WordNet

give numbers to; "You should number the pages of the thesis"
place a limit on the number of (同)keep_down
a concept of quantity involving zero and units; "every number has a unique position in the sequence"
the property possessed by a sum or total or indefinite quantity of units or individuals; "he had a number of chores to do"; "the number of parameters is small"; "the figure was about a thousand" (同)figure
a clothing measurement; "a number 13 shoe"
an item of merchandise offered for sale; "she preferred the black nylon number"; "this sweater is an all-wool number"
a numeral or string of numerals that is used for identification; "she refused to give them her Social Security number" (同)identification number
a select company of people; "I hope to become one of their number before I die"
the grammatical category for the forms of nouns and pronouns and verbs that are used depending on the number of entities involved (singular or dual or plural); "in English the subject and the verb must agree in number"
enumerate; "We must number the names of the great mathematicians" (同)list
so frightened as to be unable to move; stunned or paralyzed with terror; petrified; "too numb with fear to move"
make numb or insensitive; "The shock numbed her senses" (同)benumb, blunt, dull

PrepTutorEJDIC

〈U〉〈C〉(数えて得られる)『数,数量』 / 〈C〉(概念としての)『数,数字』 / 〈C〉『番号』 / 〈C〉(演奏会や演劇の)番組,出し物;曲目 / 〈C〉(雑誌の)号 / 〈U〉(文法で)数(すう) / 《複数形で》数の上の優勢 / 《複数形で》算数 / 〈C〉《単数形で》《話》(商品としての)洋服の1点;商品,売り物 / 〈C〉《単随形で》《俗》女の子 / …‘を'数える / (…の中に,…として)…‘を'含める,加える《+『among』(『with, as』)+『名』》 / …‘に'番号をつける / …‘の'数となる / 《しばしば受動態で》…‘の'数を制限する / 総計(…に)なる《+『in』+『名』〈数〉》
麻痺(まひ)した,感覚を失った,しびれた / …‘の'感覚を失わせる,‘を'しびれさせる
数学,数学 / (人員などの)数の優勢 / 《米》《the~》=numbers game / 《古》詞句,韻文

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2014/04/16 03:23:48」(JST)

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"Link number" redirects here. For the logic puzzle, see Numberlink.

The two curves of this (2,8)-torus link have linking number four.

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves 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 orientation of the two curves.

The linking number was introduced by 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 supercoiling.

Definition

Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions. This determines the linking number:


linking number -2	linking number -1	linking number 0

	linking number 1	linking number 2	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 (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

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 diagram. Label each crossing as positive or negative, according to the following rule:^[1]

The total number of positive crossings minus the total number of negative crossings is equal to twice the linking number. That is:

where n₁, n₂, n₃, n₄ represent the number of crossings of each of the four types. The two sums and are always equal,^[2] which leads to the following alternative formula

Note that involves only the undercrossings of the blue curve by the red, while involves only the overcrossings.

Properties and examples

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 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 simple, then the first homology group of its complement is isomorphic to 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 , define the Gauss map from the torus to the sphere by

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 is over . 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 the Gauss map (i.e. the signed number of times that the 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 γ₁ and γ₂ enables an explicit formula as a double line integral, the Gauss linking integral:

$\mbox{linking number}\,=\,\frac{1}{4\pi} \oint_{\gamma_1}\oint_{\gamma_2} \frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} \cdot (d\mathbf{r}_1 \times d\mathbf{r}_2).$

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 4π).

Generalizations

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 linked in three dimensions, any two closed manifolds of dimensions m and n may be linked in a Euclidean space of dimension . Any such link has an associated Gauss map, whose 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 products 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 minor characterization as the graphs with no Petersen family minor.

Notes

^ This is the same labeling used to compute the writhe of a knot, though in this case we only label crossings that involve both curves of the link.
^ This follows from the Jordan curve theorem if either curve is simple. For example, if the blue curve is simple, then n₁ + n₃ and n₂ + n₄ represent the number of times that the red curve crosses in and out of the region bounded by the blue curve.

References

A.V. Chernavskii (2001), "Linking coefficient", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- (2001), "Writhing number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

UpToDate Contents

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English Journal

Influence of the scaffolding protein Zonula Occludens (ZOs) on membrane channels.

Hervé JC1, Derangeon M2, Sarrouilhe D3, Bourmeyster N3.Author information 1Institut de Physiologie et Biologie Cellulaires, FRE3511, Université de Poitiers, CNRS, Poitiers, France. Electronic address: Jean.Claude.Herve@univ-poitiers.fr.2Institut du Thorax, INSERM UMR 1087, CNRS UMR 6291, Equipe "Cardiopathies et mort subite", Nantes, France.3Institut de Physiologie et Biologie Cellulaires, FRE3511, Université de Poitiers, CNRS, Poitiers, France.AbstractZonula Occludens (ZO) proteins are ubiquitous scaffolding proteins providing the structural basis for the assembly of multiprotein complexes at the cytoplasmic surface of the plasma membrane and linking transmembrane proteins to the filamentous cytoskeleton. They belong to the large family of membrane-associated guanylate kinase (MAGUK)-like proteins comprising a number of subfamilies based on domain content and sequence similarity. ZO proteins were originally described to localize specifically to tight junctions, or Zonulae Occludentes, but this notion was rapidly reconsidered since ZO proteins were found to associate with adherens junctions as well as with gap junctions, particularly with connexin-made intercellular channels, and also with a few other membrane channels. Accumulating evidence reveals that in addition to having passive scaffolding functions in organizing gap junction complexes, including connexins and cytoskeletals, ZO proteins (particularly ZO-1) also actively take part in the dynamic function as well as in the remodeling of junctional complexes in a number of cellular systems. This article is part of a Special Issue entitled: Reciprocal influences between cell cytoskeleton and membrane channels, receptors and transporters. Guest Editor: Jean Claude Hervé.
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PMID 23867773

In silico analyses of structural and allergenicity features of sapodilla (Manilkara zapota) acidic thaumatin-like protein in comparison with allergenic plant TLPs.

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PMID 24091295

Hyaluronic acid and alginate covalent nanogels by template cross-linking in polyion complex micelle nanoreactors.

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Hyaluronic acid (HA) and alginate (AL) covalent nanogels cross-linked with l-lysine ethyl ester were prepared by template chemical cross-linking of the polysaccharide in polyion complex micelle (PIC) nanoreactors. By using this method we were able to prepare HA and AL nanogels without organic solven
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Japanese Journal

Successful long-term preservation of rat sperm by freeze-drying.

Kaneko Takehito,Serikawa Tadao
PloS one 7(4), 2012-04-09
… Although the successful freeze-drying of sperm has been reported in a number of animals, the possibility of long-term preservation using this method has not yet been studied. … cross-linking in the thiol-disulfide of their protamine were highly tolerant to freeze-drying, and the fertility of freeze-dried sperm was maintained for 5 years without deterioration. …
NAID 120004057185

A Study of the Entanglement in Systems with Periodic Boundary Conditions(Statistical Physics and Topology of Polymers with Ramifications to Structure and Function of DNA and Proteins)

PANAGIOTOU Eleni,TZOUMANEKAS Christos,LAMBROPOULOU Sofia,MILLETT Kenneth C.,THEODOROU Doros N.
Progress of theoretical physics. Supplement (191), 172-181, 2011-12-16
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