Editing Jones polynomial

{{Short description|Mathematical invariant of a knot or link}}
In the [[mathematics|mathematical]] field of [[knot theory]], the '''Jones polynomial''' is a [[knot polynomial]] discovered by [[Vaughan Jones]] in 1984.<ref>{{Cite journal|last=Jones |first=Vaughan F.R. |author-link=Vaughan Jones| title=A polynomial invariant for knots via von Neumann algebra | year=1985 | journal=[[Bulletin of the American Mathematical Society]]|series= (N.S.) | volume=12 | pages=103–111|mr=0766964 | doi=10.1090/s0273-0979-1985-15304-2| doi-access=free }}</ref><ref>{{Cite journal|last=Jones |first=Vaughan F.R. |author-link=Vaughan Jones| mr=0908150| title= Hecke algebra representations of braid groups and link polynomials|journal= Annals of Mathematics| series=(2) | volume= 126 |year=1987|issue= 2|pages= 335–388|doi=10.2307/1971403|jstor=1971403 }}</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]] [[coefficient]]s.<ref>{{cite web |title=Jones Polynomials, Volume and Essential Knot Surfaces: A Survey |url=https://math.byu.edu/~jpurcell/papers/fkp-survey7.pdf |access-date=2017-07-12 |archive-date=2020-12-09 |archive-url=https://web.archive.org/web/20201209203822/https://math.byu.edu/~jpurcell/papers/fkp-survey7.pdf |url-status=dead }}</ref> 

==Definition by the bracket==
[[Image:Reidemeister move 1.png|thumb|upright|Type I Reidemeister move]]
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> by using [[Louis Kauffman]]'s [[bracket polynomial]], which we denote by <math>\langle~\rangle</math>. Here the bracket polynomial is a [[Laurent polynomial]] in the variable <math>A</math> with integer coefficients.

First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) 
:<math>X(L) = (-A^3)^{-w(L)}\langle L \rangle, </math>

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.

<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 a factor of <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 <math>+1</math> or <math>-1</math> under type I moves.

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>.

===Jones polynomial for tangles===

This construction of the Jones polynomial for [[Tangle (mathematics)|tangles]] is a simple generalization of the [[Bracket polynomial|Kauffman bracket]] of a link. The construction was developed by [[Vladimir Turaev]] and published in 1990.<ref>{{cite journal|last1=Turaev|first1=Vladimir G.|author-link=Vladimir Turaev| title=Jones-type invariants of tangles|journal=Journal of Mathematical Sciences| date=1990|volume=52|pages=2806–2807|doi=10.1007/bf01099242|s2cid=121865582|doi-access=free}}</ref>

Let <math>k</math> be a non-negative integer and <math>S_k</math> denote the set of all isotopic types of tangle diagrams, with <math>2k</math> ends, having no crossing points and no closed components (smoothings). Turaev's construction makes use of the previous construction for the Kauffman bracket and associates to each <math>2k</math>-end oriented tangle an element of the free <math>\mathrm{R}</math>-module <math>\mathrm{R}[S_k]</math>, where  <math>\mathrm{R}</math> is the [[Ring (mathematics)|ring]] of [[Laurent polynomial]]s with integer coefficients in the variable <math>t^{1/2}</math>.

==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 [[Alexander's theorem|theorem of Alexander]] 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]] <math>\operatorname{TL}_n</math> with coefficients in <math>\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.

Take the braid word <math>\sigma</math> obtained previously from <math>L</math> and compute <math>\delta^{n-1} \operatorname{tr} \rho(\sigma)</math> where <math>\operatorname{tr}</math> 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 [[Louis 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? -->

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 taking the value 1 on any diagram of the unknot and satisfies the following [[skein relation]]:

::<math> (t^{1/2} - t^{-1/2})V(L_0)  = t^{-1}V(L_{+}) - tV(L_{-}) \,</math>

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:

[[Image:Skein (HOMFLY).svg|center|200px]]

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 '''[[Chiral knot|amphicheiral 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.

