is that some path always does. Every placement of seven points in space, joined by straight segments or by any curves that do not meet, contains a knotted closed path through all seven.

This essay computes the theorem where it can be computed: for placements with straight edges, every one of the 360 paths can have its knot identified exactly, and the parity that forces a knot can be watched adding up.

Seven points and 360 paths #

A closed path through seven points, visiting each once, is a choice of cyclic order: fix the starting point, arrange the other six in any of $6! = 720$ orders, and count each path once in each direction, which leaves 360. With straight edges, each path is a polygon of seven sides, and a polygon can be knotted.

Seven sides do not allow much. A knot made of straight sticks needs a minimum number of them, and six sticks tie a trefoil found that the trefoil needs six and nothing more complicated fits in six; the figure-eight needs seven. So a path through seven points is an unknot, a trefoil or a figure-eight, and nothing else. The figure identifies each path by its Alexander polynomial — computed from the crossings of a projection, as a polynomial behind the colourings showed — once in each of two different projections, which must agree — the polynomial is unchanged by the three moves that relate any two projections of the same knot, so a disagreement would mean an error. The polynomial is $1$ for an unknot, $t^2 - t + 1$ for a trefoil and $t^2 - 3t + 1$, up to sign, for a figure-eight.

In the placement drawn, nine of the 360 paths are knotted, all trefoils. The one drawn heavy runs through the points in the order $1, 2, 3, 4, 5, 7, 6$, and its three crossings in this projection are the trefoil’s three.

The number that adds to something odd #

The invariant Conway and Gordon used is the second coefficient of a knot’s Conway polynomial, $a_2$, which can be read from the Alexander polynomial as half its second derivative at $t = 1$. It is 0 for the unknot, 1 for the trefoil and $-1$ for the figure-eight; what matters is its parity, which is also called the Arf invariant of the knot — originally defined from the quadratic form on the surface a knot bounds — and is 1 for both knots here.

Their theorem is that the sum of $a_2$ over all the closed paths through the seven points is odd, for every placement. The table computes the sum for four placements: 3, 11, 9 and 1. With only three knot types available and $a_2 = \pm 1$ for the two knotted ones, an odd sum means an odd number of knotted paths, and an odd number is at least one.

The third placement shows why the sum is used rather than a count. It has eleven trefoils and two figure-eights; the $a_2$ sum is $11 - 2 = 9$ and the count of knotted paths is 13, and both are odd. The count happens to share the sum’s parity because $a_2 = \pm 1$ for every knot seven sticks can make, but with curved edges any knot can appear, some knots have even $a_2$, and only the sum’s parity is a theorem.

Two hundred placements, and no even count #

Across two hundred random placements the counts are 1, 3, 5, and so on up to 15, with no even number anywhere — the histogram has gaps at every even value, which is the parity made visible. The most common counts are small: a random placement of seven points typically has a handful of knotted paths among 360.

Every one of the two hundred placements has at least one trefoil, not merely a knot. That is a sharper theorem for straight edges, proved by Jorge Ramírez Alfonsín in 1999 with the combinatorics of how points in space can be arranged: every straight-line placement of seven points contains a trefoil path. A figure-eight alone would satisfy the parity, and it never happens.

Why the parity cannot change #

The proof is a deformation argument, and the figure runs it. Any placement can be moved to any other by moving the points one at a time, and as long as no two edges pass through each other, no path changes its knot type. The only events that matter are the moments when an edge through the moving point passes through another edge. At such a moment, every path that uses both edges undergoes a crossing change, and its $a_2$ may change.

A crossing change alters $a_2$ by the linking number of the two-component link obtained by smoothing the crossing — the skein relation of the Conway polynomial. Conway and Gordon showed that, summed over all the paths through both edges, those linking numbers total an even number — each one that appears is counted twice — so the change in the whole sum is even. So the parity of the sum is the same for every placement. Computing it for one convenient placement, where exactly one path is a trefoil, shows the parity is odd.

In the figure, as point 1 travels across the cube, the number of knotted paths goes from 7 down to 1 and up to 15 and back, changing eleven times. The sum of $a_2$ changes too — at one stage it is 9 while the count is 15, because some paths have become figure-eights — but it is odd at every one of the 121 positions. The count never reaches nought, because to get there the sum would have to become even.

One knot, and no fewer #

The theorem promises one knotted path, and one is attained. In this placement, 359 of the 360 paths are unknots and the path drawn is a trefoil. Moving any point a little does not change that. Moving it far enough to pass one edge through another changes the count by an even number, so from one the count can only go up — to three, five or more — since one is already the least odd number.

That minimal placements exist says the theorem is sharp in its conclusion: the count of knotted paths cannot be forced higher than one by the parity argument alone. Whether some other argument forces more knotting in larger complete graphs is a different question, and the answer is yes — larger graphs are forced to contain more knotted cycles and more complicated knots — but no argument of the same simplicity is known to measure how much.

Six points are not enough #

Six points behave differently. A path through six points is a hexagon, and six sticks can make a trefoil, so some placements of six points do contain a knotted path. But three quarters of random placements contain none, and a placement with none is a proof that six points are not forced to knot. The complete graph on six points is intrinsically linked — every placement links two triangles — but not intrinsically knotted. Seven is the smallest number of points that is.

The distinction is sharper than it looks. Linking needs two disjoint cycles, and six points are the fewest that can hold two disjoint triangles. Knotting needs one cycle long enough to knot, and with straight edges that means six sides at least; but six points in a random position usually have all their hexagons unknotted, and the seventh point supplies enough paths — 360 rather than 60 — for the parity to take hold.

