In knot theory, the Kauffman polynomial is a 2variable 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 righthanded (resp. lefthanded) 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 ChernSimons gauge theories for SO(N) in the same way that the HOMFLY polynomial is related to ChernSimons gauge theories for SU(N).^{[2]}
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Introduction to the Kauffman Bracket

Trefoil knot
Transcription
References
 ^ Kauffman, Louis (1990). "An Invariant of Regular Isotopy" (PDF). Transactions of the American Mathematical Society. 318 (2): 417–471. doi:10.1090/S00029947199009588957. Retrieved 20160902.
 ^ Witten, Edward (1989). "Quantum field theory and the Jones polynomial". Comm. Math. Phys. 121 (3): 351–399. doi:10.1007/BF01217730. Retrieved 20160902.
Further reading
 Louis Kauffman, On Knots, (1987), ISBN 0691084351