Kauffman polynomial
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:
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
- ^ 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. MR 0958895.
- ^ Witten, Edward (1989). "Quantum field theory and the Jones polynomial". Communications in Mathematical Physics. 121 (3): 351–399. doi:10.1007/BF01217730. MR 0990772.
Further reading
- Kauffman, Louis (1987). On Knots. Annals of Mathematics Studies. Vol. 115. Princeton, NJ: Princeton University Press. ISBN 0-691-08435-1. MR 0907872.
External links
- "Kauffman polynomial", Encyclopedia of Mathematics
- "The Kauffman Polynomial", The Knot Atlas.
- v
- t
- e
- Figure-eight (41)
- Three-twist (52)
- Stevedore (61)
- 62
- 63
- Endless (74)
- Carrick mat (818)
- Perko pair (10161)
- Conway knot (11n34)
- Kinoshita–Terasaka knot (11n42)
- (−2,3,7) pretzel (12n242)
- Whitehead (52
1) - Borromean rings (63
2) - L10a140
- Composite knots
- Granny
- Square
- Knot sum
and operations
- Alexander–Briggs notation
- Conway notation
- Dowker–Thistlethwaite notation
- Flype
- Mutation
- Reidemeister move
- Skein relation
- Tabulation
- Category
- Commons
This knot theory-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e
This polynomial-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e