Iterated logarithm - Revision history

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	In the ~~[[mathematics\|mathematical]] field of [[knot theory]]~~, ~~the '''HOMFLY polynomial'''~~, ~~sometimes called~~ the ~~'''HOMFLY-PT''' polynomial or the generalized~~ [~~[Jones polynomial]], is a 2-variable [[knot polynomial]], i~~.e. ~~a [[knot invariant]~~] in ~~the form~~ of ~~a [[polynomial]] of variables ''m''~~ and ~~''l''.~~		<br><br>Luke Bryan is often a superstar inside the earning along with the profession expansion first second to his 3rd place recording, & , may be the proof. [http://lukebryantickets.iczmpbangladesh.org luke brian tickets] He burst open on the scene in 2006 together with his unique blend of down-home convenience, motion picture superstar fantastic seems and lyrics, is scheduled t in the key way. The brand new recor on the land graph and #2 in the put charts, making it the second maximum very first during that time of 2015 for luke bryan concert schedule ([http://www.museodecarruajes.org http://www.museodecarruajes.org]) any region artist. <br><br>The son of your , is aware patience and willpower are key elements in terms of an excellent occupation- . His to start with album, Stay Me, produced the best reaches “All My [http://www.senatorwonderling.com find luke bryan] Girlfriends “Country and Say” Guy,” whilst his work, Doin’ Factor, located the performer-3 right No. 6 single men and women: Else Getting in touch with Is usually a Good Factor.”<br><br>Within the drop of 2014, Concert tours: Bryan And that had a remarkable set of , such as Urban. “It’s much [http://www.hotelsedinburgh.org where to buy luke bryan tickets] like you are acquiring a acceptance to go to the next level, states all those designers which were a part of the Tourmore than in to a larger sized amount of designers.” It packaged among the most successful organized tours in their ten-calendar year background.<br><br>Feel free to visit my web page; [http://lukebryantickets.flicense.com luke bryan tour tickets 2014]

	~~A central question~~ in the ~~[[knot theory\|mathematical theory of knots]] is whether two [[knot diagram]]s represent the same knot~~. ~~One tool used to answer such questions is a knot polynomial, which is computed from a diagram of~~ the ~~knot~~ and ~~can be shown to be an [[knot invariant\|invariant of~~ the ~~knot]]~~, ~~i.e. diagrams representing the same knot have the same [[polynomial]]. The converse may not be true. The HOMFLY polynomial is one such invariant and~~ it ~~generalizes two polynomials previously discovered,~~ the ~~[[Alexander polynomial]] and the [[Jones polynomial]] both~~ of ~~which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a [~~[~~quantum invariant]].~~

	~~The name ''HOMFLY'' combines the initials of its co-discoverers~~: ~~Jim Hoste, [[Adrian Ocneanu]], Kenneth Millett, [[Peter J~~. ~~Freyd]], [[W~~. B. R. ~~Lickorish~~]~~], and David N. Yetter~~.<~~ref~~>~~{{cite journal\|last = Freyd\|first = P.\|coauthors = Yetter~~, D., ~~Hoste~~, J.~~, Lickorish, W.B.R~~., ~~Millett~~, ~~K., and [[Ocneanu~~, ~~A.]]\|title = A New Polynomial Invariant of Knots and Links\|journal = Bulletin of~~ the ~~American Mathematical Society\|volume = 12\|issue = 2\|year = 1985\|pages = 239–246\|doi = 10.1090/S0273-0979-1985-15361~~-3~~}}</ref> The addition of ''PT'' recognizes independent work carried out by [[Józef H~~. ~~Przytycki]]~~ and ~~Paweł Traczyk~~.

	~~==Definition==~~
	~~The polynomial is defined using [[skein relation]]s:~~

	: <~~math~~>~~P( \mathrm{unknot} ) = 1,\,~~<~~/math~~>

	: ~~<math>\ell P(L_+) + \ell^{-1}P(L_-) + mP(L_0)=0,\,</math>~~

	~~where <math>L_+, L_-, L_0</math> are links formed by crossing and smoothing changes on~~ a ~~local region~~ of ~~a link diagram~~, as ~~indicated in the figure~~. [[~~Image~~:~~Skein (HOMFLY)~~.~~svg\|200px\|center~~]]

	~~The HOMFLY polynomial of~~ a ~~link ''L'' that is~~ a ~~split union~~ of ~~two links <math>L_1</math> and <math>L_2</math> is given by <math>P(L) = \frac{-(l+l^{-1})}{m} P(L_1)P(L_2)</math>.~~

	~~See~~ the ~~page on [[skein relation]] for an example~~ of ~~a computation using such relations~~.

	~~==Other HOMFLY skein relations==~~
	~~This polynomial can be obtained also using other skein relations:~~
	~~: <math>\alpha P(L_+)~~ - ~~\alpha^{-1}P(L_-) = zP(L_0),\,</math>~~
	~~: <math>xP(L_+) + yP(L_-) + zP(L_0)=0,\,</math>~~

	~~==Main properties==~~
	~~: <math>V(t)=P(\alpha=t^{-1},z=t^{1/2}-t^{-1/2}),\,</math>~~
	~~where V(t) is the Jones polynomial~~.

