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In mathematics, the linking number is a numerical
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
that describes the linking of two
closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
s in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
. Intuitively, the linking number represents the number of times that each curve winds around the other. In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, the linking number is always an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, but may be positive or negative depending on the
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
of the two curves (this is not true for curves in most
3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
s, where linking numbers can also be fractions or just not exist at all). The linking number was introduced by
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in the form of the linking integral. It is an important object of study in
knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
,
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
, and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mul ...
, and has numerous applications in mathematics and
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
, including
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, and the study of
DNA supercoil DNA supercoiling refers to the amount of twist in a particular DNA strand, which determines the amount of strain on it. A given strand may be "positively supercoiled" or "negatively supercoiled" (more or less tightly wound). The amount of a st ...
ing.

# 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: Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as
regular homotopy In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. Similar to homotopy classes, one defines two imm ...
, 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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of 3-space minus a circle is the integers, corresponding to linking number. This can be seen via the
Seifert–Van Kampen theorem In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in t ...
(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.

There is an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a thr ...
. Label each crossing as ''positive'' or ''negative'', according to the following rule: The total number of positive crossings minus the total number of negative crossings is equal to ''twice'' the linking number. That is: :$\text=\frac$ where ''n''1, ''n''2, ''n''3, ''n''4 represent the number of crossings of each of the four types. The two sums $n_1 + n_3\,\!$ and $n_2 + n_4\,\!$ are always equal,This follows from the
Jordan curve theorem In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exteri ...
if either curve is simple. For example, if the blue curve is simple, then ''n''1 + ''n''3 and ''n''2 + ''n''4 represent the number of times that the red curve crosses in and out of the region bounded by the blue curve.
which leads to the following alternative formula :$\text \,=\, n_1 - n_4 \,=\, n_2 - n_3.$ The formula $n_1-n_4$ involves only the undercrossings of the blue curve by the red, while $n_2-n_3$ involves only the overcrossings.

# Properties and examples

* 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 Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
: taking the
mirror image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of t ...
. * The
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...
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 In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
). * More generally, if either of the curves is simple, then the first
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolo ...
of its complement is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to Z. In this case, the linking number is determined by the homology class of the other curve. * In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...
, the linking number is an example of a
topological quantum number In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are ...
. It is related to
quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
.

# Gauss's integral definition

Given two non-intersecting differentiable curves $\gamma_1, \gamma_2 \colon S^1 \rightarrow \mathbb^3$, define the
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
map $\Gamma$ from the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not t ...
to the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
by :$\Gamma\left(s,t\right) = \frac$ 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 $\gamma_1$ is over $\gamma_2$. 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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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''1 and ''γ''2 enables an explicit formula as a double
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, a ...
, the Gauss linking integral: : 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).

# In quantum field theory

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, Gauss' integral definition arises when computing the expectation value of the
Wilson loop In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representat ...
observable in $U\left(1\right)$ Chern–Simons
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
. Explicitly, the abelian Chern–Simons action for a gauge potential one-form $A$ on a three-
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
$M$ is given by : $S_ = \frac \int_M A \wedge dA$ We are interested in doing the Feynman path integral for Chern–Simons in $M = \mathbb^3$: : Here, $\epsilon$ is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet regularization or
renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
is needed. Therefore, the topological invariance of right hand side ensures that the result of the path integral will be a topological invariant. The only thing left to do is provide an overall normalization factor, and a natural choice will present itself. Since the theory is Gaussian and abelian, the path integral can be done simply by solving the theory classically and substituting for $A$. The classical equations of motion are : $\varepsilon^ \partial_\mu A_\nu = \frac J^\lambda$ Here, we have coupled the Chern–Simons field to a source with a term $-J_\mu A^\mu$ in the Lagrangian. Obviously, by substituting the appropriate $J$, we can get back the Wilson loops. Since we are in 3 dimensions, we can rewrite the equations of motion in a more familiar notation: : $\vec \times \vec = \frac \vec$ Taking the curl of both sides and choosing Lorenz gauge $\partial^\mu A_\mu = 0$, the equations become : $\nabla^2 \vec = - \frac \vec \times \vec$ From electrostatics, the solution is : $A_\lambda\left(\vec\right) = \frac \int d^3 \vec \, \frac$ The path integral for arbitrary $J$ is now easily done by substituting this into the Chern–Simons action to get an effective action for the $J$ field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e. $J = J_1 + J_2$, with : $J_i^\mu \left(x\right) = \int_ dx_i^\mu \delta^3 \left(x - x_i \left(t\right)\right)$ Since the effective action is quadratic in $J$, it is clear that there will be terms describing the self-interaction of the particles, and these are uninteresting since they would be there even in the presence of just one loop. Therefore, we normalize the path integral by a factor precisely cancelling these terms. Going through the algebra, we obtain : where : which is simply Gauss' linking integral. This is the simplest example of a
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
, where the path integral computes topological invariants. This also served as a hint that the nonabelian variant of Chern–Simons theory computes other knot invariants, and it was shown explicitly by
Edward Witten Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, ...
that the nonabelian theory gives the invariant known as the Jones polynomial. The Chern-Simons gauge theory lives in 3 spacetime dimensions. More generally, there exists higher dimensional topological quantum field theories. There exists more complicated multi-loop/string-braiding statistics of 4-dimensional gauge theories captured by the link invariants of exotic topological quantum field theories in 4 spacetime dimensions.

# Generalizations

* Just as closed curves can be linked in three dimensions, any two
closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example ...
s of dimensions ''m'' and ''n'' may be linked in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension $m + n + 1$. Any such link has an associated Gauss map, whose degree is a generalization of the linking number. * Any
framed knot In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of ...
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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
, the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commuta ...
is a far-reaching algebraic generalization of the linking number, with the
Massey product In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist. Massey triple product L ...
s being the algebraic analogs for the Milnor invariants. * A
linkless embedding In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is ...
of an
undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
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.