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

F
(
K
)
(
a
,
z
)
=
a
−
w
(
K
)
L
(
K
)
where
w
(
K
)
is the writhe of the link diagram and
L
(
K
)
is a polynomial in *a* and *z* defined on link diagrams by the following properties:

L
(
O
)
=
1
(O is the unknot)
L
(
s
r
)
=
a
L
(
s
)
,
L
(
s
ℓ
)
=
a
−
1
L
(
s
)
.
*L* is unchanged under type II and III Reidemeister moves
Here
s
is a strand and
s
r
(resp.
s
ℓ
) 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).