Matpic
mathematics, in pictures
Concept

Knot — where it appears

A closed curve in space, considered up to deformations that never pass one strand through another.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as reidemeister moves, tricolourability — the same set of essays touches all of them, so they are one junction rather than several.

the trefoil

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 3 colours5 colours7 coloursdeterminantthe unknot3 — all one colour5 — all one colour7 — all one colour1the trefoil95 — all one colour7 — all one colour3the figure-eight knot3 — all one colour257 — all one colour5every count is taken by trying every assignment of colours to arcs and keeping the ones the rule allowsa knot takes p colours in more than one way exactly when p divides its determinant, and the determinant is computedhere from the crossings by exact whole-number elimination

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.

InvariantReidemeister movesTopological invariantTricolourabilityCounting argumentCrossing numberDecidabilityEquivalenceKnot determinantLinear systemModular arithmeticOrientation

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