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]
^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.
and 26 Related for: Kauffman polynomial information
In knot theory, the Kauffmanpolynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as F ( K ) (...
for the introduction and development of the bracket polynomial and the Kauffmanpolynomial. Kauffman was valedictorian of his graduating class at Norwood...
knots in knot polynomials. Alexander polynomial Bracket polynomial HOMFLY polynomial Jones polynomialKauffmanpolynomial Graph polynomial, a similar class...
mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links. Although it is not...
diagram. We will define the Jones polynomial V ( L ) {\displaystyle V(L)} by using Louis Kauffman's bracket polynomial, which we denote by ⟨ ⟩ {\displaystyle...
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander...
Jones polynomial. Also known as the Kauffman bracket. Conway polynomial uses Skein relations. Homfly polynomial or HOMFLYPT polynomial. Jones polynomial assigns...
the same as the Alexander, Conway, and Jones polynomials of the knot 10132. However, the Kauffmanpolynomial can be used to distinguish between these two...
(z)=z^{2}+1.} The Jones polynomial is V ( q ) = q − 1 + q − 3 − q − 4 , {\displaystyle V(q)=q^{-1}+q^{-3}-q^{-4},} and the Kauffmanpolynomial of the trefoil is...
theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant...
tracktable, while the Jones polynomial is also known to be #P-hard. Computing the first coefficient of the HOMFLYPT and Kauffmanpolynomial is in P. Additive approximation...
Louis Kauffman, Kunio Murasugi (村杉 邦男), and Morwen Thistlethwaite in 1987, using the Jones polynomial. A geometric proof, not using knot polynomials, was...
{\displaystyle V(q)=-q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3},\,} and the Kauffmanpolynomial is L ( a , z ) = a z 5 + z 5 a − 1 + 2 a 2 z 4 + 2 z 4 a − 2 + 4...
the Jones polynomial in 1984. This led to other knot polynomials such as the bracket polynomial, HOMFLY polynomial, and Kauffmanpolynomial. Jones was...
Kauffman, Louis H. "KNOTS". homepages.math.uic.edu. Retrieved 2020-06-09. Litjens, Bart (August 16, 2011). "Knot theory and the Alexander polynomial"...
invariants, especially the colored Tutte polynomial, which generalizes the Tutte polynomial of a signed graph of Kauffman (1989). There has also been study of...
Alexander polynomials. A variant of the Alexander polynomial, the Alexander–Conway polynomial, is a polynomial in the variable z with integer coefficients (Lickorish...
this shows that the Arf invariant of a slice knot vanishes. Kauffman (1987) p.74 Kauffman (1987) pp.75–78 Robertello, Raymond, An Invariant of Knot Corbordism...
freedom of passing a strand through infinity. Ambient isotopy Knot polynomialKauffman, Louis H. (1990). "An invariant of regular isotopy". Transactions...
graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise [D] by a series of degree...
^{2}-4z^{2}\alpha ^{2}-4z^{4}\alpha ^{2}-z^{6}\alpha ^{2},\,} and the Kauffmanpolynomial is F ( a , z ) = 1 + 2 z − 2 + a − 2 z − 2 + a 2 z − 2 − 2 a − 1...
behavior Louis Kauffman, professor of mathematics; known for the introduction and development of the bracket polynomial and Kauffmanpolynomial in knot theory;...
Unsolved problem in mathematics: Can unknots be recognized in polynomial time? (more unsolved problems in mathematics) In mathematics, the unknotting problem...
problem in mathematics: [Extension of Jones polynomial to general 3-manifolds.] Can the original Jones polynomial, which is defined for 1-links in the 3-sphere...
important invariants can be defined in this way, including the Jones polynomial. The type I move is the only move that affects the writhe of the diagram...
particularly simple and common example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants...
Global Information
