Jones polynomial

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 {displaystyle t^{1/2}} with integer coefficients.Suppose we have an oriented link L {displaystyle L}  , given as a knot diagram. We will define the Jones polynomial, V ( L ) {displaystyle V(L)}  , using Kauffman's bracket polynomial, which we denote by ⟨   ⟩ {displaystyle langle ~ angle }  . Note that here the bracket polynomial is a Laurent polynomial in the variable A {displaystyle A}   with integer coefficients.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.The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following skein relation:For a positive integer N a N-colored Jones polynomial V N ( L , t ) {displaystyle V_{N}(L,t)}   can be defined as the Jones polynomial for N cables of the knot L {displaystyle L}   as depicted in the right figure. It is associated with an ( N + 1 ) {displaystyle (N+1)}  -dimensional irreducible representation of SU ⁡ ( 2 ) {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 V N ( L , t ) {displaystyle V_{N}(L,t)}   has the following properties:As first shown by Edward Witten, the Jones polynomial of a given knot γ {displaystyle gamma }   can be obtained by considering Chern–Simons theory on the three-sphere with gauge group S U ( 2 ) {displaystyle mathrm {SU} (2)}  , and computing the vacuum expectation value of a Wilson loop W F ( γ ) {displaystyle W_{F}(gamma )}  , associated to γ {displaystyle gamma }  , and the fundamental representation F {displaystyle F}   of S U ( 2 ) {displaystyle mathrm {SU} (2)}  .