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[[Image:Knot table.svg|thumb|[[Prime knot]]s are organized by the crossing number invariant.]]
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In the [[mathematics|mathematical]] field of [[knot theory]], a '''knot invariant''' is a quantity (in a broad sense) defined for each [[knot (mathematics)|knot]] which is the same for equivalent knots. The equivalence is often given by [[ambient isotopy]] but can be given by [[homeomorphism]]. Some invariants are indeed numbers, but invariants can range from the simple, such as a yes/no answer, to those as complex as a [[homology theory]] . Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics.
 
From the modern perspective, it is natural to define a knot invariant from a [[knot diagram]]. Of course, it must be unchanged (that is to say, invariant) under the [[Reidemeister move]]s. [[Tricolorability]] is a particularly simple example. Other examples are [[knot polynomial]]s, such as the [[Jones polynomial]], which are currently among the most useful invariants for distinguishing knots from one another, though at the time of writing it is not known whether there exists a knot polynomial which distinguishes all knots from each other, or even which distinguishes just the [[unknot]] from all other knots.
 
Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the [[crossing number (knot theory)|crossing number]], which is the minimum number of crossings for any diagram of the knot, and the [[bridge number]], which is the minimum number of bridges for any diagram of the knot.
 
Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, [[knot genus]] is particularly tricky to compute, but can be effective (for instance, in distinguishing [[mutation (knot theory)|mutants]]).
 
The [[knot complement|complement of a knot]] itself (as a [[topological space]]) is known to be a "complete invariant" of the knot by the [[Gordon–Luecke theorem]] in the sense that it distinguishes the given knot from all other knots up to [[ambient isotopy]] and [[mirror image (knot theory)|mirror image]]. Some invariants associated with the knot complement include the [[knot group]] which is just the [[fundamental group]] of the complement. The [[knot quandle]] is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic.
 
By [[Mostow rigidity theorem|Mostow–Prasad rigidity]], the hyperbolic structure on the complement of a [[hyperbolic link]] is unique, which means the [[hyperbolic volume (knot)|hyperbolic volume]] is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at [[knot table|knot tabulation]].
 
In recent years, there has been much interest in [[homology theory|homological]] invariants of knots which [[categorification|categorify]] well-known invariants. [[Floer homology#Heegaard Floer homology|Heegaard Floer homology]] is a [[homology theory]] whose [[Euler characteristic]] is the [[Alexander polynomial]] of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called [[Khovanov homology]] whose Euler characteristic is the [[Jones polynomial]]. This has recently been shown to be useful in obtaining bounds on [[slice genus]] whose earlier proofs required [[gauge theory]]. [[Mikhail Khovanov|Khovanov]] and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.
 
There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the [[Fary–Milnor theorem]] states that if the [[total curvature]] of a knot ''K'' in <math>\mathbb{R}^3</math> satisfies
 
:<math>\oint_K \kappa \,ds \leq 4\pi,</math>
 
where <math>\kappa(p)</math> is the [[Parametric curve#Curvature|curvature]] at ''p'', then ''K'' is an unknot. Therefore, for knotted curves,
 
:<math>\oint_K \kappa\,ds > 4\pi.\,</math>
 
An example of a "physical" invariant is [[ropelength]], which is the amount of 1-inch diameter rope needed to realize a particular knot type.
 
==Other invariants==
* [[Linking number]]
* [[Finite type invariant]] (or Vassiliev or Vassiliev–Goussarov invariant)
* [[Stick number]]
 
==Further reading==
*{{Cite book |last=Rolfsen |first=Dale |title=Knots and Links |location=Providence, RI |publisher=AMS |year=2003 |isbn=0-8218-3436-3 }}
*{{Cite book |last=Adams |first=Colin Conrad |title=The Knot Book: an Elementary Introduction to the Mathematical Theory of Knots |location=Providence, RI |publisher=AMS |edition=Repr., with corr |year=2004 |isbn=0-8218-3678-1 }}
*{{Cite book |last=Burde |first=Gerhard |last2=Zieschang |first2=Heiner |title=Knots |location=New York |publisher=De Gruyter |edition=2nd rev. and extended |year=2002 |isbn=3-11-017005-1 }}
 
==External links==
*J. C. Cha and C. Livingston. "[http://www.indiana.edu/~knotinfo/ KnotInfo: Table of Knot Invariants]", ''Indiana.edu''. {{Accessed|09:10, 18 April 2013 (UTC)}}
 
{{Knot theory|state=collapsed}}
 
[[Category:Knot invariants| ]]

Latest revision as of 07:33, 22 October 2014

We will discover effortless methods to speed up computer by creating the many from the built inside tools in the Windows too as downloading the Service Pack updates-speed up your PC plus fix error. Simply follow a limited guidelines to swiftly create the computer swiftly than ever.

However registry is easily corrupted plus damaged whenever we are utilizing your computer. Overtime, without right repair, it will be loaded with errors and wrong or even lost information which might create your system unable to function correctly or apply a certain task. And when your system could not discover the correct info, it will likely not learn what to do. Then it freezes up! That is the real cause of the trouble.

One of the many overlooked factors a computer might slow down is because the registry has become corrupt. The registry is essentially the computer's operating program. Anytime you're running a computer, entries are being made and deleted from the registry. The effect this has is it leaves false entries in your registry. So, a computer's resources must function about these false entries.

Always see with it that we have installed antivirus, anti-spyware plus anti-adware programs and have them updated on a regular basis. This can help stop windows XP running slow.

In a word, to accelerate windows XP, Vista business, it's quite important to disable several business items plus clean and optimize the registry. We can follow the procedures above to disable unwanted programs. To optimize the registry, I suggest we use a fix it utilities software. Because it's really risky for we to edit the registry by yourself.

2)Fix your Windows registry to speed up PC- The registry is a complex section of your computer which holds different types of information within the details we do on the laptop each day. Coincidentally, over time the registry can become cluttered with info and/or could receive some kind of virus. This is especially important and we MUST get this issue fixed right away, otherwise you run the risk of the computer being permanently damage and/or the sensitive info (passwords, etc.) can be stolen.

Your disk demands area inside order to run smoothly. By freeing up some area from your disk, you are able to speed up your PC a bit. Delete all file in the temporary internet files folder, recycle bin, clear shortcuts and icons from a desktop which we never use and remove programs you never employ.

Fortunately, there's a easy method to fix most the computer errors. You just have to be able to fix corrupt registry files on a computer. And to do that, we can merely utilize a tool recognised as a registry cleaner. These easy pieces of software really scan from the PC plus fix every corrupt file which might cause a problem to Windows. This allows the computer to employ all files it wants, that not merely speeds it up - nevertheless also stops all of the mistakes on your program as well.

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