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In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable $t^{1/2}$ with integer coefficients.

Definition by the bracket

Type I Reidemeister move

Suppose we have an oriented link $L$, given as a knot diagram. We will define the Jones polynomial, $V(L)$, using Kauffman's bracket polynomial, which we denote by $\langle ~\rangle$. Note that here the bracket polynomial is a Laurent polynomial in the variable $A$ with integer coefficients.

First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)

$X(L)=(-A^{3})^{-w(L)}\langle L\rangle$,

where $w(L)$ denotes the writhe of $L$ in its given diagram. The writhe of a diagram is the number of positive crossings ($L_{+}$ in the figure below) minus the number of negative crossings ($L_{-}$). The writhe is not a knot invariant.

$X(L)$ is a knot invariant since it is invariant under changes of the diagram of $L$ by the three Reidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by multiplication by $-A^{\pm 3}$ under a type I Reidemeister move. The definition of the $X$ polynomial given above is designed to nullify this change, since the writhe changes appropriately by +1 or -1 under type I moves.

Now make the substitution $A=t^{-1/4}$ in $X(L)$ to get the Jones polynomial $V(L)$. This results in a Laurent polynomial with integer coefficients in the variable $t^{1/2}$.

Definition by braid representation

Jones' original formulation of his polynomial came from his study of operator algebras. In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which originally arose while studying certain models, e.g. the Potts model, in statistical mechanics.

Let a link L be given. A theorem of Alexander's states that it is the trace closure of a braid, say with n strands. Now define a representation $\rho$ of the braid group on n strands, B_n, into the Temperley–Lieb algebra TL_n with coefficients in $\mathbb {Z} [A,A^{-1}]$ and $\delta =-A^{2}-A^{-2}$. The standard braid generator $\sigma _{i}$ is sent to $A\cdot e_{i}+A^{-1}\cdot 1$, where $1,e_{1},\dots ,e_{n-1}$ are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.

Take the braid word $\sigma$ obtained previously from L and compute $\delta ^{n-1}tr\rho (\sigma )$ where tr is the Markov trace. This gives $\langle L\rangle$, where $\langle$ $\rangle$ is the bracket polynomial. This can be seen by considering, as Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra.

An advantage of this approach is that one can pick similar representations into other algebras, such as the R-matrix representations, leading to "generalized Jones invariants".

Properties

The Jones polynomial is characterized by the fact that it takes the value 1 on any diagram of the unknot and satisfies the following skein relation:

$(t^{1/2}-t^{-1/2})V(L_{0})=t^{-1}V(L_{+})-tV(L_{-})\,$

where $L_{+}$, $L_{-}$, and $L_{0}$ are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below:

The definition of the Jones polynomial by the bracket makes it simple to show that for a knot $K$, the Jones polynomial of its mirror image is given by substitution of $t^{-1}$ for $t$ in $V(K)$. Thus, an amphichiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein relation for an example of a computation using these relations.

Link with Chern–Simons theory

As first shown by Edward Witten, the Jones polynomial of a given knot $\gamma$ can be obtained by considering Chern–Simons theory on the three-sphere with gauge group SU(2), and computing the vacuum expectation value of a Wilson loop $W_{F}(\gamma )$, associated to $\gamma$, and the fundamental representation $F$ of $\mathrm {SU} (2)$.

Open problems

• Is there a nontrivial knot with Jones polynomial equal to that of the unknot? It is known that there are nontrivial links with Jones polynomial equal to that of the corresponding unlinks by the work of Morwen Thistlethwaite.

See also

• HOMFLY polynomial
• Khovanov homology
• Alexander polynomial

Notes

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References

• Vaughan Jones, The Jones Polynomial
• Colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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