Jones polynomial

The Jones polynomial is one of the most important invariants of knots and links, which is in the knot theory, a branch of topology investigated. It is a Laurent polynomial in.

It was discovered in 1984 by Vaughan FR Jones, who for 1990 was awarded the Fields Medal, among others.

Defined by Kauffman bracket

Be entanglement. The Kauffman Klammerpolynom is a in a graph of associated Laurent polynomial. The normalized Kauffman polynomial is then defined by the formula, wherein the twisting of L respectively. is invariant under Reidemeister moves and therefore defines an invariant of tangles. The Jones polynomial is obtained by in substituted.

Defined by Zopfgruppendarstellungen

Let L be an entanglement. By a theorem of Alexander L is the conclusion of a braid with n components. A representation of the braid group Bn in the Temperley -Lieb algebra with coefficients in TLn and is defined by maps on which the producers of the Temperley -Lieb algebra are the producers.

Be the L associated to braid. Compute the Markov trace is. This gives the Klammerpolynom, from then as in the previous section, the Jones polynomial can be calculated.

Defined by Skein relations

It is the Jones polynomial ( clearly ) be characterized by that the trivial node assigns the value 1 and satisfies the following skein relation:

Where, and oriented link diagrams that differ within a small area in the image below and outside of this area are the same.

Defined by Chern - Simons theory

The Jones polynomial can be defined by means of Chern - Simons theory.

Distinguishability of nodes by means of Jones polynomial

It is an open question whether the unknot is the only knot with trivial Jones polynomial. There are in any case different nodes with the same Jones polynomial, for example, have mutations of a node the same Jones polynomial.