Editing Jones polynomial (section)

==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".