Matpic
mathematics, in pictures
Concept

Knot — where it appears

A closed curve in space, considered up to deformations that never pass one strand through another. Telling two apart means finding an invariant, since no amount of failed rearranging proves that a rearrangement does not exist.

Named by 17 essays across 2 fields — each of them below, with the objects they name alongside it.

the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

Three moves, and what they cannot undo

A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.

topology · Knots How many colourings each knot allows. Three knots, and the number of ways their arcs can be coloured with three, five and seven colours under the crossing rule, beside the determinant computed separately from the same crossings.

Colours that count more than three

Three colours prove the trefoil is knotted and say nothing at all about the figure-eight, which refuses them exactly as an unknotted loop does. The repair is to stop colouring and start counting — with five colours, or seven, and with the arithmetic done modulo the number of them.

topology · Knots the Hopf link, with every crossing signed. A diagram of the Hopf link with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Two loops and one number

Give each crossing between two closed curves a sign, add them up, halve — and the answer does not depend on how the curves were drawn, how they are pushed about, or which way the picture was projected.

topology · Linking number a loop that dips through and back, and the punctures of the disc. A link drawn with a shaded disc spanning the first loop, seen at an angle, with every place the second loop passes through the disc marked with the direction it was travelling in.

Zero can mean two different things

The linking number counts how often one loop pierces a surface the other one bounds. Two punctures of opposite sign add to nothing, and a loop that never goes through adds to nothing as well — so the answer zero is two pictures wearing one number.

topology · Linking number A ray through a knotted tube, crossing it 5 times. A closed surface in space — a tube round a trefoil knot — with a point, a ray from it and every crossing of the surface marked; the parity of the count says which side of the surface the point is on.

Two pieces, in every dimension

A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.

topology · Jordan curve Alexander's horned sphere at stage 3. A tree of clasped pairs of horns, 7 of them, each pair's two circles passing once through the other's disc; the horns shrink geometrically and their tips converge.

A ball whose outside is not one

Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.

topology · Jordan curve The Alexander matrix of the trefoil. The trefoil with its 3 arcs numbered and its 3 crossings lettered, beside the 3 by 3 matrix they give. A minor of the matrix is the Alexander polynomial t − 1 + t⁻¹, whose value at −1 is the determinant 3.

A polynomial behind the colourings

The figure-eight knot and the cinquefoil both have determinant five, so they admit exactly the same colourings, and every counting argument treats them as one. Put a variable where the colouring rule has a two and the determinant becomes a polynomial — and the two knots come apart.

topology · Knots The Seifert circles of the trefoil. The trefoil with an orientation, cut at each of its 3 crossings and reconnected the way the orientation allows. The 6 segments form 2 circles, and the surface built from them has genus 1.

The surface a knot bounds

Every knot is the edge of a surface with two sides, and Seifert found a way to build one from any diagram: smooth the crossings, fill the circles that result with discs, and join them with twisted bands. Counting the handles gives an upper bound on how complicated the knot is, the Alexander polynomial gives a lower one, and for the simplest knots the two meet.

topology · Knots The trefoil and its mirror image. The trefoil beside its reflection, with writhe 3 and −3. The Jones polynomials are t + t³ − t⁴ and −t⁻⁴ + t⁻³ + t⁻¹; the Alexander polynomial of both is t − 1 + t⁻¹.

A polynomial that tells left from right

The trefoil and its mirror image have the same colourings, the same determinant and the same Alexander polynomial, and the first proof that they differ was a hard argument about groups. Smooth every crossing both ways, count the circles in each of the resulting pictures, and add up the counts with the right weights: the total changes when the knot is reflected.

topology · Knots The two extreme states of the figure-eight knot. The figure-eight knot, then the state with every crossing smoothed the A way (3 circles) and the state with every crossing smoothed the B way (3 circles). The bracket spans 16 powers of A against 16 for four times the crossings.

The crossings an alternating knot cannot lose

Peter Guthrie Tait drew knots for years and believed, without proof, that a diagram whose crossings alternate over and under, and which has no twist that can be undone, is already drawn with the fewest crossings the knot allows. The proof took a century, and when it came it needed only the two most extreme ways of smoothing the diagram and Euler's count of the regions of a map.

topology · Knots Signature against unknotting, for every knot to seven crossings. A table of the fourteen prime knots with up to seven crossings, listing determinant, signature, the lower bound on crossing changes it gives, the number of changes found by search, and whether the two agree.

