In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients. Definition by the bracket Suppose we have an oriented link , given as a knot diagram. We will define the Jones polynomial, , using Louis Kauffman's bracket polynomial, which we denote by . Here the bracket polynomial is a Laurent polynomial in the variable with integer coefficients. First, we define the auxiliary polynomial (also known as the normalized bracket polynomial) where denotes the writhe of in its given diagram. The writhe of a diagram is the number of positive crossings ( in the figure below) minus the number of negative crossings (). The writhe is not a knot invariant. is a knot invariant since it is invariant under changes of the diagram of by the three Reidemeister moves. 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 under a type I Reidemeister move. The definition of the polynomial given above is designed to nullify this change, since the writhe changes appropriately by or under type I moves. Now make the substitution in to get the Jones polynomial . This results in a Laurent polynomial with integer coefficients in the variable . Jones polynomial for tangles This construction of the Jones polynomial for tangles is a simple generalization of the Kauffman bracket of a link. The construction was developed by Vladimir Turaev and published in 1990. Let be a non-negative integer and denote the set of all isotopic types of tangle diagrams, with 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 -end oriented tangle an element of the free -module , where is the ring of Laurent polynomials with integer coefficients in the variable . 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