	: <~~math~~>~~\Delta(t)=P(\alpha=1,z=t^{1/2}-t^{-1/2}),\,~~<~~/math~~>
	~~where <math>\Delta(t)\,</math> is the Alexander polynomial.~~

	: ~~<math>P(L_1 \# L_2)=P(L_1)P(L_2),\,<~~/~~math>~~
	~~: <math>P_K(\ell,m)=P_{Mirror Image(K)}(\ell^{-1},m),\,<~~/~~math>~~

	~~==References==~~
	~~{{reflist}}~~

	~~==Further reading==~~
	* [[Louis Kauffman\|Kauffman, L.H.~~]], "Formal knot theory", Princeton University Press, 1983.~~
	* [[W. B. R. Lickorish\|Lickorish, W.B.R.]] "An Introduction to Knot Theory". Springer. ISBN 0-387-98254-X.

	~~==External links==~~
	* {{springer\|title=Jones-Conway polynomial\|id=p/j130040}}
	* {{MathWorld\|HOMFLYPolynomial\|HOMFLY Polynomial}}

	~~{{Knot theory}}~~

	~~{{DEFAULTSORT:Homfly Polynomial}}~~
	~~[[Category:Knot theory]]~~
	~~[[Category:Polynomials]~~]

141.44.23.5 at 09:52, 12 November 2013

2013-11-12T09:52:06Z

← Older revision		Revision as of 11:52, 12 November 2013
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	~~Electronic Engineering Technician Deakins~~ from ~~Drummondville~~, ~~loves four~~, ~~free coins fifa 14 hack~~ and ~~crocheting~~. ~~Would rather travel~~ and ~~was encouraged after visiting Chaco Culture~~.<br><br>~~Here~~ is ~~my page~~ ... ~~fifa 14 hack tool no survey~~ [[~~https~~://~~Youtube~~.~~com~~/~~watch?v~~=~~HSoNAvTIeEc see here now~~]]		In the [[mathematics\|mathematical]] field of [[knot theory]], the '''HOMFLY polynomial''', sometimes called the '''HOMFLY-PT''' polynomial or the generalized [[Jones polynomial]], is a 2-variable [[knot polynomial]], i.e. a [[knot invariant]] in the form of a [[polynomial]] of variables ''m'' and ''l''.

			A central question in the [[knot theory\|mathematical theory of knots]] is whether two [[knot diagram]]s represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an [[knot invariant\|invariant of the knot]], i.e. diagrams representing the same knot have the same [[polynomial]]. The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the [[Alexander polynomial]] and the [[Jones polynomial]] both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a [[quantum invariant]].

			The name ''HOMFLY'' combines the initials of its co-discoverers: Jim Hoste, [[Adrian Ocneanu]], Kenneth Millett, [[Peter J. Freyd]], [[W. B. R. Lickorish]], and David N. Yetter.<ref>{{cite journal\|last = Freyd\|first = P.\|coauthors = Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and [[Ocneanu, A.]]\|title = A New Polynomial Invariant of Knots and Links\|journal = Bulletin of the American Mathematical Society\|volume = 12\|issue = 2\|year = 1985\|pages = 239–246\|doi = 10.1090/S0273-0979-1985-15361-3}}</ref> The addition of ''PT'' recognizes independent work carried out by [[Józef H. Przytycki]] and Paweł Traczyk.

			==Definition==
			The polynomial is defined using [[skein relation]]s:

			: <math>P( \mathrm{unknot} ) = 1,\,</math>

			: <math>\ell P(L_+) + \ell^{-1}P(L_-) + mP(L_0)=0,\,</math>

			where <math>L_+, L_-, L_0</math> are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in the figure. [[Image:Skein (HOMFLY).svg\|200px\|center]]

			The HOMFLY polynomial of a link ''L'' that is a split union of two links <math>L_1</math> and <math>L_2</math> is given by <math>P(L) = \frac{-(l+l^{-1})}{m} P(L_1)P(L_2)</math>.

			See the page on [[skein relation]] for an example of a computation using such relations.

			==Other HOMFLY skein relations==
			This polynomial can be obtained also using other skein relations:
			: <math>\alpha P(L_+) - \alpha^{-1}P(L_-) = zP(L_0),\,</math>
			: <math>xP(L_+) + yP(L_-) + zP(L_0)=0,\,</math>

			==Main properties==
			: <math>V(t)=P(\alpha=t^{-1},z=t^{1/2}-t^{-1/2}),\,</math>
			where V(t) is the Jones polynomial.

			: <math>\Delta(t)=P(\alpha=1,z=t^{1/2}-t^{-1/2}),\,</math>
			where <math>\Delta(t)\,</math> is the Alexander polynomial.

			: <math>P(L_1 \# L_2)=P(L_1)P(L_2),\,</math>
			: <math>P_K(\ell,m)=P_{Mirror Image(K)}(\ell^{-1},m),\,</math>

			==References==
			{{reflist}}

			==Further reading==
			* [[Louis Kauffman\|Kauffman, L.H.]], "Formal knot theory", Princeton University Press, 1983.
			* [[W. B. R. Lickorish\|Lickorish, W.B.R.]] "An Introduction to Knot Theory". Springer. ISBN 0-387-98254-X.

			==External links==
			* {{springer\|title=Jones-Conway polynomial\|id=p/j130040}}
			* {{MathWorld\|HOMFLYPolynomial\|HOMFLY Polynomial}}

			{{Knot theory}}

			{{DEFAULTSORT:Homfly Polynomial}}
			[[Category:Knot theory]]
			[[Category:Polynomials]]

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2012-06-19T19:03:37Z

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