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This page explains commonly used terms related to knots.

• List of knots

A friction hitch is a kind of knot used to attach one rope to another in a way that is easily adjusted. These knots are commonly used in climbing as part of single-rope technique, doubled-rope technique and as “ratchets” to capture progress on a moving rope, most typically in a mechanical advantage system such as a Z-drag. These hitches are a simple and cheap alternative to mechanical ascenders.

• List of hitch knots
• List of knots

• Son of a Hitch: A Genealogy of Arborists’ Climbing Hitches

• Climber’s Friction Hitch Line up

There are many types of knots that are commonly used in the pursuit of rock climbing, ice climbing, and general mountaineering, the most popular of which are listed below.

• Rock climbing knots Knots used in climbing and mountaineering
• Grog’s Index of Climbing Knots

A binding knot is a knot that may be used to keep an object or multiple loose objects together, using a string or a rope that passes at least once around them. There are various binding knots, divided into two types. Friction knots are held in place by the friction between the windings of line. Knotted-ends knots are held in place by the two ends of the line being knotted together.

Stopping may be either a temporary whipping or seizing, the commonest variety consisting of a few round turns finished off with a reef knot. The purpose of a whipping is to prevent the end of a rope from fraying. A seizing holds several objects together.

Whipping and seizing are binding knots, but are more complex since they contain many turns, like a lashing.

• Boa knot
• Bottle sling
• Constrictor knot
• Corned beef knot
• Granny knot
• Grief knot^[2]
• Ground-line hitch
• Miller’s knot
• Packer’s knot
• Reef knot
• Strangle knot
• Surgeon’s knot
• Thief knot
• Jamming knot
• Sheet bend
• Sheepshank
• Common whipping

• List of bend knots
• List of knots
• Rope splicing

In knot theory, a Lissajous knot is a knot defined by parametric equations of the form

${\displaystyle x=\cos(n_{x}t+\phi _{x}),\qquad y=\cos(n_{y}t+\phi _{y}),\qquad z=\cos(n_{z}t+\phi _{z}),}$

where ${\displaystyle n_{x}}$, ${\displaystyle n_{y}}$, and ${\displaystyle n_{z}}$ are integers and the phase shifts ${\displaystyle \phi _{x}}$, ${\displaystyle \phi _{y}}$, and ${\displaystyle \phi _{z}}$ may be any real numbers.^[1]

The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves.

Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajous
knot isotopically into a billiard curve inside a cube, the simplest case of so-called billiard knots.
Billiard knots can also be studied in other domains, for instance in a cylinder^[2] or in a (flat) solid torus (Lissajous-toric knot).

Because a knot cannot be self-intersecting, the three integers ${\displaystyle n_{x},n_{y},n_{z}}$ must be pairwise relatively prime, and none of the quantities

${\displaystyle n_{x}\phi _{y}-n_{y}\phi _{x},\quad n_{y}\phi _{z}-n_{z}\phi _{y},\quad n_{z}\phi _{x}-n_{x}\phi _{z}}$

may be an integer multiple of pi. Moreover, by making a substitution of the form ${\displaystyle t’=t+c}$, one may assume that any of the three phase shifts ${\displaystyle \phi _{x}}$, ${\displaystyle \phi _{y}}$, ${\displaystyle \phi _{z}}$ is equal to zero.

Here are some examples of Lissajous knots,^[3] all of which have ${\displaystyle \phi _{z}=0}$:

• 
  Three-twist knot
  ${\displaystyle (n_{x},n_{y},n_{z})=(3,2,7)}$
  ${\displaystyle (\phi _{x},\phi _{y})=(0.7,0.2)}$
• 
  Stevedore knot
  ${\displaystyle (n_{x},n_{y},n_{z})=(3,2,5)}$
  ${\displaystyle (\phi _{x},\phi _{y})=(1.5,0.2)}$
• 
  Square knot
  ${\displaystyle (n_{x},n_{y},n_{z})=(3,5,7)}$
  ${\displaystyle (\phi _{x},\phi _{y})=(0.7,1.0)}$
• 
  8₂₁ knot
  ${\displaystyle (n_{x},n_{y},n_{z})=(3,4,7)}$
  ${\displaystyle (\phi _{x},\phi _{y})=(0.1,0.7)}$

