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mathematics, in pictures
Concept

Writhe

The sum of the signs of a knot diagram's crossings, each counted plus or minus one according to how the two directed strands cross. Adding or removing a twist changes it by one, which makes it the correction that turns the Kauffman bracket into an invariant.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The trefoil and its mirror image. The trefoil beside its reflection, with writhe 3 and −3. The Jones polynomials are t + t³ − t⁴ and −t⁻⁴ + t⁻³ + t⁻¹; the Alexander polynomial of both is t − 1 + t⁻¹.

A polynomial that tells left from right

The trefoil and its mirror image have the same colourings, the same determinant and the same Alexander polynomial, and the first proof that they differ was a hard argument about groups. Smooth every crossing both ways, count the circles in each of the resulting pictures, and add up the counts with the right weights: the total changes when the knot is reflected.

topology · Knots The two extreme states of the figure-eight knot. The figure-eight knot, then the state with every crossing smoothed the A way (3 circles) and the state with every crossing smoothed the B way (3 circles). The bracket spans 16 powers of A against 16 for four times the crossings.

The crossings an alternating knot cannot lose

Peter Guthrie Tait drew knots for years and believed, without proof, that a diagram whose crossings alternate over and under, and which has no twist that can be undone, is already drawn with the fewest crossings the knot allows. The proof took a century, and when it came it needed only the two most extreme ways of smoothing the diagram and Euler's count of the regions of a map.

topology · Knots A ribbon on a lifted figure eight: link 1, twist 0.47, writhe 0.53. A closed ribbon drawn as its core curve and one edge, joined by short ties. The edges have linking number 1; the twist 0.467 and writhe 0.533 add to it.

A whole number split into two that are not

Run a ribbon round a closed loop and its two edges link a whole number of times. That number is shared between two quantities that are nothing like whole numbers: how far the ribbon twists about its core, and how far the core coils about itself. Bend the loop and the twist and the coiling trade continuously, to three decimal places, while their sum stays fixed — the arithmetic behind a coiled telephone cord and a supercoiled loop of DNA.

topology · Linking number A trefoil knot made of six straight sticks. Hexagon with corners (6, 2, 3), (7, 7, 10), (1, 6, 4), (2, 1, 7), (8, 10, 5), (0, 9, 9); three crossings from above, determinant 3, Alexander polynomial t² − t + 1.

Six sticks tie a trefoil, and five cannot

Build a knot from straight sticks joined end to end and ask for the fewest. A trefoil takes six, and the six corners can be whole-number points in a box ten units wide. Five sticks can cross one another five times in a picture, as often as a cinquefoil needs, and still tie nothing — the five crossings always twist three one way and two the other. The fewest sticks is a measure of how knotted a knot is that no diagram shows directly.

topology · Knots

Named alongside it

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

InvariantKnotCrossing numberJones polynomialKauffman bracketAlternating knotChiralityEuler characteristicFramingIntegralKnot determinantLinking number

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