How many changes undo a knot

Cut the string at a crossing, pass it through the other strand and join it up again, and any knot can be undone by doing that often enough. The fewest changes needed is the unknotting number, and proving that fewer will not do needs a number that each change can move only a little. The signature moves by at most two per change — enough to settle thirteen of the fourteen knots up to seven crossings, and not the fourteenth.

topology · Knots A ribbon on a lifted figure eight: link 1, twist 0.47, writhe 0.53. A closed ribbon drawn as its core curve and one edge, joined by short ties. The edges have linking number 1; the twist 0.467 and writhe 0.533 add to it.

A whole number split into two that are not

Run a ribbon round a closed loop and its two edges link a whole number of times. That number is shared between two quantities that are nothing like whole numbers: how far the ribbon twists about its core, and how far the core coils about itself. Bend the loop and the twist and the coiling trade continuously, to three decimal places, while their sum stays fixed — the arithmetic behind a coiled telephone cord and a supercoiled loop of DNA.

topology · Linking number Six points in space and the triangles that link, seed 48. A projection of K₆ with straight edges, the under-strands broken at crossings. 1 of the ten pairs of disjoint triangles are linked; the pair 126 and 345 is coloured.

Six points in space and a pair that must link

Put six points anywhere in space and join every pair with a straight segment. Split the six into two triangles — there are ten ways — and at least one of the ten pairs of triangles is linked like two rings of a chain. No placement avoids it. The reason is a parity: moving an edge through another changes exactly two of the ten linking numbers, so their sum stays odd whatever is done.

topology · Linking number A trefoil knot made of six straight sticks. Hexagon with corners (6, 2, 3), (7, 7, 10), (1, 6, 4), (2, 1, 7), (8, 10, 5), (0, 9, 9); three crossings from above, determinant 3, Alexander polynomial t² − t + 1.

Six sticks tie a trefoil, and five cannot

Build a knot from straight sticks joined end to end and ask for the fewest. A trefoil takes six, and the six corners can be whole-number points in a box ten units wide. Five sticks can cross one another five times in a picture, as often as a cinquefoil needs, and still tie nothing — the five crossings always twist three one way and two the other. The fewest sticks is a measure of how knotted a knot is that no diagram shows directly.

topology · Knots How often a closed random polygon is certainly knotted, by length. 10: 0.8% (mean crossings 2.8); 20: 1.5% (mean crossings 7.9); 40: 6.3% (mean crossings 21.2); 60: 10.5% (mean crossings 35.1); 80: 20.8% (mean crossings 51.3); 100: 22.0% (mean crossings 66.9); 130: 32.3% (mean crossings 94.9); 160: 45.7% (mean crossings 122.6); 200: 47.0% (mean crossings 154.5); 250: 51.7% (mean crossings 205.5).

Almost every long loop is knotted

Close a random walk into a loop and ask whether it is knotted. With ten steps almost never; with a hundred, more than one time in five it can be proved knotted by a single number; with two hundred and fifty, more than half. The chance of staying unknotted falls exponentially with length — Frisch, Wasserman and Delbrück guessed it for polymer rings around 1961, and it was proved in 1988 — because a knot needs only one small tangle somewhere, and a long loop has room for many.

topology · Knots A knot drawn with three sines: 5₂ as a Lissajous knot. Frequencies 2, 3, 7 with phases 0.1, 0.7, 0.3; 7 crossings from above; Alexander polynomial 2t − 3 + 2t⁻¹; determinant 7.

Three sines tie a knot

Let a point move with a different sine wave in each of the three directions of space, the three frequencies whole numbers with no common factor, and it traces a closed curve that is usually knotted. With frequencies 2, 3 and 7 the curve is the knot 5₂. The trefoil, the simplest knot of all, can never be made this way: the half-turn that shifting time by half a period performs forces a condition on the knot's polynomial that the trefoil fails.

analysis · Circular functions Seven points in space and a knotted path through all of them, seed 12. A projection of K₇ with straight edges, faint, and one Hamiltonian cycle drawn heavy with its crossings broken: a trefoil. 9 of the 360 Hamiltonian cycles are knotted.

Seven points and a knot they cannot avoid

Put seven points anywhere in space and join every pair with a straight segment. There are 360 closed paths that visit all seven points once each, and at least one of them is knotted — however the points are placed. The reason is a parity, as it was for six points and a linked pair: a number read off each path's knot adds up, over all 360, to something odd. Six points are not enough; seven always are.

topology · Linking number

Named alongside it

The objects these essays reach for when they reach for this one.

InvariantKnot determinantReidemeister movesCrossing numberLinking numberOrientationWritheJones polynomialParityProjectionTopological invariantChirality

All concepts