Generator (category theory): Difference between revisions
en>Helpful Pixie Bot m Fixed header Reference => References (Build J2) |
en>Mark viking →Examples: Added wl |
||
Line 1: | Line 1: | ||
{{distinguish|Kauffman bracket}} | |||
In [[knot theory]], the '''Kauffman polynomial''' is a 2-variable [[knot polynomial]] due to [[Louis Kauffman]]. It is initially defined on a [[link (knot theory)|link]] diagram as | |||
:<math>F(K)(a,z)=a^{-w(K)}L(K)\,</math> | |||
where <math>w(K)</math> is the [[writhe]] of the link diagram and <math>L(K)</math> is a polynomial in ''a'' and ''z'' defined on link diagrams by the following properties: | |||
*<math>L(O) = 1</math> (O is the unknot) | |||
*<math>L(s_r)=aL(s), \qquad L(s_\ell)=a^{-1}L(s).</math> | |||
*''L'' is unchanged under type II and III [[Reidemeister move]]s | |||
Here <math>s</math> is a strand and <math>s_r</math> (resp. <math>s_\ell</math>) 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]]: | |||
:[[Image:Kauffman poly.png|400px]] | |||
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 theory|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) (see Witten's article | |||
"Quantum field theory and the Jones polynomial", in Commun. Math. Phys.) | |||
==Further reading== | |||
*[[Louis Kauffman]], ''On Knots'', (1987), ISBN 0-691-08435-1 | |||
==External links== | |||
*[http://eom.springer.de/k/k120040.htm Springer EoM entry for Kauffman polynomial] | |||
*{{Knot Atlas|The_Kauffman_Polynomial}} | |||
{{Knot theory}} | |||
[[Category:Knot theory]] | |||
[[Category:Polynomials]] | |||
{{knottheory-stub}} |
Revision as of 02:17, 14 September 2013
Template:Distinguish 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
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) (see Witten's article "Quantum field theory and the Jones polynomial", in Commun. Math. Phys.)
Further reading
- Louis Kauffman, On Knots, (1987), ISBN 0-691-08435-1