Kauffman polynomial

In <a href="/facts/Knot_theory/mb7rtqVP">knot theory</a>, the Kauffman polynomial is a 2-variable <a href="/facts/Knot_polynomial/fB770Bw0">knot polynomial</a> due to <a href="/facts/Louis_Kauffman/lep7X1gB">Louis Kauffman</a>. It is initially defined on a <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a> diagram as

F
 (
 K
 )
 (
 a
 ,
 z
 )
 =
 
 a
 
 −
 w
 (
 K
 )
 
 
 L
 (
 K
 )
 
 
 
 {\displaystyle F(K)(a,z)=a^{-w(K)}L(K)\,}
 
,
where 
 
 
 
 w
 (
 K
 )
 
 
 {\displaystyle w(K)}
 
 is the <a href="/facts/Writhe/trAFn7Rd">writhe</a> of the link diagram and 
 
 
 
 L
 (
 K
 )
 
 
 {\displaystyle L(K)}
 
 is a polynomial in a and z defined on link diagrams by the following properties:

<ul><li>
 
 
 
 L
 (
 O
 )
 =
 1
 
 
 {\displaystyle L(O)=1}
 
 (O is the unknot).</li>
<li>
 
 
 
 L
 (
 
 s
 
 r
 
 
 )
 =
 a
 L
 (
 s
 )
 ,
 
 L
 (
 
 s
 
 ℓ
 
 
 )
 =
 
 a
 
 −
 1
 
 
 L
 (
 s
 )
 .
 
 
 {\displaystyle L(s_{r})=aL(s),\qquad L(s_{\ell })=a^{-1}L(s).}
 
</li>
<li>L is unchanged under type II and III <a href="/facts/Reidemeister_move/0jQmU28C">Reidemeister moves</a>.</li></ul>
Here 
 
 
 
 s
 
 
 {\displaystyle s}
 
 is a strand and 
 
 
 
 
 s
 
 r
 
 
 
 
 {\displaystyle s_{r}}
 
 (resp. 
 
 
 
 
 s
 
 ℓ
 
 
 
 
 {\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 <a href="/facts/Skein_relation/7LbCnnLp">skein relation</a>:

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 <a href="/facts/Regular_isotopy/h6V6LJP0">regular isotopy</a> invariant of unoriented links. It follows easily that F is an <a href="/facts/Ambient_isotopy/b90TwzUM">ambient isotopy</a> invariant of oriented links.
The <a href="/facts/Jones_polynomial/c73vElt9">Jones polynomial</a> is a special case of the Kauffman polynomial, as the L polynomial specializes to the <a href="/facts/Bracket_polynomial/gWCWYSM1">bracket polynomial</a>. The Kauffman polynomial is related to <a href="/facts/Chern%E2%80%93Simons_theory/AQ3Kayn3">Chern–Simons gauge theories</a> for SO(N) in the same way that the <a href="/facts/HOMFLY_polynomial/0n0mc1Be">HOMFLY polynomial</a> is related to Chern–Simons gauge theories for SU(N).

Kauffman polynomial open-in-new

Kauffman polynomial