Editing Jones polynomial (section)

==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>.