- Bracket polynomial
In the mathematical field of
knot theory , the bracket polynomial (also known as the Kauffman bracket) is apolynomial invariant offramed link s. Although it is not an invariant of knots or links (as it is not invariant under type IReidemeister move s), a suitably "normalized" version yields the famousknot invariant called theJones polynomial . The bracket polynomial plays an important role in unifying the Jones polynomial with otherquantum invariant s. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of3-manifold s.The bracket polynomial was discovered by
Louis Kauffman in 1987.Definition
The bracket polynomial of any (unoriented) link diagram "L", denoted <"L">, is characterized by the three rules:
*
= 1, where O is the standard diagram of the unknot
*
* langle O cup L angle = (-A^2 - A^{-2}) langle L angleThe pictures in the second rule represent brackets of the link diagrams which differ inside a disc as shown but are identical outside. The third rule means that removing a circle disjoint from the rest of the diagram multiplies the bracket of the remaining diagram by "-A2 - A-2".
References
*Louis H. Kauffman, "State models and the Jones polynomial." Topology 26 (1987), no. 3, 395--407. (introduces the bracket polynomial)
External links
* [http://mathworld.wolfram.com/BracketPolynomial.html mathworld]
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