Matpic
mathematics, in pictures
Generator

the trefoil

A generator in the topology library, called 32 times across 6 essays. Below: what it draws at its defaults 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.

At its defaults

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.

show: "colours"

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.

show: "colour"

the trefoil, coloured with 3 colours. The 3 arcs of the trefoil coloured with the numbers 0 to 2 so that at every crossing twice the over-strand equals the sum of the two under-strands, modulo 3. 9 colourings obey the rule.

show: "moves"

The three legal moves. Reidemeister's three moves: undoing a twist, pulling two strands apart, and sliding a strand across a crossing. Two diagrams are the same knot exactly when a sequence of these turns one into the other.

show: "table"

Three knots, in order of crossings. The unknot, the trefoil and the figure-eight knot, with 0, 3, 4 crossings. No amount of moving the string turns one into another.

show: "tricolour"

Three colours, and the knot that refuses them. The trefoil's three arcs can be given three different colours, and at each crossing all three meet — which the rule allows. The unknot has one arc and so only ever gets one colour, so the two cannot be the same knot.

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

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

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

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

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