|
|
Line 1: |
Line 1: |
| {{redirects|Link number|the logic puzzle|Numberlink}}
| | At the time a association struggle begins, you will see The specific particular War Map, the new map of this conflict area area association battles booty place. Welcoming territories will consistently often be on the left, by having the adversary association on the inside the right. boondocks anteroom on the war map represents some type of war base.<br><br> |
| [[Image:3D-Link.PNG|thumb|right|The two curves of this (2,8)-[[torus knot|torus link]] have linking number four.]]
| |
| In [[mathematics]], the '''linking number''' is a numerical [[invariant (mathematics)|invariant]] that describes the linking of two [[closed curve]]s in [[three-dimensional space]]. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an [[integer]], but may be positive or negative depending on the [[curve orientation|orientation]] of the two curves.
| |
|
| |
|
| The linking number was introduced by [[Carl Friedrich Gauss|Gauss]] in the form of the '''linking integral'''. It is an important object of study in [[knot theory]], [[algebraic topology]], and [[differential geometry]], and has numerous applications in [[mathematics]] and [[science]], including [[quantum mechanics]], [[electromagnetism]], and the study of [[DNA supercoil]]ing.
| | Beginning nearly enough gems to get another developer. Don''t waste some of the gems in any way on rush-building anything, as if it all can save you the group you are going to actually eventually obtain enough for free extra gems to grab that extra builder without even cost. Particularly, the customer can get free gallstones for clearing obstructions adore rocks and trees, when you are done you clear them outside they come back in addition to you may re-clear these to get more jewelry.<br><br>When you find yourself getting a online round for your little one, look for one typically enables numerous customers to do with each other. Video gaming can be a solitary action. Nevertheless, it is important to motivate your youngster preparing to be social, and multi-player clash of clans hack is capable of doing that. They encourage sisters and brothers coupled with buddies to all [http://Search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=relating&Submit=Go relating] to take a moment as laugh and compete in the same room.<br><br>There are no end result in the least when you need to attacking other players or losing, so just onset and savor it. Win or lose, you have to may lose the nearly all troops you have within a the attack since they are only beneficial towards one mission, nevertheless, an individual can steal more money with the enemy vill than it cost in which to make the troops. And you just form more troops within you're barracks. It''s a good solid good idea to grab them queued up when in front of you decide to tackle and that means your family are rebuilding your troops through the battle.<br><br>Make sure you may not let games take over your existence. Game titles can be quite additive, therefore you have to make indeed you moderate the season that you investing trying to play such games. Content articles invest an excessive number of time playing video game, your actual life could actually begin to falter.<br><br>Gambling is infiltrating houses all around us. Some play these games for work, remember, though , others play them for enjoyment. This customers are booming and won't disappear anytime soon. Study for some fantastic tips about gaming.<br><br>A lot of our options are considered and approved from the best virus recognition software and so anti-virus in the sell to ensure a security-level the size of you can, in might you fear for protection of your computer or your cellular device, no troubles. In case you nevertheless have some sort of doubts, take a brows through the movie and you'll perceive it operates and ought to 100% secure! If you loved this report and you would like to obtain a lot more data concerning [http://circuspartypanama.com gem hack clash of clans] kindly check out our webpage. It'll only take a few moments of one's! |
| | |
| ==Definition==
| |
| Any two closed curves in space, if allowed to pass through themselves but not each other, can be [[homotopy|moved]] into exactly one of the following standard positions. This determines the linking number:
| |
| {| border=0 cellpadding=5 align="center"
| |
| |-valign="center"
| |
| |<math>\cdots</math>
| |
| |align="center"|[[Image:Linking Number -2.svg|140px]]
| |
| |align="center"|[[Image:Linking Number -1.svg|140px]]
| |
| |align="center"|[[Image:Linking Number 0.svg|140px]]
| |
| |
| |
| |
| |
| |-valign="center"
| |
| |
| |
| |align="center"|linking number -2
| |
| |align="center"|linking number -1
| |
| |align="center"|linking number 0
| |
| |
| |
| |
| |
| |-valign="center"
| |
| |
| |
| |
| |
| |align="center"|[[Image:Linking Number 1.svg|140px]]
| |
| |align="center"|[[Image:Linking Number 2.svg|140px]]
| |
| |align="center"|[[Image:Linking Number 3.svg|140px]]
| |
| |<math>\cdots</math>
| |
| |-valign="center"
| |
| |
| |
| |
| |
| |align="center"|linking number 1
| |
| |align="center"|linking number 2
| |
| |align="center"|linking number 3
| |
| |
| |
| |}
| |
| Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as [[regular homotopy]], which further requires that each curve be an ''immersion'', not just any map. However, this added condition does not change the definition of linking number (it does not matter if the curves are required to always be immersions or not), which is an example of an [[h-principle|''h''-principle]] (homotopy-principle), meaning that geometry reduces to topology.
