- Jones polynomial
In the mathematical field of

knot theory , the**Jones polynomial**is aknot polynomial discovered byVaughan Jones in 1983. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link aLaurent polynomial in the variable $t^\{1/2\}$ with integer coefficients.**Definition by the bracket**Suppose we have an

oriented link $L$, given as aknot diagram . We will define the Jones polynomial, $V(L)$, using Kauffman'sbracket polynomial , which we denote by $langle~\; angle$. 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) $X(L)\; =\; (-A^3)^\{-w(L)\}langle\; L\; angle$, 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 move s. 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 , instatistical 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 $ho$ of the braid group on "n" strands, "B

_{n}", into theTemperley-Lieb algebra "TL_{n}" with coefficients in $mathbb\; Z\; [A,\; A^\{-1\}]$ and $delta\; =\; -A^2\; -\; A^\{-2\}$. A standard braid generator $sigma\_i$ gets sent to $A\; e\_i\; +\; A^\{-1\}\; 1$, 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 $delta^\{n-1\}\; tr\; ho(sigma)$ where "tr" is the

Markov trace . This gives $<\; L\; >$, where "< >" 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 oriented link diagrams that are identical except in a small region where they differ by crossing change or smoothing as 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

**amphicheiral knot**, a knot equivalent to its mirror image, haspalindromic entries in its Jones polynomial.See the page 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 γ, can be obtained by consideringChern-Simons theory on the three-sphere withgauge group SU(2), and computing thevacuum expectation value of aWilson loop "W"_{"F"}(γ), associated to γ, and thefundamental representation "F" of 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 correspondingunlink s by the work ofMorwen Thistlethwaite .**ee also***

HOMFLY polynomial

*Khovanov homology **References***

Vaughan Jones , [*http://math.berkeley.edu/~vfr/jones.pdf "The Jones Polynomial"*]

*Colin Adams, "The Knot Book", American Mathematical Society, ISBN 0-8050-7380-9

*Louis H. Kauffman , "State models and the Jones polynomial." Topology 26 (1987), no. 3, 395--407. (explains the definition by bracket polynomial and its relation to Jones' formulation by braid representation)

*W. B. R. Lickorish , "An introduction to knot theory." Graduate Texts in Mathematics, 175. Springer-Verlag, New York, (1997). ISBN 0-387-98254-X

*Morwen Thistlethwaite , "Links with trivial Jones polynomial." J. Knot Theory Ramifications 10 (2001), no. 4, 641–643.

*Eliahou, Shalom; Kauffman, Louis H.;Thistlethwaite, Morwen B. "Infinite families of links with trivial Jones polynomial", Topology 42 (2003), no. 1, 155–169.**External links*** [

*http://www.math.uic.edu/~kauffman/tj.pdf Links with trivial Jones polynomial*] byMorwen Thistlethwaite

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