Editing Jones polynomial (section)

===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>.