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.
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).
- 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.
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