en>Kiran.yala: /* Example */

2014-11-08T02:14:18Z

Example

en>DexDor: reword (WP:REFER), rm Terminology cat (this article is not about language) (see WP:Categories_for_discussion/Log/2013_January_25#Category:Electronics_terminology)

2013-02-10T21:25:05Z

reword (WP:REFER), rm Terminology cat (this article is not about language) (see WP:Categories_for_discussion/Log/2013_January_25#Category:Electronics_terminology)

← Older revision		Revision as of 23:25, 10 February 2013
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	~~Alyson~~ is ~~what my husband enjoys~~ to ~~call me but I don~~'~~t like when people use my full title~~. ~~Invoicing~~ is ~~my profession~~. One of the ~~very best issues~~ in the ~~world for him~~ is ~~performing ballet and he~~'~~ll be starting something else along~~ with it. ~~Ohio~~ is ~~where his home~~ is and ~~his family enjoys it~~.<br><br>~~Feel free~~ to ~~surf to my site~~ :: [http://si.~~dgmensa~~.~~org~~/xe/~~index~~.~~php?document_srl~~=~~48014~~&~~mid~~=~~c0102 free tarot readings~~]		In [[physical knot theory]], each realization of a [[link (knot theory)\|link]] or [[knot (mathematics)\|knot]] has an associated '''ropelength'''. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are called '''ideal knots''' and '''ideal links''' respectively.

			[[Image:Ideal Trefoil.png\|thumb\|upright\|alt=A numeric approximation of an ideal trefoil.\|A numeric approximation of an ideal [[Trefoil knot\|trefoil]].]]

			== Definition ==
			The ropelength of a knot curve ''C'' is defined as the ratio <math>{\scriptstyle L(C)\, =\, \operatorname{Len}(C)/ \tau(C)}</math>, where Len(''C'') is the length of ''C'' and τ(''C'') is the [[knot thickness\|thickness]] of the link defined by ''C''.

			== Ropelength minimizers ==
			One of the earliest knot theory questions was posed in the following terms:

			<blockquote>
			''Can I tie a knot on a foot-long rope that is one inch thick?''
			</blockquote>

			In our terms we are asking if there is a knot with ropelength 12. This question has been answered, and it was shown to be impossible: an argument using [[quadrisecant]]s shows that the ropelength of any nontrivial knot has to be at least 15.66.<ref name="Quad">{{citation
			\| last1 = Denne \| first1 = Elizabeth
			\| last2 = Diao \| first2 = Yuanan
			\| last3 = Sullivan \| first3 = John M.
			\| doi = 10.2140/gt.2006.10.1
			\| journal = [[Geometry & Topology]]
			\| mr = 2207788
			\| pages = 1–26
			\| title = Quadrisecants give new lower bounds for the ropelength of a knot
			\| url = http://msp.warwick.ac.uk/gt/2006/10/p001.xhtml
			\| volume = 10
			\| year = 2006}}.</ref> However, the search for the answer has spurred a lot of research on both theoretical and computational ground. It has been shown that for each link type there is a ropelength minimizer although it is only of class ''C''<sup> 1, 1</sup>.<ref>{{citation
			\| last1 = Gonzalez \| first1 = O.
			\| last2 = Maddocks \| first2 = J. H.
			\| last3 = Schuricht \| first3 = F.
			\| last4 = von der Mosel \| first4 = H.
			\| doi = 10.1007/s005260100089
			\| issue = 1
			\| journal = Calculus of Variations and Partial Differential Equations
			\| mr = 1883599
			\| pages = 29–68
			\| title = Global curvature and self-contact of nonlinearly elastic curves and rods
			\| volume = 14
			\| year = 2002}}</ref><ref>{{citation
			\| last1 = Cantarella \| first1 = Jason
			\| last2 = Kusner \| first2 = Robert B.
			\| last3 = Sullivan \| first3 = John M.
			\| doi = 10.1007/s00222-002-0234-y
			\| issue = 2
			\| journal = [[Inventiones Mathematicae]]
			\| mr = 1933586
			\| pages = 257–286
			\| title = On the minimum ropelength of knots and links
			\| url = http://torus.math.uiuc.edu/jms/Papers/thick/ropelen.pdf
			\| volume = 150
			\| year = 2002}}.</ref> For the simplest nontrivial knot, the trefoil knot, computer simulations have shown that its ropelength is at most 16.372.<ref name="Quad"/>

