Formal power series: Difference between revisions

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clarified power series raised to a power formula
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{{Infobox knot theory
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| name=              Unknot
| practical name=    Torus
| image=            Blue Unknot.png
| caption=         
| arf invariant=   
| bridge number=    0
| crossing number=  0
| linking number=    0
| stick number=      3
| unknotting number= 0
| conway_notation=  -
| ab_notation=      0<sub>1</sub>
| dowker notation=  -
| thistlethwaite=   
| last crossing=   
| last order=       
| next crossing=    3
| next order=        1
  | alternating=     
| class=            torus
| fibered=          fibered
| prime=            prime
| slice=            slice
| symmetry=        fully amphichiral
| tricolorable=    tricolorable
| twist=           
}}
 
[[Image:unknots.svg|right|165px|thumb|Two simple diagrams of the unknot]]
[[Image:thistlethwaite_unknot.svg|right|200px|thumb|A tricky unknot diagram by [[Morwen Thistlethwaite]]]]
 
The '''unknot''' arises in the [[knot theory|mathematical theory of knots]]. Intuitively, the unknot is a closed loop of rope without a [[knot]] in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. [[ambient isotopy|ambient-isotoped]], to the '''standard unknot''', i.e. the embedding of the [[circle]] as a geometrically round circle. The unknot is also called the '''trivial knot'''. An unknot is the [[identity element]] with respect to the [[knot sum]] operation.  
 
== Unknotting problem ==
{{main|Unknotting problem}}
 
Deciding if a particular knot is the unknot was a major driving force behind [[knot invariant]]s, since it was thought this approach would possibly give an efficient algorithm to [[unknotting problem|recognize the unknot]] from some presentation such as a knot diagram. Currently there are several well-known unknot recognition algorithms (not using invariants), but they are either known to be inefficient or have no efficient implementation. It is not known whether many of the current invariants, such as [[finite type invariant]]s, are a complete invariant of the unknot, but [[Floer homology#Heegaard Floer homology|knot Floer homology]] is known to detect the unknot. Even if they were, the problem of computing them efficiently remains.
 
== Examples ==
 
Many useful practical knots are actually the unknot, including all knots which can be tied in the [[bight (knot)|bight]].<ref name="knotty">{{cite web|url=http://www.volkerschatz.com/knots/knots.html|title=Knotty topics|author=Volker Schatz|accessdate=2007-04-23}}</ref> Other noteworthy unknots are those that consist of rigid line segments connected by universal joints at their endpoints (linkages), that yet cannot be reconfigured into a convex polygon, thus acquiring the name [[stuck unknot|''stuck'' unknots]].<ref>[[Godfried Toussaint]], "A new class of stuck unknots in Pol-6," ''Contributions to Algebra and Geometry'', Vol. 42, No. 2, 2001, pp. 301-306.</ref>
 
==Invariants==
The [[Alexander-Conway polynomial]] and [[Jones polynomial]] of the unknot are trivial:
 
:<math>\Delta(t) = 1,\quad \nabla(z) = 1,\quad V(q) = 1.</math>
 
No other knot with 10 or fewer [[crossing number (knot theory)|crossings]] has trivial Alexander polynomial, but the [[Kinoshita-Terasaka knot]] and [[Conway knot]] (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.
 
The [[knot group]] of the unknot is an infinite [[cyclic group]], and the [[knot complement]] is [[homeomorphism|homeomorphic]] to a [[solid torus]].
 
==See also==
*[[Knot (mathematics)]]
*[[Unlink]]
 
==References==
{{reflist}}
 
==External links==
*{{Knot Atlas|0 1|Unknot|date=May 7, 2013}}
*{{MathWorld|Unknot}}
 
{{Knot theory|state=collapsed}}

Revision as of 08:10, 18 February 2014

When cleansing, drying out is actually a critical point. Here's more regarding Houston roofing company review our own web site. Your fluids have got to be clean and fresh before you put your car into storage. When water backs up behind the dam, it creates a body of water, which then creates the reservoir. When the mold spores get into the walls you could be living next to them and not even knowing it. Laminate Countertop Repair - Examine the countertop for any areas that are lifting.

When it comes to water damage what you don�t know really can hurt you. In this instance any water damage restoration company insures mould remediation. However, mostly the companies follow S500, the IICRA method of water damage restoration. Article Source: Corkern is a writer and promoter of. You want to make sure you dry the entire floor out as much as possible.

Chemical delivery systems are a much better choice to protect your data center. You should check to see if any other food in your refrigerator is contaminated with mold and throw it away immediately, especially if it is something that does not come in a jar. The putty powder is used on the surfaced with a soft cloth. I keep a drink near me at all times and fortunately I have never spilled anything onto my beloved laptop, but there are many incidents where others have. In any natural pond ecosystem, you'll find pond aeration, but a pond aeration system adds more O2 to an oxygen depleted environment.

You can either submit your work and join their crew or you can begin your own site and solicit business that way. Love and loss, redemption, restored trust, salvation -- these overarching ideas provide the belief that drives the plot of any bestseller, should it be fiction or non-fiction, Christian or secular. There are a lot of unfortunate events that can damage your most important investments and put them at risk. Next include a copy of the signed work authorization and signed certificate of satisfaction. It is east of Flint and known as the county seat of Lapeer.

Our bodies are engineered much like finely precisioned automobiles. Higher moisture amounts, if left unwatched all through drinking water harm repair can result in the feasible formation of mildew at later phases. Dont wait for the floods to occur to hire restoration contractors. Amanda Rockler (Central Maryland: Frederick, Howard, and Montgomery Counties) arockler@umd. This will take some time to wear off unless chemical sprays are applied excessively.

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