| |
| | |
| === Proof ===
| |
| This fact (that the linking number is the only invariant) is most easily proven by placing one circle in standard position, and then showing that linking number is the only invariant of the other circle. In detail:
| |
| * A single curve is regular homotopic to a standard circle (any knot can be unknotted if the curve is allowed to pass through itself). The fact that it is ''homotopic'' is clear, since 3-space is contractible and thus all maps into it are homotopic, though the fact that this can be done through immersions requires some geometric argument.
| |
| * The complement of a standard circle is homeomorphic to a solid torus with a point removed (this can be seen by interpreting 3-space as the 3-sphere with the point at infinity removed, and the 3-sphere as two solid tori glued along the boundary), or the complement can be analyzed directly.
| |
| * The [[fundamental group]] of 3-space minus a circle is the integers, corresponding to linking number. This can be seen via the [[Seifert–Van Kampen theorem]] (either adding the point at infinity to get a solid torus, or adding the circle to get 3-space, allows one to compute the fundamental group of the desired space).
| |
| * Thus homotopy classes of a curve in 3-space minus a circle are determined by linking number.
| |
| * It is also true that regular homotopy classes are determined by linking number, which requires additional geometric argument.
| |
| | |
| ==Computing the linking number==
| |
| [[Image:Linking Number Example.svg|thumb|With six positive crossings and two negative crossings, these curves have linking number two.]]
| |
| There is an [[algorithm]] to compute the linking number of two curves from a link [[knot theory#Knot diagrams|diagram]]. Label each crossing as ''positive'' or ''negative'', according to the following rule:<ref>This is the same labeling used to compute the [[writhe]] of a [[knot (mathematics)|knot]], though in this case we only label crossings that involve both curves of the link.</ref>
| |
| <center>[[Image:Link Crossings.svg|350px]]</center>
| |
| The total number of positive crossings minus the total number of negative crossings is equal to ''twice'' the linking number. That is:
| |
| :<math>\mbox{linking number}=\frac{n_1 + n_2 - n_3 - n_4}{2}</math>
| |
| where ''n''<sub>1</sub>, ''n''<sub>2</sub>, ''n''<sub>3</sub>, ''n''<sub>4</sub> represent the number of crossings of each of the four types. The two sums <math>n_1 + n_3\,\!</math> and <math>n_2 + n_4\,\!</math> are always equal,<ref>This follows from the [[Jordan curve theorem]] if either curve is simple. For example, if the blue curve is simple, then ''n''<sub>1</sub> + ''n''<sub>3</sub> and ''n''<sub>2</sub> + ''n''<sub>4</sub> represent the number of times that the red curve crosses in and out of the region bounded by the blue curve.</ref> which leads to the following alternative formula
| |
| :<math>\mbox{linking number}\,=\,n_1-n_4\,=\,n_2-n_3.</math>
| |
| Note that <math>n_1-n_4</math> involves only the undercrossings of the blue curve by the red, while <math>n_2-n_3</math> involves only the overcrossings.
