JonesPolynomial

The Jones polynomial of a knot. A knot named takes the closed form; anything else goes through the TEMPERLEY–LIEB algebra rather than a matrix representation, where each crossing becomes its two smoothings and each closed loop is worth .

WikipediaJones polynomial↗MathWorldJonesPolynomial↗WikidataQ1528291↗nLabJones polynomial↗Britannicatopic/Jones-polynomial↗

JonesPolynomial(knot), for a knot however it is named

TorusKnot(p, q) as a knot — no braid, no bracket, no diagrams

FigureEightKnot()the one twist knot with a braid word already on file

KauffmanBracket(knot) in — defined for links too

BracketInvariant(knot), already an invariant

Domain: Braids and knots

Details

• and — the two images ARE the two smoothings of a crossing
• The raw bracket is not an invariant: one positive crossing on an unknot leaves behind, which the writhe correction divides out
• follows by substituting ; the SIGN of that exponent is the handedness convention, and getting it backwards mirrors every answer
• A link with an even number of components needs a square root of , so `JonesPolynomial` declines and the bracket is what to ask for
• The knot determinant is both and — one from Burau matrices, the other from diagrams, which is how each checks the other
• The Temperley–Lieb diagrams used here are the ones the diagram-algebra package catalogues, with the same product

Examples

See also: AlexanderPolynomial, Braid, LorenzBraid