There are infinitely many different Lissajous knots,^[4] and other examples with 10 or fewer crossings include the 7₄ knot, the 8₁₅ knot, the 10₁ knot, the 10₃₅ knot, the 10₅₈ knot, and the composite knot 5₂^* # 5₂,^[1] as well as the 9₁₆ knot, 10₇₆ knot, the 10₉₉ knot, the 10₁₂₂ knot, the 10₁₄₄ knot, the granny knot, and the composite knot 5₂ # 5₂.^[5] In addition, it is known that every twist knot with Arf invariant zero is a Lissajous knot.^[6]

Lissajous knots are highly symmetric, though the type of symmetry depends on whether or not the numbers ${\displaystyle n_{x}}$, ${\displaystyle n_{y}}$, and ${\displaystyle n_{z}}$ are all odd.

Consequences

The symmetry of a Lissajous knot puts severe constraints on the Alexander polynomial. In the odd case, the Alexander
polynomial of the Lissajous knot must be a perfect square.^[9] In the even case, the Alexander polynomial must be a perfect square modulo 2.^[10] In addition, the Arf invariant of a Lissajous knot must be zero. It follows that:

• The trefoil knot and figure-eight knot are not Lissajous.
• No torus knot can be Lissajous.
• No fibered 2-bridge knot can be Lissajous.

In knot theory, a Lissajous-toric knot is a knot defined by parametric equations of the form:

${\displaystyle x(t)=(2+\sin qt)\cos Nt,\qquad y(t)=(2+\sin qt)\sin Nt,\qquad z(t)=\cos p(t+\phi ),}$

where ${\displaystyle N}$, ${\displaystyle p}$, and ${\displaystyle q}$ are integers, the phase shift ${\displaystyle \phi }$ is a real number
and the parameter ${\displaystyle t}$ varies between 0 and ${\displaystyle 2\pi }$.^[1]

For ${\displaystyle p=q}$ the knot is a torus knot.

In braid form these knots can be defined in a square solid torus (i.e. the cube ${\displaystyle [-1,1]^{3}}$ with identified top and bottom) as

${\displaystyle x(t)=\sin 2\pi qt,\qquad y(t)=\cos 2\pi p(t+\phi ),\qquad z(t)=2(Nt-\lfloor Nt\rfloor )-1,\qquad t\in [0,1]}$.

The projection of this Lissajous-toric knot onto the x-y-plane is a Lissajous curve.

Replacing the sine and cosine functions in the parametrization by a triangle wave transforms a Lissajous-toric
knot isotopically into a billiard curve inside the solid torus. Because of this property Lissajous-toric knots are also called billiard knots in a solid torus.^[2]

Lissajous-toric knots were first studied as billiard knots and they share many properties with billiard knots in a cylinder.^[3]
They also occur in the analysis of singularities of minimal surfaces with branch points^[4] and in the study of
the Three-body problem.^[5]

The knots in the subfamily with ${\displaystyle p=q\cdot l}$, with an integer ${\displaystyle l\geq 1}$, are known as ′Lemniscate knots′.^[6] Lemniscate knots have period ${\displaystyle q}$ and are fibred. The knot shown on the right is of this type (with ${\displaystyle l=5}$).

Lissajous-toric knots are denoted by ${\displaystyle K(N,q,p,\phi )}$. To ensure that the knot is traversed only once in the parametrization
the conditions ${\displaystyle \gcd(N,q)=\gcd(N,p)=1}$ are needed. In addition, singular values for the phase, leading to self-intersections, have to be excluded.

The isotopy class of Lissajous-toric knots surprisingly does not depend on the phase ${\displaystyle \phi }$ (up to mirroring).
If the distinction between a knot and its mirror image is not important, the notation ${\displaystyle K(N,q,p)}$ can be used.

The properties of Lissajous-toric knots depend on whether ${\displaystyle p}$ and ${\displaystyle q}$ are coprime or ${\displaystyle d=\gcd(p,q)>1}$. The main properties are:

• Interchanging ${\displaystyle p}$ and ${\displaystyle q}$:

${\displaystyle K(N,q,p)=K(N,p,q)}$ (up to mirroring).

• Ribbon property:

If ${\displaystyle p}$ and ${\displaystyle q}$ are coprime, ${\displaystyle K(N,q,p)}$ is a symmetric union and therefore a ribbon knot.

