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Wikipedia / Jones polynomial

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 with integer coefficients.

Definition by the bracket

Suppose we have an oriented link, given as a knot diagram by using Louis Kauffman's bracket polynomial. Here the bracket polynomial is a Laurent polynomial with integer coefficients.

First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)

denotes the writhe). The writhe is not a knot invariant.

by the three Reidemeister moves under type I moves.

.

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.

is the ring of Laurent polynomials.

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 link L be given. A theorem of Alexander states that it is the trace closure of a braid, say with n strands. of the braid group on n strands, Bn, into the Temperley–Lieb algebra. are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.

is the Markov trace 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:

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:

. 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 of the property to tangles.

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

is a generalisation of the Jones polynomial. It is the Reshetikhin–Turaev invariant-irreducible representation of the quantum group. One thinks of the strands of a link as being "colored" by a representation, hence the name.

is the Reshetikhin–Turaev invariant, colored Jones polynomials satisfy the following two properties:

These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.

can be given a combinatorial description using the Jones-Wenzl idempotents, as follows:

  • ;

  • view it as an element of the Temperley-Lieb algebra;

  • parallel strands.

-colored Jones polynomial. See appendix H of for further details.

Relationship to other theories

As first shown by Edward Witten, 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, and the fundamental representation.

of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be the Vassiliev invariant. 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 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.

Detection of the unknot

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. The simplest such example has 15 essential crossings and consists of a trefoil knot linked to a figure-eight knot. Every prime knot with up to 24 crossings, of which there are over three hundred million, is known to have a non-trivial Jones polynomial. This was determined by computing the Jones polynomial of several trillion knot diagrams, and verifying those diagrams with trivial Jones polynomials corresponded to trivial knots. It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot. While no nontrivial knot is known to have the same Jones polynomial as the unknot, two or more knots may have the same Jones polynomial and require other invariants to distinguish them. Examples include the Conway knot and the Kinoshita-Terasaka knot, with 11 crossings.

See also

Notes

References

  • (explains the definition by bracket polynomial and its relation to Jones' formulation by braid representation)