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| In the mathematical field of [[knot theory]], the '''Jones polynomial''' is a [[knot polynomial]] discovered by [[Vaughan Jones]] in 1984.<ref>{{Cite journal|last=Jones |first=V.F.R. |title=A polynomial invariant for knots via von Neumann algebra | year=1985 | journal=Bull. Amer. Math. Soc.(N.S.) | volume=12 | pages=103–111}}</ref> Specifically, it is an [[knot invariant|invariant]] of an oriented [[knot (mathematics)|knot]] or [[link (knot theory)|link]] which assigns to each oriented knot or link a [[Laurent polynomial]] in the variable <math>t^{1/2}</math> with integer coefficients.<ref>JONES POLYNOMIALS, VOLUME AND ESSENTIAL KNOT
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| SURFACES: A SURVEY [http://math.byu.edu/~jpurcell/papers/fkp-survey7.pdf]</ref>
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| ==Definition by the bracket==
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| [[Image:Reidemeister move 1.png|thumb|upright|Type I Reidemeister move]]
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| Suppose we have an [[Link (knot theory)|oriented link]] <math>L</math>, given as a [[knot diagram]]. We will define the Jones polynomial, <math>V(L)</math>, using Kauffman's [[bracket polynomial]], which we denote by <math>\langle~\rangle</math>. Note that here the bracket polynomial is a Laurent polynomial in the variable <math>A</math> with integer coefficients.
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| First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)
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| :<math>X(L) = (-A^3)^{-w(L)}\langle L \rangle </math>,
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| where <math>w(L)</math> denotes the [[writhe]] of <math>L</math> in its given diagram. The writhe of a diagram is the number of positive crossings (<math>L_{+}</math> in the figure below) minus the number of negative crossings (<math>L_{-}</math>). The writhe is not a knot invariant.
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| <math>X(L)</math> is a knot invariant since it is invariant under changes of the diagram of <math>L</math> 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 <math>-A^{\pm 3}</math> under a type I Reidemeister move. The definition of the <math>X</math> polynomial given above is designed to nullify this change, since the writhe changes appropriately by +1 or -1 under type I moves.
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| Now make the substitution <math>A = t^{-1/4} </math> in <math>X(L)</math> to get the Jones polynomial <math>V(L)</math>. This results in a Laurent polynomial with integer coefficients in the variable <math>t^{1/2}</math>.
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| ==Definition by braid representation==
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| 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]].
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| Let a link ''L'' be given. A theorem of [[James Waddell Alexander II|Alexander]]'s states that it is the trace closure of a braid, say with ''n'' strands.<!-- trace closure here is the one that is NOT the plat closure --> Now define a representation <math>\rho</math> of the braid group on ''n'' strands, ''B<sub>n</sub>'', into the [[Temperley–Lieb algebra]] ''TL<sub>n</sub>'' with coefficients in <math>\mathbb Z [A, A^{-1}]</math> and <math>\delta = -A^2 - A^{-2}</math>.<!-- the defn of the algebra here is not the same as currently in the Temperly–lieb article, but is another standard one; that article should either be changed or mention the alternative --> The standard braid generator <math>\sigma_i</math> is sent to <math>A\cdot e_i + A^{-1}\cdot 1</math>, where <math>1, e_1, \dots, e_{n-1}</math> are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.
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| Take the braid word <math>\sigma</math> obtained previously from ''L'' and compute <math>\delta^{n-1} tr \rho(\sigma)</math> where ''tr'' is the [[Markov trace]]. This gives <math>\langle L \rangle</math>, where ''<math>\langle</math> <math>\rangle</math>'' is the bracket polynomial. This can be seen by considering, as Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra.<!-- Diagram algebra is what Kauffman says in his article, but I think by now there is a more standard name for this...maybe Kauffman diagrams? -->
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| 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".
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| ==Properties==
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| 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]]:
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| ::<math> (t^{1/2} - t^{-1/2})V(L_0) = t^{-1}V(L_{+}) - tV(L_{-}) \,</math>
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| where <math>L_{+}</math>, <math>L_{-}</math>, and <math>L_{0}</math> 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:
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| [[Image:Skein (HOMFLY).svg|center|200px]]
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| The definition of the Jones polynomial by the bracket makes it simple to show that for a knot <math>K</math>, the Jones polynomial of its mirror image is given by substitution of <math>t^{-1}</math> for <math>t</math> in <math>V(K)</math>. 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.
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| ==Link with Chern–Simons theory==
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| As first shown by [[Edward Witten]], the Jones polynomial of a given knot <math>\gamma</math> 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]] <math>W_F(\gamma)</math>, associated to <math>\gamma</math>, and the [[fundamental representation]] <math>F</math> of <math>\mathrm{SU}(2)</math>.
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| ==Open problems==
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| *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 [[unlink]]s by the work of [[Morwen Thistlethwaite]].
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| ==See also==
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| *[[HOMFLY polynomial]]
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| *[[Khovanov homology]]
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| *[[Alexander polynomial]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *[[Vaughan Jones]], [http://math.berkeley.edu/~vfr/jones.pdf ''The Jones Polynomial'']
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| *[[Colin Adams (mathematician)|Colin Adams]], ''The Knot Book'', American Mathematical Society, ISBN 0-8050-7380-9
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| *{{cite journal|last=H. Kauffman|first=Louis|title=State models and the jones polynomial|journal=Topology|year=1987|volume=26|issue=3|pages=395–407|doi=10.1016/0040-9383(87)90009-7|url=http://www.sciencedirect.com/science/article/pii/0040938387900097|accessdate=22 December 2012}} (explains the definition by bracket polynomial and its relation to Jones' formulation by braid representation)
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| *{{cite book|last=Lickorish|first=W. B. Raymond|title=An introduction to knot theory|year=1997|publisher=Springer|location=New York; Berlin; Heidelberg; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo|isbn=978-0-387-98254-0|page=175|url=http://www.springer.com/mathematics/geometry/book/978-0-387-98254-0}}
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| *{{cite journal|last=THISTLETHWAITE|first=MORWEN|title=LINKS WITH TRIVIAL JONES POLYNOMIAL|journal=Journal of Knot Theory and Its Ramifications|year=2001|volume=10|issue=04|pages=641–643|doi=10.1142/S0218216501001050|url=http://www.worldscientific.com/doi/abs/10.1142/S0218216501001050}}
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| *{{cite journal|last=Eliahou|first=Shalom|coauthors=Kauffman, Louis H.; Thistlethwaite, Morwen B.|title=Infinite families of links with trivial Jones polynomial|journal=Topology|year=2003|volume=42|issue=1|pages=155–169|doi=10.1016/S0040-9383(02)00012-5|url=http://www.sciencedirect.com/science/article/pii/S0040938302000125}}
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| ==External links==
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| * {{springer|title=Jones-Conway polynomial|id=p/j130040}}
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| * [http://www.math.uic.edu/~kauffman/tj.pdf Links with trivial Jones polynomial] by [[Morwen Thistlethwaite]]
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| {{Knot theory}}
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| [[Category:Knot theory]]
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| [[Category:Polynomials]]
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The author is known by the title of Numbers Wunder. Since she was eighteen she's been operating as a meter reader but she's usually wanted her own business. South Dakota is her birth location but she needs to move because of her family members. One of the issues she loves most is to do aerobics and now she is trying to make cash with it.
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