Matpic
mathematics, in pictures
Concept

Chirality

The property of an object that cannot be moved onto its own mirror image, as a left hand cannot become a right. The trefoil knot has it and the figure-eight knot does not, and invariants such as the Jones polynomial and the signature tell the two hands apart.

Named by 2 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 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.

InvariantJones polynomialKnotKauffman bracketKnot determinantPolynomialQuadratic formReidemeister movesUnknotting numberWrithe

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