Matpic
mathematics, in pictures
Concept

Tricolourability

The property of a knot diagram that its arcs can be given three colours with the three at every crossing all alike or all different. It is a knot invariant because the three Reidemeister moves preserve it, which is what makes it able to tell the trefoil from the unknot.

Named by 2 essays across one field — 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

Named alongside it

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

InvariantKnotReidemeister movesTopological invariantCounting argumentCrossing numberDecidabilityEquivalenceKnot determinantLinear systemModular arithmeticOrientation

All concepts