Bracket polynomial
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In the 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 an invariant of knots or links (as it is not invariant under type I Reidemeister moves), a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of 3-manifolds.
The bracket polynomial was discovered by Louis Kauffman in 1987.
[edit] Definition
The bracket polynomial of any (unoriented) link diagram L, denoted <L>, is characterized by the three rules:
- <O> = 1, where O is the standard diagram of the unknot
- <L> = A <||> + A-1<=>
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. <||> and <=> refer to the brackets of the two diagrams obtained by "smoothing" (removing) a single crossing as in the figure:
[edit] References
- Louis H. Kauffman, State models and the Jones polynomial. Topology 26 (1987), no. 3, 395--407. (introduces the bracket polynomial)
[edit] External links
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