To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Kauffman polynomial

In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman.[1] It is initially defined on a link diagram as

${\displaystyle F(K)(a,z)=a^{-w(K)}L(K)\,}$

where ${\displaystyle w(K)}$ is the writhe of the link diagram and ${\displaystyle L(K)}$ is a polynomial in a and z defined on link diagrams by the following properties:

• ${\displaystyle L(O)=1}$ (O is the unknot)
• ${\displaystyle L(s_{r})=aL(s),\qquad L(s_{\ell })=a^{-1}L(s).}$
• L is unchanged under type II and III Reidemeister moves

Here ${\displaystyle s}$ is a strand and ${\displaystyle s_{r}}$ (resp. ${\displaystyle s_{\ell }}$) is the same strand with a right-handed (resp. left-handed) curl added (using a type I Reidemeister move).

Additionally L must satisfy Kauffman's skein relation:

The pictures represent the L polynomial of the diagrams which differ inside a disc as shown but are identical outside.

Kauffman showed that L exists and is a regular isotopy invariant of unoriented links. It follows easily that F is an ambient isotopy invariant of oriented links.

The Jones polynomial is a special case of the Kauffman polynomial, as the L polynomial specializes to the bracket polynomial. The Kauffman polynomial is related to Chern-Simons gauge theories for SO(N) in the same way that the HOMFLY polynomial is related to Chern-Simons gauge theories for SU(N).[2]

• 1/2
Views:
1 082
1 688
• Introduction to the Kauffman Bracket
• Trefoil knot

## References

1. ^ Kauffman, Louis (1990). "An Invariant of Regular Isotopy" (PDF). Transactions of the American Mathematical Society. 318 (2): 417–471. doi:10.1090/S0002-9947-1990-0958895-7. Retrieved 2016-09-02.
2. ^ Witten, Edward (1989). "Quantum field theory and the Jones polynomial". Comm. Math. Phys. 121 (3): 351–399. doi:10.1007/BF01217730. Retrieved 2016-09-02.