Matpic
mathematics, in pictures
Concept

Reidemeister moves

The three local changes to a knot diagram that generate every deformation of the knot itself. Any two diagrams of the same knot are joined by a sequence of them, which is what makes checking an invariant a finite task.

Named by 3 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 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

Named alongside it

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

KnotTopological invariantCrossing numberInvariantOrientationTricolourabilityCounting argumentDecidabilityDeformationEquivalenceKnot determinantLinear system

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