• Periodicity:

If ${\displaystyle d=\gcd(p,q)>1}$, the Lissajous-toric knot has period ${\displaystyle d}$ and the factor knot is a ribbon knot.

• Strongly positive amphicheirality:

If ${\displaystyle p}$ and ${\displaystyle q}$ have different parity, then ${\displaystyle K(N,q,p)}$ is strongly positive amphicheiral.

• Period 2:

If ${\displaystyle p}$ and ${\displaystyle q}$ are both odd, then ${\displaystyle K(N,q,p)}$ has period 2 (for even ${\displaystyle N}$) or is freely 2-periodic (for odd ${\displaystyle N}$).

In the following table a systematic overview of the possibilities to build billiard rooms from the interval and the circle (interval with identified boundaries) is given:

Billiard room	Billiard knots
${\displaystyle [-1,1]^{3}}$	Lissajous knots
${\displaystyle [-1,1]^{2}\times \mathbb {S} ^{1}}$	Lissajous-toric knots
${\displaystyle [-1,1]\times \mathbb {S} ^{1}\times \mathbb {S} ^{1}}$	Torus knots
${\displaystyle \mathbb {S} ^{1}\times \mathbb {S} ^{1}\times \mathbb {S} ^{1}}$	(room not embeddable into ${\displaystyle \mathbb {R} ^{3}}$)

In the case of Lissajous knots reflections at the boundaries occur in all of the three cube’s dimensions.
In the second case reflections occur in two dimensions and we have a uniform movement in the third dimension.
The third case is nearly equal to the usual movement on a torus, with an additional triangle wave movement in the first dimension.

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 an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three-dimensional analogue of the planar graphs.^[1] Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding.

Flat embeddings are automatically linkless, but not vice versa.^[2] The complete graph K₆, the Petersen graph, and the other five graphs in the Petersen family do not have linkless embeddings.^[1] Every graph minor of a linklessly embeddable graph is again linklessly embeddable,^[3] as is every graph that can be reached from a linklessly embeddable graph by YΔ- and ΔY-transformations.^[2] The linklessly embeddable graphs have the Petersen family graphs as their forbidden minors,^[4] and include the planar graphs and apex graphs.^[2] They may be recognized, and a flat embedding may be constructed for them, in O(n²).^[5]

When the circle is mapped to three-dimensional Euclidean space by an injective function (a continuous function that does not map two different points of the circle to the same point of space), its image is a closed curve.
Two disjoint closed curves that both lie on the same plane are unlinked, and more generally a pair of disjoint closed curves is said to be unlinked when there is a continuous deformation of space that moves them both onto the same plane, without either curve passing through the other or through itself. If there is no such continuous motion, the two curves are said to be linked. For example, the Hopf link is formed by two circles that each pass through the disk spanned by the other. It forms the simplest example of a pair of linked curves, but it is possible for curves to be linked in other more complicated ways. If two curves are not linked, then it is possible to find a topological disk in space, having the first curve as its boundary and disjoint from the second curve. Conversely if such a disk exists then the curves are necessarily unlinked.

The linking number of two closed curves in three-dimensional space is a topological invariant of the curves: it is a number, defined from the curves in any of several equivalent ways, that does not change if the curves are moved continuously without passing through each other. The version of the linking number used for defining linkless embeddings of graphs is found by projecting the embedding onto the plane and counting the number of crossings of the projected embedding in which the first curve passes over the second one, modulo 2.^[2] The projection must be “regular”, meaning that no two vertices project to the same point, no vertex projects to the interior of an edge, and at every point of the projection where the projections of two edges intersect, they cross transversally; with this restriction, any two projections lead to the same linking number.
The linking number of the unlink is zero, and therefore, if a pair of curves has nonzero linking number, the two curves must be linked. However, there are examples of curves that are linked but that have zero linking number, such as the Whitehead link.

