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| In the [[mathematics|mathematical]] field of [[knot theory]], the '''bracket polynomial''' (also known as the '''Kauffman bracket''') is a [[polynomial]] invariant of [[framed link]]s. Although it is not an invariant of knots or links (as it is not invariant under type I [[Reidemeister move]]s), 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 invariant]]s. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to invariants of [[3-manifold]]s.
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| The bracket polynomial was discovered by [[Louis Kauffman]] in 1987.
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| ==Definition==
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| The bracket polynomial of any (unoriented) link diagram ''L'', denoted <''L''>, is a polynomial in the variable <math>A</math>, characterized by the three rules:
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| * <O> = 1, where O is the standard diagram of the unknot
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| * [[Image:kauffman_bracket2.png|275px]]
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| * <math> \langle O \cup L \rangle = (-A^2 - A^{-2}) \langle L \rangle </math>
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| The 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 adding a circle disjoint from the rest of the diagram multiplies the bracket of the remaining diagram by ''-A<sup>2</sup> - A<sup>-2</sup>''.
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| ==Further reading==
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| *Louis H. Kauffman, ''State models and the Jones polynomial.'' Topology 26 (1987), no. 3, 395--407. (introduces the bracket polynomial)
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| ==External links==
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| *{{MathWorld|BracketPolynomial|Bracket Polynomial}}
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| {{Knot theory}}
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| [[Category:Knot theory]]
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| [[Category:Polynomials]]
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| {{knottheory-stub}}
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Latest revision as of 01:26, 18 December 2014
I am Oscar and I completely dig that title. To collect cash is 1 of the issues I love most. My working day occupation is a meter reader. Years in the past we moved to North Dakota.
Check out my blog post: std testing at home (just click the up coming website)