Another remarkable property of this invariant states that the Jones polynomial of an alternating link is an [[alternating polynomial]]. This property was proved by [[Morwen Thistlethwaite]]<ref>{{cite journal|last=Thistlethwaite|first=Morwen B.|author-link=Morwen Thistlethwaite| title=A spanning tree expansion of the Jones polynomial|journal=[[Topology (journal)|Topology]]| date=1987|volume=26|issue=3|pages=297–309|doi=10.1016/0040-9383(87)90003-6|doi-access=free}}</ref> in 1987. Another proof of this last property is due to [[Hernando Burgos-Soto]], who also gave an extension of the property to tangles.<ref>{{cite journal|last=Burgos-Soto|first=Hernando|author-link=Hernando Burgos-Soto | title=The Jones polynomial and the planar algebra of alternating links|journal=Journal of Knot Theory and Its Ramifications|date=2010|volume=19|issue=11|pages=1487–1505|doi=10.1142/s0218216510008510|arxiv=0807.2600|s2cid=13993750}}</ref>

The Jones polynomial is not a complete invariant. There exist an infinite number of non-equivalent knots that have the same Jones polynomial. An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi.<ref>{{Cite book |last=Murasugi |first=Kunio |title=Knot theory and its applications |publisher=Birkhäuser Boston, MA |year=1996 |isbn=978-0-8176-4718-6 |pages=227}}</ref>

== Colored Jones polynomial ==

For a positive integer <math>N</math>, the <math>N</math>-colored Jones polynomial <math>V_N(L,t)</math> is a generalisation of the Jones polynomial. It is the [[Reshetikhin–Turaev invariant]] associated with the <math>(N+1)</math>-irreducible representation of the [[quantum group]] <math>U_q(\mathfrak{sl}_2)</math>. In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional) of <math>U_q(\mathfrak{sl}_2)</math>. One thinks of the strands of a link as being "colored" by a representation, hence the name.

More generally, given a link  <math>L</math> of <math>k</math> components and representations <math>V_1,\ldots,V_k</math> of <math>U_q(\mathfrak{sl}_2)</math>, the <math>(V_1,\ldots,V_k)</math>-colored Jones polynomial <math>V_{V_1,\ldots,V_k}(L,t)</math> is the [[Reshetikhin–Turaev invariant]] associated to <math>V_1,\ldots,V_k</math> (here we assume the components are ordered). Given two representations <math>V</math> and <math>W</math>, colored Jones polynomials satisfy the following two properties:<ref>{{Cite book|arxiv = 1211.6075|doi = 10.1090/conm/613/12235|chapter = Lectures on Knot Homology and Quantum Curves|title = Topology and Field Theories|series = Contemporary Mathematics|year = 2014|last1 = Gukov|first1 = Sergei|last2 = Saberi|first2 = Ingmar|volume = 613|pages = 41–78|isbn = 9781470410155|s2cid = 27676682}}</ref>
:*<math>V_{V\oplus W}(L,t)=V_V(L,t)+V_W(L,t)</math>,
:*<math>V_{V\otimes W}(L,t) = V_{V,W}(L^2,t)</math>, where  <math>L^2</math> denotes the [[Satellite knot|2-cabling]] of  <math>L</math>.
These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.

Let <math>K</math> be a knot. Recall that by viewing a diagram of <math>K</math> as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial of <math>K</math>. Similarly, the <math>N</math>-colored Jones polynomial of <math>K</math> can be given a combinatorial description using the [[Jones-Wenzl idempotents]], as follows:
:*consider the <math>N</math>-cabling <math>K^N</math> of <math>K</math>;
:*view it as an element of the Temperley-Lieb algebra;
:*insert the Jones-Wenzl idempotents on some <math>N</math> parallel strands.
The resulting element of <math>\mathbb{Q}(t)</math> is the <math>N</math>-colored Jones polynomial. See appendix H of <ref>Ohtsuki, Quantum Invariants: A Study of Knots, 3-manifolds, and Their Sets</ref> for further details.

==Relationship to other theories==

===Link with Chern–Simons theory===
As first shown by [[Edward Witten]],<ref>{{Cite journal |last=Witten |first=Edward |year=1989 |title=Quantum Field Theory and the Jones Polynomial |url=http://www.maths.ed.ac.uk/~aar/papers/witten.pdf |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 }}</ref> 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]] <math>\mathrm{SU}(2)</math>, 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>.

===Link with quantum knot invariants===

By substituting <math>e^h</math> for the variable <math>t</math> of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be the [[Vassiliev invariant]] of the knot <math>K</math>. In order to unify the Vassiliev invariants (or, finite type invariants), [[Maxim Kontsevich]] constructed the [[Kontsevich integral]]. The value of the Kontsevich integral, which is the infinite sum of 1, 3-valued [[chord diagram (mathematics)|chord diagram]]s, named the Jacobi chord diagrams, reproduces the Jones polynomial along with the <math>\mathfrak{sl}_2</math> weight system studied by [[Dror Bar-Natan]].