An embedding of a graph into three-dimensional space consists of a mapping from the vertices of the graph to points in space, and from the edges of the graph to curves in space, such that each endpoint of each edge is mapped to an endpoint of the corresponding curve, and such that the curves for two different edges do not intersect except at a common endpoint of the edges.
Any finite graph has a finite (though perhaps exponential) number of distinct simple cycles, and if the graph is embedded into three-dimensional space then each of these cycles forms a simple closed curve. One may compute the linking number of each disjoint pair of curves formed in this way; if all pairs of cycles have zero linking number, the embedding is said to be linkless.^[6]

In some cases, a graph may be embedded in space in such a way that, for each cycle in the graph, one can find a disk bounded by that cycle that does not cross any other feature of the graph. In this case, the cycle must be unlinked from all the other cycles disjoint from it in the graph. The embedding is said to be flat if every cycle bounds a disk in this way.^[7] A flat embedding is necessarily linkless, but there may exist linkless embeddings that are not flat: for instance, if G is a graph formed by two disjoint cycles, and it is embedded to form the Whitehead link, then the embedding is linkless but not flat.

A graph is said to be intrinsically linked if, no matter how it is embedded, the embedding is always linked. Although linkless and flat embeddings are not the same, the graphs that have linkless embeddings are the same as the graphs that have flat embeddings.^[8]

As Sachs (1983) showed, each of the seven graphs of the Petersen family is intrinsically linked: no matter how each of these graphs is embedded in space, they have two cycles that are linked to each other. These graphs include the complete graph K₆, the Petersen graph, the graph formed by removing an edge from the complete bipartite graph K_4,4, and the complete tripartite graph K_3,3,1.

Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space. If a graph is planar, this is the only way to embed it flatly and linklessly into space: every flat embedding can be continuously deformed to lie on a flat plane. And conversely, every nonplanar linkless graph has multiple linkless embeddings.^[2]

An apex graph, formed by adding a single vertex to a planar graph, also has a flat and linkless embedding: embed the planar part of the graph on a plane, place the apex above the plane, and draw the edges from the apex to its neighbors as line segments. Any closed curve within the plane bounds a disk below the plane that does not pass through any other graph feature, and any closed curve through the apex bounds a disk above the plane that does not pass through any other graph feature.^[2]

If a graph has a linkless or flat embedding, then modifying the graph by subdividing or unsubdividing its edges, adding or removing multiple edges between the same pair of points, and performing YΔ- and ΔY-transformations that replace a degree-three vertex by a triangle connecting its three neighbors or the reverse all preserve flatness and linklessness.^[2] In particular, in a cubic planar graph (one in which all vertices have exactly three neighbors, such as the cube) it is possible to make duplicates of any independent set of vertices by performing a YΔ-transformation, adding multiple copies of the resulting triangle edges, and then performing the reverse ΔY-transformations.

If a graph G has a linkless or flat embedding, then every minor of G (a graph formed by contraction of edges and deletion of edges and vertices) also has a linkless or flat embedding. Deletions cannot destroy the flatness of an embedding, and a contraction can be performed by leaving one endpoint of the contracted edge in place and rerouting all the edges incident to the other endpoint along the path of the contracted edge. Therefore, by the Robertson–Seymour theorem, the linklessly embeddable graphs have a forbidden graph characterization as the graphs that do not contain any of a finite set of minors.^[3]

The set of forbidden minors for the linklessly embeddable graphs was identified by Sachs (1983): the seven graphs of the Petersen family are all minor-minimal intrinsically linked graphs. However, Sachs was unable to prove that these were the only minimal linked graphs, and this was finally accomplished by Robertson, Seymour & Thomas (1995).

The forbidden minor characterization of linkless graphs leads to a polynomial time algorithm for their recognition, but not for actually constructing an embedding. Kawarabayashi, Kreutzer & Mohar (2010) described a linear time algorithm that tests whether a graph is linklessly embeddable and, if so, constructs a flat embedding of the graph. Their algorithm finds large planar subgraphs within the given graph such that, if a linkless embedding exists, it has to respect the planar embedding of the subgraph. By repeatedly simplifying the graph whenever such a subgraph is found, they reduce the problem to one in which the remaining graph has bounded treewidth, at which point it can be solved by dynamic programming.

