Matpic
mathematics, in pictures
Generator

the trefoil

A generator in the topology library, called 72 times across 13 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

knot is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

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.

the figure-eight knot, coloured with 5 colours

the figure-eight knot, coloured with 5 colours. The 4 arcs of the figure-eight knot coloured with the numbers 0 to 4 so that at every crossing twice the over-strand equals the sum of the two under-strands, modulo 5. 25 colourings obey the rule.

The Alexander matrix of the figure-eight knot

The Alexander matrix of the figure-eight knot. The figure-eight knot with its 4 arcs numbered and its 4 crossings lettered, beside the 4 by 4 matrix they give. A minor of the matrix is the Alexander polynomial −t + 3 − t⁻¹, whose value at −1 is the determinant 5.

Four knots, their determinants and their Alexander polynomials

Four knots, their determinants and their Alexander polynomials. A table of 4 knots with crossing number, determinant, Alexander polynomial and the polynomial's value at −1. The figure-eight knot and the cinquefoil have the same determinant and different polynomials.

Alexander polynomials around the unit circle

Alexander polynomials around the unit circle. The symmetric Alexander polynomials of the trefoil, the figure-eight knot, the cinquefoil evaluated at t = e^(iθ) for θ from 0 to π. All start at 1 and end at plus or minus their determinants.

One crossing, smoothed two ways

One crossing, smoothed two ways. A crossing with its four corners marked A and B, then the A smoothing, which joins the two A corners into one region, and the B smoothing, which joins the two B corners. The bracket of a diagram is A times the bracket with the A smoothing plus A inverse times the bracket with the B smoothing.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Topology

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

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

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

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

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

Linked, and no two of them are

Three rings that cannot be pulled apart, in which every pair comes apart the moment the third is removed. Every pairwise linking number is zero, so the number cannot see it — and what does see it is a word in two letters that refuses to cancel.

 Topology

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

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

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

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

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

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

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.

The whole library · What the figures prove