===Link with the volume conjecture===

By numerical examinations on some hyperbolic knots, [[Rinat Kashaev]] discovered that substituting the [[primitive n-th root of unity|''n''-th root of unity]] into the parameter of the [[#Colored Jones polynomial|colored Jones polynomial]] corresponding to the ''n''-dimensional representation, and limiting it as ''n'' grows to infinity, the limit value would give the [[hyperbolic volume]] of the [[knot complement]]. (See [[Volume conjecture]].)

===Link with Khovanov homology===

In 2000 [[Mikhail Khovanov]] constructed a certain chain complex for knots and links and showed that the homology induced from it is a knot invariant (see [[Khovanov homology]]). The Jones polynomial is described as the [[Euler characteristic]] for this homology.

==Detection of the unknot==
It is an [[open problem|open question]] whether there is 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]].<ref>{{Cite journal |last=Thistlethwaite |first=Morwen |date=2001-06-01 |title=Links with trivial jones polynomial |url=https://www.worldscientific.com/doi/abs/10.1142/S0218216501001050 |journal=Journal of Knot Theory and Its Ramifications |volume=10 |issue=4 |pages=641–643 |doi=10.1142/S0218216501001050 |issn=0218-2165}}</ref> It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot.<ref>{{Cite journal|title = Khovanov homology is an unknot-detector|journal = Publications Mathématiques de l'IHÉS|date = 2011-02-11|issn = 0073-8301|pages = 97–208|volume = 113|issue = 1|doi = 10.1007/s10240-010-0030-y|first1 = P. B.|last1 = Kronheimer|first2 = T. S.|last2 = Mrowka|arxiv = 1005.4346|s2cid = 119586228}}</ref>


==See also==
{{Div col|colwidth=25em}}
*[[HOMFLY polynomial]]
*[[Alexander polynomial]]
*[[Volume conjecture]]
*[[Chern–Simons theory]]
*[[Quantum group]]
{{Div col end}}

==Notes==
{{Reflist}}

==References==
*{{cite book|author-link=Colin Adams (mathematician)|first=Colin |last=Adams| title=The Knot Book|publisher=[[American Mathematical Society]] |isbn=0-8050-7380-9|date=2000-12-06 }}
*{{cite web|first= Vaughan|last= Jones|author-link=Vaughan Jones|url=http://math.berkeley.edu/~vfr/jones.pdf |title=The Jones Polynomial}}
*{{cite journal|last=Jones|first=Vaughan|author-link=Vaughan Jones|title=Hecke algebra representations of braid groups and link polynomials|journal=[[Annals of Mathematics]]|year=1987|volume=126|issue=2|pages=335–388|doi=10.2307/1971403|jstor=1971403}}
*{{cite journal|last=Kauffman|first=Louis H.|author-link=Louis Kauffman|title=State models and the Jones polynomial|journal=[[Topology (journal)|Topology]]|year=1987|volume=26|issue=3|pages=395–407|doi=10.1016/0040-9383(87)90009-7|doi-access=free}} (explains the definition by bracket polynomial and its relation to Jones' formulation by braid representation)
*{{cite book|last=Lickorish|first=W. B. Raymond|author-link=W. B. R. Lickorish| 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=https://www.springer.com/mathematics/geometry/book/978-0-387-98254-0}} 
*{{cite journal|last=Thistlethwaite|first=Morwen|author-link=Morwen Thistlethwaite|title=Links with trivial Jones polynomial|journal=[[Journal of Knot Theory and Its Ramifications]]|year=2001|volume=10|issue=4|pages=641–643|doi=10.1142/S0218216501001050}}
*{{cite journal|last1=Eliahou|first1=Shalom|last2=Kauffman|first2= Louis H. |author2-link=Louis Kauffman|last3=Thistlethwaite|first3= Morwen B.|author3-link=Morwen Thistlethwaite |title=Infinite families of links with trivial Jones polynomial|journal=[[Topology (journal)|Topology]]|year=2003|volume=42|issue=1|pages=155–169|doi=10.1016/S0040-9383(02)00012-5|doi-access=free}}
*{{cite journal|author-link=Józef H. Przytycki|last=Przytycki|first=Józef H.|title=Skein modules of 3-manifolds|journal=[[Bulletin of the Polish Academy of Sciences]]|year=1991|volume=39|issue=1–2|pages=91–100|arxiv=math/0611797 }}

==External links==
* {{springer|title=Jones-Conway polynomial|id=p/j130040}}
* [http://www.math.uic.edu/~kauffman/tj.pdf Links with trivial Jones polynomial] by [[Morwen Thistlethwaite]]
* {{Knot Atlas|The Jones Polynomial}}

{{Knot theory}}

[[Category:Knot theory]]
[[Category:Polynomials]]
[[Category:Knot invariants]]