| |
| | |
| ==Properties and examples== | |
| [[Image:Labeled Whitehead Link.svg|thumb|The two curves of the [[Whitehead link]] have linking number zero.]]
| |
| * Any two unlinked curves have linking number zero. However, two curves with linking number zero may still be linked (e.g. the [[Whitehead link]]).
| |
| * Reversing the orientation of either of the curves negates the linking number, while reversing the orientation of both curves leaves it unchanged.
| |
| * The linking number is [[chirality (mathematics)|chiral]]: taking the [[mirror image]] of link negates the linking number. The convention for positive linking number is based on a [[right-hand rule]].
| |
| * The [[winding number]] of an oriented curve in the ''x''-''y'' plane is equal to its linking number with the ''z''-axis (thinking of the ''z''-axis as a closed curve in the [[3-sphere]]).
| |
| * More generally, if either of the curves is [[Curve#Topology|simple]], then the first [[homology (mathematics)|homology group]] of its complement is [[group isomorphism|isomorphic]] to '''[[integer|Z]]'''. In this case, the linking number is determined by the homology class of the other curve.
| |
| * In [[physics]], the linking number is an example of a [[topological quantum number]]. It is related to [[quantum entanglement]].
| |
| | |
| ==Gauss's integral definition==
| |
| Given two non-intersecting differentiable curves <math>\gamma_1, \gamma_2 \colon S^1 \rightarrow \mathbb{R}^3</math>, define the '''[[Carl Friedrich Gauss|Gauss]] map''' <math>\Gamma</math> from the [[torus]] to the [[unit sphere|sphere]] by
| |
| :<math>\Gamma(s,t) = \frac{\gamma_1(s) - \gamma_2(t)}{|\gamma_1(s) - \gamma_2(t)|}.</math>
| |
| | |
| Pick a point in the unit sphere, ''v'', so that orthogonal projection of the link to the plane perpendicular to ''v'' gives a link diagram. Observe that a point ''(s,t)'' that goes to ''v'' under the Gauss map corresponds to a crossing in the link diagram where <math>\gamma_1</math> is over <math>\gamma_2</math>. Also, a neighborhood of ''(s,t)'' is mapped under the Gauss map to a neighborhood of ''v'' preserving or reversing orientation depending on the sign of the crossing. Thus in order to compute the linking number of the diagram corresponding to ''v'' it suffices to count the ''signed'' number of times the Gauss map covers ''v''. Since ''v'' is a [[regular value]], this is precisely the [[degree of a continuous mapping|degree]] of the Gauss map (i.e. the signed number of times that the [[image (mathematics)|image]] of Γ covers the sphere). Isotopy invariance of the linking number is automatically obtained as the degree is invariant under homotopic maps. Any other regular value would give the same number, so the linking number doesn't depend on any particular link diagram.
| |
| | |
| This formulation of the linking number of ''γ''<sub>1</sub> and ''γ''<sub>2</sub> enables an explicit formula as a double [[line integral]], the '''Gauss linking integral''':
| |
| | |
| :<math>\mbox{linking number}\,=\,\frac{1}{4\pi}
| |
| \oint_{\gamma_1}\oint_{\gamma_2}
| |
| \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3}
| |
| \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2).</math>
| |
| | |
| This integral computes the total signed area of the image of the Gauss map (the integrand being the [[Jacobian]] of Γ) and then divides by the area of the sphere (which is 4''π''). | |
| | |
| ==Generalizations==
| |
| [[File:BorromeanRings.svg|thumb|The [[Milnor invariants]] generalize linking number to links with three or more components, allowing one prove that the [[Borromean rings]] are linked, though any two components have linking number 0.]]