How often a path through seven points knots #

The knotted paths are rare. Nine of 360 is two and a half per cent, and the typical placement has fewer. That fits what is known about random polygons in general: almost every long loop is knotted, but short ones almost never are, and a polygon of seven sides is about as short as a knotted polygon can be. A seven-sided polygon with its corners chosen at random in a cube is knotted only a small fraction of the time.

What the theorem says is that the 360 paths of a placement are not independent random polygons. They share their 21 edges, each edge lying on 120 of the paths, and the sharing is what the parity sees. A single random heptagon can easily be unknotted; 360 heptagons built from the same 21 segments cannot all be. The rarity of knotting and its inevitability are both true, and they are about different things: one about a random path, the other about the whole family of paths at once.

The rarity also explains why the count is usually small. Most placements have one, three or five knotted paths; getting fifteen requires a placement in which one point sits somewhere unusual relative to the others, and the moving-point figure shows such a stretch — a few positions where many paths through point 1 are tangled at once.

Flat, linked, knotted #

The theorem is the third in a sequence. In the plane, two graphs will not lie flat: $K_5$ and $K_{3,3}$ cannot be drawn without two edges crossing, and Kuratowski’s theorem says they are the only obstructions. In space every graph can be drawn without crossings, and the question moves up a level — not whether edges cross, but whether cycles link, which is what the linking number measures, and $K_6$ is the smallest graph that must contain a linked pair. Then it moves up again, to whether a single cycle knots, and $K_7$ is the smallest complete graph that must.

Each step asks for more and needs a bigger graph. Five points force a crossing in the plane; six force a link in space; seven force a knot. The three obstructions are measured by three different invariants — a count of crossings modulo two, a linking number, a Conway coefficient — and each theorem is a parity argument about the right invariant, summed over the right family of sub-objects.

Which graphs must knot #

Conway and Gordon’s theorem makes $K_7$ the first intrinsically knotted graph: one that contains a knotted cycle however it is placed in space. The natural question is which graphs are intrinsically knotted, and it has a precise shape. If a graph is intrinsically knotted, so is every graph that contains it as a minor — a graph obtained by deleting and contracting edges, the operation five spokes squeezed into $K_5$ used to find $K_5$ inside the Petersen graph. By the graph minor theorem of Neil Robertson and Paul Seymour, the minor-minimal intrinsically knotted graphs therefore form a finite list.

For linking, the list is known: Robertson, Seymour and Robin Thomas proved in 1995 that the minor-minimal intrinsically linked graphs are exactly the seven graphs of the Petersen family, $K_6$ among them. For knotting, the list is not known. Many minor-minimal intrinsically knotted graphs have been found: $K_7$ and the graphs obtained from it by replacing triangles with three-pointed stars, $K_{3,3,1,1}$, which Thomas Foisy proved intrinsically knotted in 2002, and many more. Nobody knows how many there are.

Where the question comes from #

Conway and Gordon’s paper, Knots and links in spatial graphs, appeared in the Journal of Graph Theory in 1983, and Horst Sachs proved the linking half independently around the same time. Together they started a subject — spatial graph theory — that asks which properties a graph forces on every way of placing it in space, the three-dimensional counterpart of asking which graphs can be drawn flat.

Part of the interest came from chemistry. A molecule whose atoms and bonds form a graph sits in space in some particular way, and chemists synthesising molecules shaped like complete graphs, strips of rings twisted into Möbius bands, or interlocked rings needed to know which shapes could be built without knots or links and which could not. A graph that is intrinsically knotted cannot be made as a molecule without a knotted ring of bonds somewhere in it, however cleverly it is assembled. Erica Flapan and others developed the topology of such molecules, including when a molecule must be different from its mirror image for topological reasons, and the theorem about seven points is the simplest statement of the kind.

What the figures cannot show #

Every placement here uses straight edges, where the knots are few and each can be identified exactly. The theorem is about every placement with any curved edges, where the paths can be any knots at all; the parity of the $a_2$ sum is the part of the argument that survives, and the figures show it only in the straight case.

The knot types are identified by the Alexander polynomial, which does not distinguish every knot from every other. For polygons of six and seven sides it does, because only three types are possible and their polynomials differ; for longer polygons the identification would need more.

And the random placements are points in a cube chosen by a seeded generator. The counts they give — at most 15 knotted paths in two hundred placements — are observations about typical placements, not bounds. Placements with more knotted paths exist, and how many the most knotted straight-line $K_7$ can have is a question about the finitely many combinatorial types of seven points in space — one that exhaustive computation can reach and random sampling cannot.

Still open: the list of graphs that must knot #

The complete list of minor-minimal intrinsically knotted graphs is unknown. It is finite, by a general theorem that gives no bound on its size and no way to generate it. For small graphs it has been worked out: among graphs with 21 edges the minor-minimal intrinsically knotted ones are known completely, and every intrinsically knotted graph has at least 21 edges. Beyond that, new examples keep being found and no characterisation is in sight — nothing like the single family of seven that settled the question for linking.

A second open direction is quantitative. $K_7$ is forced to contain one knotted cycle; larger complete graphs are forced to contain knots of greater complexity, and linked cycles with larger linking numbers. How fast the forced complexity grows with the number of points is known only within wide bounds.

A parity for knots #

The two theorems of Conway and Gordon have the same skeleton. Take a graph, list all the sub-objects of one kind — pairs of disjoint triangles in $K_6$, closed paths through every point in $K_7$ — attach to each a number from knot theory, and add. The sum’s parity cannot change under any deformation, because every event that could change it changes it by an even amount; and one placement shows the parity is odd. From that, a linked pair or a knotted path follows for every placement at once.

What the second theorem adds is that the numbers are about single curves. A linking number compares two loops; $a_2$ belongs to one, and measures, in its parity, whether a knot is there at all. That seven points in space cannot be joined up without tying a knot is not a fact about any particular placement. It is a fact about 360 numbers that cannot all be even.