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mathematics, in pictures
Concept

Jones polynomial

A polynomial in t attached to every knot by Vaughan Jones in 1984, unchanged by every deformation of the knot. Unlike the Alexander polynomial it can tell a knot from its mirror image, and its span gives the exact crossing number of every alternating knot.

Named by 3 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 Signature against unknotting, for every knot to seven crossings. A table of the fourteen prime knots with up to seven crossings, listing determinant, signature, the lower bound on crossing changes it gives, the number of changes found by search, and whether the two agree.

How many changes undo a knot

Cut the string at a crossing, pass it through the other strand and join it up again, and any knot can be undone by doing that often enough. The fewest changes needed is the unknotting number, and proving that fewer will not do needs a number that each change can move only a little. The signature moves by at most two per change — enough to settle thirteen of the fourteen knots up to seven crossings, and not the fourteenth.

topology · Knots

Named alongside it

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

InvariantKnotChiralityKauffman bracketWritheAlternating knotCrossing numberEuler characteristicKnot determinantPolynomialQuadratic formReidemeister moves

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