The problem of efficiently testing whether a given embedding is flat or linkless was posed by Robertson, Seymour & Thomas (1993a). It remains unsolved, and is equivalent in complexity to unknotting problem, the problem of testing whether a single curve in space is unknotted.^[5] Testing unknottedness (and therefore, also, testing linklessness of an embedding) is known to be in NP but is not known to be NP-complete.^[9]

The question of whether K₆ has a linkless or flat embedding was posed within the topology research community in the early 1970s by Bothe (1973). Linkless embeddings were brought to the attention of the graph theory community by Horst Sachs (1983), who posed several related problems including the problem of finding a forbidden graph characterization of the graphs with linkless and flat embeddings; Sachs showed that the seven graphs of the Petersen family (including K₆) do not have such embeddings. As Nešetřil & Thomas (1985) observed, linklessly embeddable graphs are closed under graph minors, from which it follows by the Robertson–Seymour theorem that a forbidden graph characterization exists. The proof of the existence of a finite set of obstruction graphs does not lead to an explicit description of this set of forbidden minors, but it follows from Sachs’ results that the seven graphs of the Petersen family belong to the set. These problems were finally settled by Robertson, Seymour & Thomas (1995),^[18] who showed that the seven graphs of the Petersen family are the only minimal forbidden minors for these graphs. Therefore, linklessly embeddable graphs and flat embeddable graphs are both the same set of graphs, and are both the same as the graphs that have no Petersen family minor.

Sachs (1983) also asked for bounds on the number of edges and the chromatic number of linkless embeddable graphs. The number of edges in an n-vertex linkless graph is at most 4n − 10: maximal apex graphs with n > 4 have exactly this many edges,^[1] and Mader (1968) proved a matching upper bound on the more general class of K₆-minor-free graphs. Nešetřil & Thomas (1985) observed that Sachs’ question about the chromatic number would be resolved by a proof of Hadwiger’s conjecture that any k-chromatic graph has as a minor a k-vertex complete graph. The proof by Robertson, Seymour & Thomas (1993c) of the case k = 6 of Hadwiger’s conjecture is sufficient to settle Sachs’ question: the linkless graphs can be colored with at most five colors, as any 6-chromatic graph contains a K₆ minor and is not linkless, and there exist linkless graphs such as K₅ that require five colors. The snark theorem implies that every cubic linklessly embeddable graph is 3-edge-colorable.

Linkless embeddings started being studied within the algorithms research community in the late 1980s through the works of Fellows & Langston (1988) and Motwani, Raghunathan & Saran (1988). Algorithmically, the problem of recognizing linkless and flat embeddable graphs was settled once the forbidden minor characterization was proven: an algorithm of Robertson & Seymour (1995) can be used to test in polynomial time whether a given graph contains any of the seven forbidden minors.^[19] This method does not construct linkless or flat embeddings when they exist, but an algorithm that does construct an embedding was developed by van der Holst (2009), and a more efficient linear time algorithm was found by Kawarabayashi, Kreutzer & Mohar (2010).

A final question of Sachs (1983) on the possibility of an analogue of Fáry’s theorem for linkless graphs appears not to have been answered: when does the existence of a linkless or flat embedding with curved or piecewise linear edges imply the existence of a linkless or flat embedding in which the edges are straight line segments?

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. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all).

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.

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.

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:

${\displaystyle {\text{linking number}}={\frac {n_{1}+n_{2}-n_{3}-n_{4}}{2}}}$

where n₁, n₂, n₃, n₄ represent the number of crossings of each of the four types. The two sums ${\displaystyle n_{1}+n_{3}\,\!}$ and ${\displaystyle n_{2}+n_{4}\,\!}$ are always equal,^[2] which leads to the following alternative formula

${\displaystyle {\text{linking number}}\,=\,n_{1}-n_{4}\,=\,n_{2}-n_{3}.}$

The formula ${\displaystyle n_{1}-n_{4}}$ involves only the undercrossings of the blue curve by the red, while ${\displaystyle n_{2}-n_{3}}$ involves only the overcrossings.

• 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.

