link L be given. A theorem of Alexander states that it is the trace closure of a braid, say with n strands. Now define a representation of the braid group on n strands, Bn, into the Temperley–Lieb algebra with coefficients in and . The standard braid generator is sent to , where are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation. Take the braid word obtained previously from and compute where is the Markov trace. This gives , where is the bracket polynomial. This can be seen by considering, as Louis Kauffman did, the Temperley–Lieb algebra as a particular diagram algebra. 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". Properties The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the following skein relation: where , , and are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below: The definition of the Jones polynomial by the bracket makes it simple to show that for a knot , the Jones polynomial of its mirror image is given by substitution of for in . Thus, an amphicheiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial. See the article on skein relation for an example of a computation using these relations. Another remarkable property of this invariant states that the Jones polynomial of an alternating link is an alternating polynomial. This property was proved by Morwen Thistlethwaite in 1987. Another proof of this last property is due to Hernando Burgos-Soto, who also gave an extension to tangles of the property. The Jones polynomial is not a complete invariant. There exist an infinite number of non-equivalent knots that have the same Jones polynomial. An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi. Colored Jones polynomial For a positive integer , the -colored Jones polynomial is a generalisation of the Jones polynomial. It is the Reshetikhin–Turaev invariant associated with the -irreducible representation of the quantum group . In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible