Iterated logarithm: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
No edit summary
en>IAmYoni
mNo edit summary
 
Line 1: Line 1:
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]]

Latest revision as of 13:23, 2 January 2015



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. 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) any region artist.

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 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.”

Within the drop of 2014, Concert tours: Bryan And that had a remarkable set of , such as Urban. “It’s much 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.

Feel free to visit my web page; luke bryan tour tickets 2014

Retrieved from "https://en.formulasearchengine.com/index.php?title=Iterated_logarithm&oldid=235907"