Given two non-intersecting differentiable curves ${\displaystyle \gamma _{1},\gamma _{2}\colon S^{1}\rightarrow \mathbb {R} ^{3}}$, define the Gauss map ${\displaystyle \Gamma }$ from the torus to the sphere by

${\displaystyle \Gamma (s,t)={\frac {\gamma _{1}(s)-\gamma _{2}(t)}{|\gamma _{1}(s)-\gamma _{2}(t)|}}}$

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 ${\displaystyle \gamma _{1}}$ is over ${\displaystyle \gamma _{2}}$. 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 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:

${\displaystyle {\begin{aligned}\operatorname {link} (\gamma _{1},\gamma _{2})&={\frac {1}{4\pi }}\oint _{\gamma _{1}}\oint _{\gamma _{2}}{\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|^{3}}}\cdot (d\mathbf {r} _{1}\times d\mathbf {r} _{2})\\[4pt]&={\frac {1}{4\pi }}\int _{S^{1}\times S^{1}}{\frac {\det \left({\dot {\gamma }}_{1}(s),{\dot {\gamma }}_{2}(t),\gamma _{1}(s)-\gamma _{2}(t)\right)}{\left|\gamma _{1}(s)-\gamma _{2}(t)\right|^{3}}}\,ds\,dt\end{aligned}}}$

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).

In quantum field theory, Gauss’s integral definition arises when computing the expectation value of the Wilson loop observable in ${\displaystyle U(1)}$ Chern–Simons gauge theory. Explicitly, the abelian Chern–Simons action for a gauge potential one-form ${\displaystyle A}$ on a three-manifold ${\displaystyle M}$ is given by

${\displaystyle S_{CS}={\frac {k}{4\pi }}\int _{M}A\wedge dA}$

We are interested in doing the Feynman path integral for Chern–Simons in ${\displaystyle M=\mathbb {R} ^{3}}$:

${\displaystyle Z[\gamma _{1},\gamma _{2}]=\int {\mathcal {D}}A_{\mu }\exp \left({\frac {ik}{4\pi }}\int d^{3}x\varepsilon ^{\lambda \mu \nu }A_{\lambda }\partial _{\mu }A_{\nu }+i\int _{\gamma _{1}}dx^{\mu }\,A_{\mu }+i\int _{\gamma _{2}}dx^{\mu }\,A_{\mu }\right)}$

Here, ${\displaystyle \epsilon }$ is the antisymmetric symbol. Since the theory is just Gaussian, no ultraviolet regularization or renormalization 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 ${\displaystyle A}$.

The classical equations of motion are

${\displaystyle \varepsilon ^{\lambda \mu \nu }\partial _{\mu }A_{\nu }={\frac {2\pi }{k}}J^{\lambda }}$

Here, we have coupled the Chern–Simons field to a source with a term ${\displaystyle -J_{\mu }A^{\mu }}$ in the Lagrangian. Obviously, by substituting the appropriate ${\displaystyle J}$, 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:

${\displaystyle {\vec {\nabla }}\times {\vec {A}}={\frac {2\pi }{k}}{\vec {J}}}$

Taking the curl of both sides and choosing Lorenz gauge ${\displaystyle \partial ^{\mu }A_{\mu }=0}$, the equations become

${\displaystyle \nabla ^{2}{\vec {A}}=-{\frac {2\pi }{k}}{\vec {\nabla }}\times {\vec {J}}}$

From electrostatics, the solution is

${\displaystyle A_{\lambda }({\vec {x}})={\frac {1}{2k}}\int d^{3}{\vec {y}}\,{\frac {\varepsilon _{\lambda \mu \nu }\partial ^{\mu }J^{\nu }({\vec {y}})}{|{\vec {x}}-{\vec {y}}|}}}$

The path integral for arbitrary ${\displaystyle J}$ is now easily done by substituting this into the Chern–Simons action to get an effective action for the ${\displaystyle J}$ field. To get the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e. ${\displaystyle J=J_{1}+J_{2}}$, with

${\displaystyle J_{i}^{\mu }(x)=\int _{\gamma _{i}}dx_{i}^{\mu }\delta ^{3}(x-x_{i}(t))}$

Since the effective action is quadratic in ${\displaystyle J}$, 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

${\displaystyle Z[\gamma _{1},\gamma _{2}]=\exp {\left({\frac {2\pi i}{k}}\Phi [\gamma _{1},\gamma _{2}]\right)},}$

where

${\displaystyle \Phi [\gamma _{1},\gamma _{2}]={\frac {1}{4\pi }}\int _{\gamma _{1}}dx^{\lambda }\int _{\gamma _{2}}dy^{\mu }\,{\frac {(x-y)^{\nu }}{|x-y|^{3}}}\varepsilon _{\lambda \mu \nu },}$

which is simply Gauss’s linking integral. This is the simplest example of a topological quantum field theory, 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 that the nonabelian theory gives the invariant known as the Jones polynomial. ^[3]

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. ^[4]

• 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 ${\displaystyle m+n+1}$. 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.

