Jones polynomial

Type I Reidemeister move

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

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

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

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

${\displaystyle X(L)}$ is a knot invariant since it is invariant under changes of the diagram of ${\displaystyle L}$ 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 ${\displaystyle -A^{\pm 3}}$ under a type I Reidemeister move. The definition of the ${\displaystyle X}$ polynomial given above is designed to nullify this change, since the writhe changes appropriately by ${\displaystyle +1}$ or ${\displaystyle -1}$ under type I moves.

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

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

Take the braid word ${\displaystyle \sigma }$ obtained previously from ${\displaystyle L}$ and compute ${\displaystyle \delta ^{n-1}\operatorname {tr} \rho (\sigma )}$ where ${\displaystyle \operatorname {tr} }$ is the Markov trace. This gives ${\displaystyle \langle L\rangle }$, where ${\displaystyle \langle }$ ${\displaystyle \rangle }$ is the bracket polynomial. This can be seen by considering, as Louis 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".

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

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

where ${\displaystyle L_{+}}$, ${\displaystyle L_{-}}$, and ${\displaystyle L_{0}}$ 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 ${\displaystyle K}$, the Jones polynomial of its mirror image is given by substitution of ${\displaystyle t^{-1}}$ for ${\displaystyle t}$ in ${\displaystyle V(K)}$. Thus, an 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 [5] in 1987. Another proof of this last property is due to Hernando Burgos-Soto, who also gave an extension to tangles[6] of the property.

N colored Jones Polynomial: The N cables of L are parallel with each other along the knot L and each is colored a different color.

For a positive integer N a N-colored Jones polynomial ${\displaystyle V_{N}(L,t)}$ can be defined as the Jones polynomial for N cables of the knot ${\displaystyle L}$ as depicted in the right figure. It is associated with an ${\displaystyle (N+1)}$-dimensional irreducible representation of ${\displaystyle \operatorname {SU} (2)}$ . The label N stands for coloring. Like the ordinary Jones polynomial it can be defined by Skein relation and is a Laurent polynomial in one variable t . The N-colored Jones polynomial ${\displaystyle V_{N}(L,t)}$ has the following properties:

• ${\displaystyle V_{X\oplus Y}(L,t)=V_{X}(L,t)+V_{Y}(L,t)}$ where ${\displaystyle X,Y}$ are two representation space.
• ${\displaystyle V_{X\otimes Y}(L,t)}$ equals the Jones polynomial of the 2-cables of L with two components labeled by ${\displaystyle X}$ and ${\displaystyle Y}$ . So the N-colored Jones polynomial equals the original Jones polynomial of the N cables of ${\displaystyle L}$ .
• The original Jones polynomial appears as a special case: ${\displaystyle V(L,t)=V_{1}(L,t)}$ .

• 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.

Problem（Extension of Jones polynomial to general 3-manifolds）　

``The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space ${\displaystyle \mathbb {R} ^{3}}$). Can you define the Jones polynomial for 1-links in any 3-manifold?’’

Such an approach was proposed by Józef H. Przytycki under the name of skein modules. In particular, the Kauffman bracket skein module and HOMFLYPT skein module. [7]

See section 1.1 of this paper[8] for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots.[9] It is open in the other cases. Witten’s path integral for the Jones polynomial is written formally for links in any compact 3-manifold, but the calculus is not done even at the physics level in any other case than the 3-sphere (the 3-ball, or the 3-space ${\displaystyle \mathbb {R} ^{3}}$). This problem is also open at the physics level. In the case of the Alexander polynomial, this problem is solved.

• HOMFLY polynomial
• Alexander polynomial

• Adams, Colin (2000-12-06). The Knot Book. American Mathematical Society. ISBN 0-8050-7380-9.
• Jones, Vaughan. "The Jones Polynomial" (PDF).
• Jones, Vaughan (1987). "Hecke algebra representations of braid groups and link polynomials". Annals of Mathematics. 126 (2): 335–388. doi:10.2307/1971403.
• Kauffman, Louis H. (1987). "State models and the Jones polynomial". Topology. 26 (3): 395–407. doi:10.1016/0040-9383(87)90009-7. (explains the definition by bracket polynomial and its relation to Jones' formulation by braid representation)
• Lickorish, W. B. Raymond (1997). An introduction to knot theory. New York; Berlin; Heidelberg; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer. p. 175. ISBN 978-0-387-98254-0.
• Thistlethwaite, Morwen (2001). "Links with trivial Jones polynomial". Journal of Knot Theory and Its Ramifications. 10 (4): 641–643. doi:10.1142/S0218216501001050.
• Eliahou, Shalom; Kauffman, Louis H.; Thistlethwaite, Morwen B. (2003). "Infinite families of links with trivial Jones polynomial". Topology. 42 (1): 155–169. doi:10.1016/S0040-9383(02)00012-5.
• Przytycki, Józef H. (1991). "Skein modules of 3-manifolds". Bulletin of the Polish Academy of Sciences. 39 (1–2): 91–100. arXiv:math/0611797.

• "Jones-Conway polynomial", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Links with trivial Jones polynomial by Morwen Thistlethwaite
• "The Jones Polynomial", The Knot Atlas.