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

where is the writhe of the link diagram and is a polynomial in *a* and *z* defined on link diagrams by the following properties:

- (O is the unknot)
*L*is unchanged under type II and III Reidemeister moves

Here is a strand and (resp. ) 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

**^**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.**^**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.

## Further reading

- Louis Kauffman,
*On Knots*, (1987), ISBN 0-691-08435-1