Samiksha Jaiswal (Editor)

Kauffman polynomial

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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 ) = a 1 L ( s ) .
  • L is unchanged under type II and III Reidemeister moves
  • Here s is a strand and s r (resp. s ) 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).

    References

    Kauffman polynomial Wikipedia


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