Vector-valued differential form - Revision history

en>TakuyaMurata: /* Basic or tensorial forms on principal bundles */

2014-12-18T00:26:37Z

Basic or tensorial forms on principal bundles

← Older revision		Revision as of 02:26, 18 December 2014
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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.~~		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.<br><br>Check out my blog post: std testing at home ([http://exchange.craftemergency.org/thankyous/can-men-be-sexually-harassed-ladies just click the up coming website])

	~~The bracket polynomial was discovered by [[Louis Kauffman]] in 1987~~.

	~~==Definition==~~
	~~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~~:

	* <O> = 1, where O is the standard diagram of the unknot
	* [~~[Image~~:~~kauffman_bracket2~~.~~png\|275px]]~~
	* <math> \langle O \cup L \rangle = (-A^2 - A^{-2}) \langle L \rangle </~~math>~~

	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>''.~~

	~~==Further reading==~~
	*Louis H. Kauffman, ''State models and the Jones polynomial.'' Topology 26 (1987), no. 3, 395--~~407. (introduces~~ the ~~bracket polynomial~~)

	~~==External links==~~
	*{{MathWorld\|BracketPolynomial\|Bracket Polynomial}}

	~~{{Knot theory}}~~

	~~[[Category:Knot theory]]~~
	~~[[Category:Polynomials]]~~

	~~{{knottheory-stub}}~~

171.65.239.16: Corrected the product rule for the covariant exterior derivative

2013-11-10T11:14:24Z

Corrected the product rule for the covariant exterior derivative

← Older revision		Revision as of 13:14, 10 November 2013
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	~~The writer~~ is ~~recognized by the title~~ of ~~Numbers Lint~~. ~~Minnesota has always~~ ~~home std test been his home but his wife wants them to~~ move~~. Since she was eighteen she'~~s ~~been working as~~ a [~~http://bossip~~.~~com/634680/celebrities-accused-~~of-~~carrying-stds3920/ receptionist~~] ~~but her marketing never~~ [~~http://www.slideshare.net/realtrafficsource/celebrities-with-herpes arrives~~]. ~~One~~ of the ~~issues he loves most is ice skating but he~~ is ~~std home test having difficulties to~~ [~~http~~:~~//mcb-law~~.~~net/candidiasis~~-~~tips~~-~~you~~-~~have-to-remember~~/ ~~home std test kit] discover time for it~~.<br><br>~~Have a look at my website [http:~~/~~/mtomtech~~.co.~~kr/bbs/?document_srl=92816 over~~ the ~~counter std test] :: [http://faculty~~.~~jonahmancini~~.~~com/groups/solid~~-~~advice~~-~~for-dealing~~-~~with-a-candida-albicans/members/ home std test]~~		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.

			The bracket polynomial was discovered by [[Louis Kauffman]] in 1987.

			==Definition==
			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:

			* <O> = 1, where O is the standard diagram of the unknot
			* [[Image:kauffman_bracket2.png\|275px]]
			* <math> \langle O \cup L \rangle = (-A^2 - A^{-2}) \langle L \rangle </math>

			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>''.

			==Further reading==
			*Louis H. Kauffman, ''State models and the Jones polynomial.'' Topology 26 (1987), no. 3, 395--407. (introduces the bracket polynomial)

			==External links==
			*{{MathWorld\|BracketPolynomial\|Bracket Polynomial}}

			{{Knot theory}}

			[[Category:Knot theory]]
			[[Category:Polynomials]]

			{{knottheory-stub}}

75.24.144.36: /* Formal definition */

2012-04-11T01:58:42Z

Formal definition

New page

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