Толбағайлар теориясының математикалық инварианты: Джонс полиномы

Mathematical invariant of a knot or link
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.

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 .

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 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".

As first shown by Edward Witten, the Jones polynomial of a given knot can be obtained by considering Chern–Simons theory on the three sphere with gauge group , and computing the vacuum expectation value of a Wilson loop , associated to , and the fundamental representation of .

By substituting for the variable of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be the Vassiliev invariant of the knot In order to unify the Vassiliev invariants (or, finite type invariants), Maxim Kontsevich constructed the Kontsevich integral. The value of the Kontsevich integral, which is the infinite sum of 1, 3 valued chord diagrams, named the Jacobi chord diagrams, reproduces the Jones polynomial along with the weight system studied by Dror Bar Natan.

By numerical examinations on some hyperbolic knots, Rinat Kashaev discovered that substituting the n th root of unity into the parameter of the colored Jones polynomial corresponding to the n dimensional representation, and limiting it as n grows to infinity, the limit value would give the hyperbolic volume of the knot complement. (See Volume conjecture.)

In 2000 Mikhail Khovanov constructed a certain chain complex for knots and links and showed that the homology induced from it is a knot invariant (see Khovanov homology). The Jones polynomial is described as the Euler characteristic for this homology.

It is an open question whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. It is known that there are nontrivial links with Jones polynomial equal to that of the corresponding unlinks by the work of Morwen Thistlethwaite. It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot.