Kauffman polynomial

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In knot theory, the Kauffman polynomial is a 2-variable 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 right-handed (resp. left-handed) curl added (using a type I Reidemeister move).

Additionally L must satisfy Kauffman's skein relation:

Kauffman poly.png

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).[2]

References[edit]

  1. ^ 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. 
  2. ^ 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. 

Further reading[edit]

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