			== Dependence of ropelength on other knot invariants ==
			An extensive search has been devoted to showing relations between ropelength and other knot invariants. As an example there are well known bounds on the asymptotic dependence of ropelength on the [[Crossing number (knot theory)\|crossing number]] of a knot. It has been shown that
			:<math>L(C)=\Omega(\operatorname{Cr}(C)^{3/4})\,</math>
			and
			:<math>L(C)= O(\operatorname{Cr}(C)\log^5(\operatorname{Cr}(C)))\,</math>
			for a knot ''C'' with crossing number Cr(''C'') and ropelength ''L''(''C''), where the ''O'' and Ω are examples of [[big O notation]] and big Omega notation, respectively.

			The lower bound (big Omega) is shown with two families ((''k'', ''k''−1) torus knots and ''k''-Hopf links) that realize this bound. A former upper bound of O(Cr(''C''))<sup>3/2</sup> has been shown using Hamiltonian cycles in graphs embedded in a cubic integer lattice.<ref>{{citation
			\| last1 = Diao \| first1 = Yuanan
			\| last2 = Ernst \| first2 = Claus
			\| last3 = Yu \| first3 = Xingxing
			\| doi = 10.1016/S0166-8641(03)00182-2
			\| issue = 1-3
			\| journal = [[Topology and its Applications]]
			\| mr = 2023409
			\| pages = 7–36
			\| title = Hamiltonian knot projections and lengths of thick knots
			\| url = http://www.math.gatech.edu/~yu/Papers/hamilton10-8.pdf
			\| volume = 136
			\| year = 2004}}.</ref> The current best near-linear upper bound was established with a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice.<ref>{{citation
			\| last1 = Diao \| first1 = Yuanan
			\| last2 = Ernst \| first2 = Claus
			\| last3 = Por \| first3 = Attila
			\| last4 = Ziegler \| first4 = Uta
			\| arxiv = 0912.3282
			\| title = The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers
			\| year = 2009}}.</ref> However, no one has yet observed a knot family with super-linear length dependence ''L''(''C'') > O(Cr(''CK'')) and it is conjectured that the upper bound is in fact linear.<ref>{{citation
			\| last1 = Diao \| first1 = Yuanan
			\| last2 = Ernst \| first2 = Claus
			\| issue = 2
			\| journal = JP Journal of Geometry and Topology
			\| mr = 2105812
			\| pages = 197–208
			\| title = Realizable powers of ropelengths by non-trivial knot families
			\| url = http://www.math.uncc.edu/preprint/2003/2003_04.pdf
			\| volume = 4
			\| year = 2004}}.</ref>

			== Ropelength as a knot invariant ==
			Ropelength can be turned into a [[knot invariant]] by defining the ropelength of a knot type to be the minimum ropelength over all realizations of that knot type. So far this invariant is impractical as we have not determined that minimum for the majority of knots.

			==References==
			{{reflist\|colwidth=30em}}

			[[Category:Knot invariants]]
			[[Category:Geometric topology]]

en>Joel B. Lewis: /* Example */

2012-03-31T14:59:10Z

Example

New page

Alyson is what my husband enjoys to call me but I don't like when people use my full title. Invoicing is my profession. One of the very best issues in the world for him is performing ballet and he'll be starting something else along with it. Ohio is where his home is and his family enjoys it.<br><br>Feel free to surf to my site :: [http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 free tarot readings]

Complex gain - Revision history

en>Kiran.yala: /* Example */

en>DexDor: reword (WP:REFER), rm Terminology cat (this article is not about language) (see WP:Categories_for_discussion/Log/2013_January_25#Category:Electronics_terminology)

en>Joel B. Lewis: /* Example */