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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| ]]

Revision as of 17:36, 28 February 2014

If you think of oneself because being fat and ugly then this might be the right region for you. In this article there is a step by step representation of Ways to Lose weight. For many fat reduction may prove to be a rather difficult issue to deal with due to the fact which they simply never have the idea of how to burn the calories plus what to avoid and what to consume that might provide the nourishment without piling up the calories.

Leafy greens are high inside fiber, low inside calories, packed full of compounds, plus approximately half their digestible calories come from protein. Eating at least 10 servings of greens a day may replace a great deal of the high calorie foods inside your diet, however, leave we really as full. If you want to lose weight fast, eat your greens. If you want to lose fat faster, eat MORE greens. I highly suggest drinking a green smoothie for breakfast each morning.

In regards to the quickest method to lose weight, sleep is an often overlooked component. People simply talk about eating and functioning out to lose fat. Not everybody ever states sleeping is important for fat reduction. Yet somehow, this really is one of the many important components inside the weight-loss equation.

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