Matpic
mathematics, in pictures
Generator

knot

A generator in the topology library, called 6 times across 1 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 trefoilthe 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 trefoil

show: "table"

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

show: "moves"

The three legal movesReidemeister'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.1. twist it out↕2. pull them apart↕3. slide it across↕

show: "tricolour"

Three colours, and the knot that refuses themThe 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.three arcs, three colours: allowedone arc, one colour: not alloweda quantity that survives all three moves tells the two apart

What it checks while it draws

Collected by running the family and listening to lib/verify.js, not written here. The count is how many separate times this build put that claim to the test.

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

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.

The whole library · What the figures prove