Jones polynomial

The Jones polynomial is one of the most important invariants of knots and links that is investigated in knot theory , a branch of topology . It's a Laurent polynomial in . ${\displaystyle {\sqrt {t}}}$

It was discovered in 1984 by Vaughan FR Jones , who received the Fields Medal for it in 1990 .

Definition by Kauffman brackets

Be a tangle . The Kauffman polynomial in brackets is a Laurent polynomial associated with a diagram of in . The normalized Kauffman polynomial is then defined by the formula , wherein the twisting of from designated. is invariant under Reidemeister movements and therefore defines an invariant of links. The Jones polynomial is obtained by in groups. ${\displaystyle L}$ ${\displaystyle \langle L\rangle }$ ${\displaystyle L}$ ${\displaystyle A}$ ${\displaystyle X(L)=(-A^{3})^{-w(L)}\langle L\rangle }$ ${\displaystyle w(L)}$ ${\displaystyle L}$ ${\displaystyle X(L)}$ ${\displaystyle V(L)}$ ${\displaystyle A=t^{-1/4}}$ ${\displaystyle X(L)}$

Definition through braid group representations

Be a tangle. According to one of Alexander's sentences, it is the completion of a braid with components. A representation of the Zopf group in the Temperley – Lieb algebra with coefficients in and is defined by mapping the producer to , where the producers are the Temperley – Lieb algebra. ${\displaystyle L}$ ${\displaystyle L}$ ${\displaystyle n}$ ${\displaystyle \rho }$ ${\displaystyle B_{n}}$ ${\displaystyle TL_{n}}$ ${\displaystyle \mathbb {Z} [A,A^{-1}]}$ ${\displaystyle \delta =-A^{2}-A^{-2}}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle A\cdot e_{i}+A^{-1}\cdot 1}$ ${\displaystyle 1,e_{1},\dots ,e_{n-1}}$

Be the braid too associated. Compute where is the Markov trace . This gives the polynomial in brackets , from which the Jones polynomial can be calculated as in the previous section. ${\displaystyle \sigma }$ ${\displaystyle L}$ ${\displaystyle \delta ^{n-1}\operatorname {tr} \rho (\sigma )}$ ${\displaystyle \operatorname {tr} }$ ${\displaystyle \langle L\rangle }$

Definition by skein relations

The Jones polynomial can be characterized (unambiguously) by assigning the value 1 to the trivial node and satisfying the following Skein relation :

${\displaystyle (t^{1/2}-t^{-1/2})V(L_{0})=t^{-1}V(L_{+})-tV(L_{-})\,,}$

where , and are oriented link diagrams that differ within a small area as in the image below and are identical outside of this area. ${\displaystyle L_{+}}$ ${\displaystyle L_{-}}$ ${\displaystyle L_{0}}$

Definition by Chern-Simons theory

According to Edward Witten , the Jones polynomial can be defined using a topological quantum field theory , the Chern-Simons theory .

Applications

Kauffman , Murasugi, and Thistlethwaite used the Jones polynomial to prove one of the 19th-century Tait conjectures : for an alternating knot , each reduced diagram has the smallest possible number of intersections .

Distinguishability of nodes by means of the Jones polynomial

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

Special values

• For a knot is , for an entanglement with components is . ${\displaystyle V(1)=1}$ ${\displaystyle l\geq 2}$ ${\displaystyle V(1)={\frac {1}{2^{l-1}}}}$
• If the Arf invariant is defined, is . ${\displaystyle V(i)=(-{\sqrt {2}})^{l-1}(-1)^{Arf(L)}}$
• ${\displaystyle V(e^{\frac {2\pi i}{3}})=1}$.
• The values ​​in roots of unity are important in the Chern-Simons theory .

See also

• Alexander polynomial
• HOMFLY polynomial

literature

• Vaughan FR Jones: A polynomial invariant for knots via von Neumann algebras . In: Hyman Bass, Meyer Jerison, Calvin C. Moore (Eds.): Bulletin of the American Mathematical Society (New Series) . Vol. 12, No. 1 . American Mathematical Society, 1985, ISSN  0273-0979 , pp. 103–111 , doi : 10.1090 / S0273-0979-1985-15304-2 ( ams.org [PDF; accessed December 2, 2012]).
• Louis H. Kauffman: State models and the Jones polynomial . In: Topology . Vol. 26, No. 3 . Elsevier, 1987, ISSN  0040-9383 , pp. 395-407 , doi : 10.1016 / 0040-9383 (87) 90009-7 ( knot.kaist.ac.kr [PDF; accessed December 2, 2012]).
• Pierre de la Harpe, Michel Kervaire, Claude Weber: On the Jones polynomial . In: Enseign. Math. (2) 32 (1986) no. 3-4, pp. 271-335.
• WB Raymond Lickorish: An introduction to knot theory (=  Graduate Texts in Mathematics . Volume 175 ). Springer, New York 1997, ISBN 0-387-98254-X .

Web links

• Edward Witten : Two Lectures on the Jones Polynomial and Khovanov Homology . (PDF)
• Alan Chang: On the Jones polynomial and its applications .

Individual evidence

1. ↑ Witten, op.cit.