Concept

Kauffman bracket

A polynomial in a variable A computed from a knot diagram by smoothing every crossing in both possible ways and counting the circles each result leaves. It is unchanged by two of the three Reidemeister moves, and corrected by the writhe it becomes the Jones polynomial.

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

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

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

Named alongside it

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

Invariant Jones polynomial Knot Writhe Alternating knot Chirality Crossing number Euler characteristic Polynomial Reidemeister moves

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