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# 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]

## 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.