• Differentiable curve – Study of curves from a differential point of view
• Hopf invariant – Homotopy invariant of maps between n-spheres
• Kissing number – Geometric concept
• Writhe – Invariant of a knot diagram

• A.V. Chernavskii (2001) [1994], “Linking coefficient”, Encyclopedia of Mathematics, EMS Press
• A.V. Chernavskii (2001) [1994], “Writhing number”, Encyclopedia of Mathematics, EMS Press

In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, (Milnor 1954). Notably, the link group is not in general the fundamental group of the link complement.

The link group of an n-component link is essentially the set of (n + 1)-component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions), knotting or unknotting itself, but is not allowed to move through other components. This is a weaker condition than isotopy: for example, the Whitehead link has linking number 0, and thus is link homotopic to the unlink, but it is not isotopic to the unlink.

The link group is not the fundamental group of the link complement, since the components of the link are allowed to move through themselves, though not each other, but thus is a quotient group of the link complement’s fundamental group, since one can start with elements of the fundamental group, and then by knotting or unknotting components, some of these elements may become equivalent to each other.

The link group of the n-component unlink is the free group on n generators, ${\displaystyle F_{n}}$, as the link group of a single link is the knot group of the unknot, which is the integers, and the link group of an unlinked union is the free product of the link groups of the components.

The link group of the Hopf link, the simplest non-trivial link – two circles, linked once – is the free abelian group on two generators, ${\displaystyle \mathbf {Z} ^{2}.}$ Note that the link group of two unlinked circles is the free nonabelian group on two generators, of which the free abelian group on two generators is a quotient. In this case the link group is the fundamental group of the link complement, as the link complement deformation retracts onto a torus.

The Whitehead link is link homotopic to the unlink – though it is not isotopic to the unlink – and thus has link group the free group on two generators.

Milnor defined invariants of a link (functions on the link group) in (Milnor 1954), using the character ${\displaystyle {\bar {\mu }},}$ which have thus come to be called “Milnor’s μ-bar invariants”, or simply the “Milnor invariants”. For each k, there is a k-ary function ${\displaystyle {\bar {\mu }},}$ which defines invariants according to which k of the links one selects, in which order.

Milnor’s invariants can be related to Massey products on the link complement (the complement of the link); this was suggested in (Stallings 1965), and made precise in (Turaev 1976) and (Porter 1980).

As with Massey products, the Milnor invariants of length k + 1 are defined if all Milnor invariants of length less than or equal to k vanish. The first (2-fold) Milnor invariant is simply the linking number (just as the 2-fold Massey product is the cup product, which is dual to intersection), while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times (that is to say, the Borromean rings have a Milnor 3-fold invariant of 1 or –1, depending on order, but other 3-element links can have an invariant of 2 or more, just as linking numbers can be greater than 1).

Another definition is the following: consider a link ${\displaystyle L=L_{1}\cup L_{2}\cup L_{3}}$. Suppose that ${\displaystyle {\rm {lk}}(L_{i},L_{j})=0}$ for ${\displaystyle i,j=1,2,3}$ and ${\displaystyle i<j}$. Pick any Seifert surfaces for the respective link components, say, ${\displaystyle F_{1},F_{2},F_{3}}$, such that ${\displaystyle F_{i}\cap L_{j}=\emptyset }$ for all ${\displaystyle i\neq j}$. Then the Milnor 3-fold invariant equals minus the number of intersection points in ${\displaystyle F_{1}\cap F_{2}\cap F_{3}}$ counting with signs; (Cochran 1990).

Milnor invariants can also be defined if the lower order invariants do not vanish, but then there is an indeterminacy, which depends on the values of the lower order invariants. This indeterminacy can be understood geometrically as the indeterminacy in expressing a link as a closed string link, as discussed below (it can also be seen algebraically as the indeterminacy of Massey products if lower order Massey products do not vanish).

