Editing Jones polynomial (section)

== Colored Jones polynomial ==

For a positive integer <math>N</math>, the <math>N</math>-colored Jones polynomial <math>V_N(L,t)</math> is a generalisation of the Jones polynomial. It is the [[Reshetikhin–Turaev invariant]] associated with the <math>(N+1)</math>-irreducible representation of the [[quantum group]] <math>U_q(\mathfrak{sl}_2)</math>. In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional) of <math>U_q(\mathfrak{sl}_2)</math>. One thinks of the strands of a link as being "colored" by a representation, hence the name.

More generally, given a link  <math>L</math> of <math>k</math> components and representations <math>V_1,\ldots,V_k</math> of <math>U_q(\mathfrak{sl}_2)</math>, the <math>(V_1,\ldots,V_k)</math>-colored Jones polynomial <math>V_{V_1,\ldots,V_k}(L,t)</math> is the [[Reshetikhin–Turaev invariant]] associated to <math>V_1,\ldots,V_k</math> (here we assume the components are ordered). Given two representations <math>V</math> and <math>W</math>, colored Jones polynomials satisfy the following two properties:<ref>{{Cite book|arxiv = 1211.6075|doi = 10.1090/conm/613/12235|chapter = Lectures on Knot Homology and Quantum Curves|title = Topology and Field Theories|series = Contemporary Mathematics|year = 2014|last1 = Gukov|first1 = Sergei|last2 = Saberi|first2 = Ingmar|volume = 613|pages = 41–78|isbn = 9781470410155|s2cid = 27676682}}</ref>
:*<math>V_{V\oplus W}(L,t)=V_V(L,t)+V_W(L,t)</math>,
:*<math>V_{V\otimes W}(L,t) = V_{V,W}(L^2,t)</math>, where  <math>L^2</math> denotes the [[Satellite knot|2-cabling]] of  <math>L</math>.
These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.

Let <math>K</math> be a knot. Recall that by viewing a diagram of <math>K</math> as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial of <math>K</math>. Similarly, the <math>N</math>-colored Jones polynomial of <math>K</math> can be given a combinatorial description using the [[Jones-Wenzl idempotents]], as follows:
:*consider the <math>N</math>-cabling <math>K^N</math> of <math>K</math>;
:*view it as an element of the Temperley-Lieb algebra;
:*insert the Jones-Wenzl idempotents on some <math>N</math> parallel strands.
The resulting element of <math>\mathbb{Q}(t)</math> is the <math>N</math>-colored Jones polynomial. See appendix H of <ref>Ohtsuki, Quantum Invariants: A Study of Knots, 3-manifolds, and Their Sets</ref> for further details.