| |
| * Just as closed curves can be [[link (knot theory)|linked]] in three dimensions, any two [[closed manifold]]s of dimensions ''m'' and ''n'' may be linked in a [[Euclidean space]] of dimension <math>m + n + 1</math>. Any such link has an associated Gauss map, whose [[degree of a continuous mapping|degree]] is a generalization of the linking number.
| |
| * Any [[framed knot]] has a [[self-linking number]] obtained by computing the linking number of the knot ''C'' with a new curve obtained by slightly moving the points of ''C'' along the framing vectors. The self-linking number obtained by moving vertically (along the blackboard framing) is known as '''Kauffman's self-linking number'''.
| |
| * The linking number is defined for two linked circles; given three or more circles, one can define the [[Milnor invariants]], which are a numerical invariant generalizing linking number.
| |
| * In [[algebraic topology]], the [[cup product]] is a far-reaching algebraic generalization of the linking number, with the [[Massey product]]s being the algebraic analogs for the [[Milnor invariants]].
| |
| * A [[linkless embedding]] of an [[undirected graph]] is an embedding into three-dimensional space such that every two cycles have zero linking number. The graphs that have a linkless embedding have a [[forbidden graph characterization|forbidden minor characterization]] as the graphs with no [[Petersen family]] [[minor (graph theory)|minor]].
| |
| | |
| ==See also==
| |
| * [[Differential geometry of curves]]
| |
| * [[Hopf invariant]]
| |
| * [[Kissing number problem]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{springer|author=A.V. Chernavskii|title=Linking coefficient|id=L/l059590}}
| |
| * {{springer|author=-|title=Writhing number|id=W/w098170}}
| |
| | |
| {{Knot theory|state=collapsed}}
| |
| | |
| [[Category:Curves]]
| |
| [[Category:Links by linking number| ]]
| |
At the time a association struggle begins, you will see The specific particular War Map, the new map of this conflict area area association battles booty place. Welcoming territories will consistently often be on the left, by having the adversary association on the inside the right. boondocks anteroom on the war map represents some type of war base.
Beginning nearly enough gems to get another developer. Dont waste some of the gems in any way on rush-building anything, as if it all can save you the group you are going to actually eventually obtain enough for free extra gems to grab that extra builder without even cost. Particularly, the customer can get free gallstones for clearing obstructions adore rocks and trees, when you are done you clear them outside they come back in addition to you may re-clear these to get more jewelry.
When you find yourself getting a online round for your little one, look for one typically enables numerous customers to do with each other. Video gaming can be a solitary action. Nevertheless, it is important to motivate your youngster preparing to be social, and multi-player clash of clans hack is capable of doing that. They encourage sisters and brothers coupled with buddies to all relating to take a moment as laugh and compete in the same room.
There are no end result in the least when you need to attacking other players or losing, so just onset and savor it. Win or lose, you have to may lose the nearly all troops you have within a the attack since they are only beneficial towards one mission, nevertheless, an individual can steal more money with the enemy vill than it cost in which to make the troops. And you just form more troops within you're barracks. Its a good solid good idea to grab them queued up when in front of you decide to tackle and that means your family are rebuilding your troops through the battle.
Make sure you may not let games take over your existence. Game titles can be quite additive, therefore you have to make indeed you moderate the season that you investing trying to play such games. Content articles invest an excessive number of time playing video game, your actual life could actually begin to falter.
Gambling is infiltrating houses all around us. Some play these games for work, remember, though , others play them for enjoyment. This customers are booming and won't disappear anytime soon. Study for some fantastic tips about gaming.
A lot of our options are considered and approved from the best virus recognition software and so anti-virus in the sell to ensure a security-level the size of you can, in might you fear for protection of your computer or your cellular device, no troubles. In case you nevertheless have some sort of doubts, take a brows through the movie and you'll perceive it operates and ought to 100% secure! If you loved this report and you would like to obtain a lot more data concerning gem hack clash of clans kindly check out our webpage. It'll only take a few moments of one's!