Milnor invariants can be considered as invariants of string links, in which case they are universally defined, and the indeterminacy of the Milnor invariant of a link is precisely due to the multiple ways that a given links can be cut into a string link; this allows the classification of links up to link homotopy, as in (Habegger & Lin 1990). Viewed from this point of view, Milnor invariants are finite type invariants, and in fact they (and their products) are the only rational finite type concordance invariants of string links; (Habegger & Masbaum 2000).

The number of linearly independent Milnor invariants of length ${\displaystyle k+1}$ for m-component links is ${\displaystyle mN_{k}-N_{k+1}}$, where ${\displaystyle N_{k}}$ is the number of basic commutators of length k in the free Lie algebra on m generators, namely:

${\displaystyle N_{k}={\frac {1}{k}}\sum _{d|m}\phi (d)\left(m^{k/d}\right)}$,

where ${\displaystyle \phi }$ is the Möbius function; see for instance (Orr 1989). This number grows on the order of ${\displaystyle m^{k+1}/k^{2}}$.

Link groups can be used to classify Brunnian links.

• Knot group
• Regular homotopy

• Cochran, Tim D. (1990), “Derivatives of links: Milnor’s concordance invariants and Massey’s Products”, Memoirs of the American Mathematical Society, 84 (427), American Mathematical Society, doi:10.1090/memo/0427
• Habegger, Nathan; Lin, Xiao Song (1990), “The classification of links up to homotopy”, Journal of the American Mathematical Society, 2, 3 (2), American Mathematical Society: 389–419, doi:10.2307/1990959, JSTOR 1990959
• Habegger, Nathan; Masbaum, Gregor (2000), “The Kontsevich integral and Milnor’s invariants”, Topology, 39 (6): 1253–1289, doi:10.1016/S0040-9383(99)00041-5, MR 1783857
• Milnor, John (March 1954), “Link groups”, Annals of Mathematics, 59 (2), Annals of Mathematics: 177–195, doi:10.2307/1969685, JSTOR 1969685, MR 0071020
• Orr, Kent E. (1989), “Homotopy invariants of links”, Inventiones Mathematicae, 95 (2): 379–394, doi:10.1007/BF01393902, MR 0974908, S2CID 120916814
• Porter, Richard D. (1980), “Milnor’s μ-invariants and Massey products”, Transactions of the American Mathematical Society, 257 (1), American Mathematical Society: 39–71, doi:10.2307/1998124, JSTOR 1998124, MR 0549154
• Stallings, John R. (1965), “Homology and central series of groups”, Journal of Algebra, 2 (2): 170–181, doi:10.1016/0021-8693(65)90017-7, MR 0175956
• Turaev, Vladimir G. (1976), “The Milnor invariants and Massey products”, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Studies in Topology-II, 66: 189–203, MR 0451251

In mathematics, two links ${\displaystyle L_{0}\subset S^{n}}$ and ${\displaystyle L_{1}\subset S^{n}}$ are concordant if there exists an embedding ${\displaystyle f:L_{0}\times [0,1]\to S^{n}\times [0,1]}$ such that ${\displaystyle f(L_{0}\times \{0\})=L_{0}\times \{0\}}$ and ${\displaystyle f(L_{0}\times \{1\})=L_{1}\times \{1\}}$.

By its nature, link concordance is an equivalence relation. It is weaker than isotopy, and stronger than homotopy: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the unlink.

A function of a link that is invariant under concordance is called a concordance invariant.

The linking number of any two components of a link is one of the most elementary concordance invariants. The signature of a knot is also a concordance invariant. A subtler concordance invariant are the Milnor invariants, and in fact all rational finite type concordance invariants are Milnor invariants and their products,^[1] though non-finite type concordance invariants exist.

One can analogously define concordance for any two submanifolds ${\displaystyle M_{0},M_{1}\subset N}$. In this case one considers two submanifolds concordant if there is a cobordism between them in ${\displaystyle N\times [0,1],}$ i.e., if there is a manifold with boundary ${\displaystyle W\subset N\times [0,1]}$ whose boundary consists of ${\displaystyle M_{0}\times \{0\}}$ and ${\displaystyle M_{1}\times \{1\}.}$

This higher-dimensional concordance is a relative form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but “cobordant in N“.

• Slice knot

• J. Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific.
• Livingston, Charles, A survey of classical knot concordance, in: Handbook of knot theory, pp 319–347, Elsevier, Amsterdam, 2005. MR 2179265 ISBN 0-444-51452-X