JonesPolynomial

The Jones polynomial $V(t)$ of a knot. A knot named $T(p,q)$ 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 $\delta = -A^2 - A^{-2}$.

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

JonesPolynomial(knot)$V(t)$, for a knot however it is named

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

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

KauffmanBracket(knot)$\langle L\rangle$ in $A$ — defined for links too

BracketInvariant(knot)$(-A^3)^{-w}\langle L\rangle$, already an invariant

Typed signatures

Type	Package	Applies
(expression<Braid> | expression<FigureEightKnot> | expression<Link> | expression<PretzelKnot> | expression<TorusKnot> | expression<TwistKnot> | string) -> expression	braid	always

Domain: Braids and knots

Details

• $\sigma_i \mapsto A\cdot 1 + A^{-1}e_i$ and $\sigma_i^{-1} \mapsto A^{-1}\cdot 1 + A e_i$ — the two images ARE the two smoothings of a crossing
• The raw bracket is not an invariant: one positive crossing on an unknot leaves $-A^3$ behind, which the writhe correction divides out
• $V(t)$ follows by substituting $A = t^{1/4}$; 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 $t$, so JonesPolynomial declines and the bracket is what to ask for
• The knot determinant is both $|\Delta(-1)|$ and $|V(-1)|$ — 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

Basic #

the right-handed trefoil

live ↗

In$\mathrm{JonesPolynomial}(\mathrm{Braid}(2, \bigl\lbrack1, 1, 1\bigr\rbrack))$Out$-\frac{1}{t^4}+\frac{1}{t}+\frac{1}{t^3}$

Applications #

the shortest knotted modular geodesic, by yet another route

live ↗

In$\mathrm{JonesPolynomial}(\text{LLRLR})$Out$-\frac{1}{t^4}+\frac{1}{t}+\frac{1}{t^3}$

the figure-eight, through the braid it carries as TwistKnot(1)

live ↗

In$\mathrm{JonesPolynomial}(\mathrm{FigureEightKnot}())$Out$t^2-t-\frac{1}{t}+\frac{1}{t^2}+1$

Properties #

not yet an invariant

live ↗

In$\mathrm{KauffmanBracket}(\mathrm{Braid}(2, \bigl\lbrack1\bigr\rbrack))$Out$-A^3$

…and the writhe correction fixes it

live ↗

In$\mathrm{BracketInvariant}(\mathrm{Braid}(2, \bigl\lbrack1\bigr\rbrack))$Out$1$

About

Examples	5
Crosswalk	WikipediaMathWorldWikidatanLabBritannica

See also: AlexanderPolynomial, Braid, LorenzBraid