View source for Knot theory

{{short description|Study of mathematical knots}}
[[File:Tabela de nós matemáticos 01, crop.jpg|thumb|Examples of different knots including the [[trivial knot]] (top left) and the [[trefoil knot]] (below it)]]
[[File:TrefoilKnot 01.svg|thumb|A knot diagram of the trefoil knot, the simplest non-trivial knot]]

In [[topology]], '''knot theory''' is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by [[knot]]s 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 be undone, the simplest knot being a ring (or "[[unknot]]"). In mathematical language, a knot is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], <math>\mathbb{E}^3</math>. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of <math>\mathbb{R}^3</math> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

A complete algorithmic solution to this problem exists, which has unknown [[Computational complexity|complexity]].<ref>As first sketched using the theory of [[Haken manifold]]s by {{harvtxt|Haken|1962}}. For a more recent survey, see {{harvtxt|Hass|1998}}</ref> In practice, knots are often distinguished using a ''[[knot invariant]]'', a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include [[knot polynomials]], [[knot group]]s, and hyperbolic invariants.

The original motivation for the founders of knot theory was to create a table of knots and [[link (knot theory)|link]]s, which are knots of several components entangled with each other. More than six billion knots and links [[knot tabulation|have been tabulated]] since the beginnings of knot theory in the 19th century.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other [[3-manifold|three-dimensional spaces]] and objects other than circles can be used; see ''[[knot (mathematics)]]''. For example, a higher-dimensional knot is an [[n-sphere|''n''-dimensional sphere]] embedded in (''n''+2)-dimensional Euclidean space.

==History==
{{main|History of knot theory}}
[[File:KellsFol034rXRhoDet3.jpeg|thumb|upright=.85|Intricate Celtic knotwork in the 1200-year-old [[Book of Kells]]]]
Archaeologists have discovered that knot tying dates back to prehistoric times.  Besides their uses such as [[khipu|recording information]] and [[knot tying|tying]] objects together, knots have interested humans for their aesthetics and spiritual symbolism.  Knots appear in various forms of Chinese artwork dating from several centuries BC (see [[Chinese knotting]]). The [[endless knot]] appears in [[Tibetan Buddhism]], while the [[Borromean rings]] have made repeated appearances in different cultures, often representing strength in unity.  The [[Celtic Christianity|Celtic]] monks who created the [[Book of Kells]] lavished entire pages with intricate [[Celtic knot]]work.

[[File:Peter_Guthrie_Tait._Mezzotint_by_J._Faed_after_Sir_G._Reid._Wellcome_V0006622.jpg|thumb|left|upright|The first knot tabulator, [[Peter Guthrie Tait]]]]
A mathematical theory of knots was first developed in 1771 by [[Alexandre-Théophile Vandermonde]] who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with [[Carl Friedrich Gauss]], who defined the [[linking integral]] {{Harv|Silver|2006}}.  In the 1860s, [[William Thomson, 1st Baron Kelvin|Lord Kelvin]]'s [[Vortex theory of the atom|theory that atoms were knots in the aether]] led to [[Peter Guthrie Tait]]'s creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the [[Tait conjectures]].  This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of [[topology]].

These topologists in the early part of the 20th century—[[Max Dehn]], [[James Waddell Alexander II|J. W. Alexander]], and others—studied knots from the point of view of the [[knot group]] and invariants from [[Homology (mathematics)|homology]] theory such as the [[Alexander polynomial]].  This would be the main approach to knot theory until a series of breakthroughs transformed the subject.

In the late 1970s, [[William Thurston]] introduced [[hyperbolic geometry]] into the study of knots with the [[geometrization conjecture|hyperbolization theorem]]. Many knots were shown to be [[hyperbolic knot]]s, enabling the use of geometry in defining new, powerful [[knot invariant]]s. The discovery of the [[Jones polynomial]] by [[Vaughan Jones]] in 1984 {{Harv|Sossinsky|2002|pp=71–89}}, and subsequent contributions from [[Edward Witten]], [[Maxim Kontsevich]], and others, revealed deep connections between knot theory and mathematical methods in [[statistical mechanics]] and [[quantum field theory]]. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as [[quantum group]]s and [[Floer homology]].

In the last several decades of the 20th century, scientists became interested in studying [[physical knot theory|physical knots]] in order to understand knotting phenomena in [[DNA]] and other polymers. Knot theory can be used to determine if a molecule is [[chirality (chemistry)|chiral]] (has a "handedness") or not {{Harv|Simon|1986}}. [[tangle (mathematics)|Tangle]]s, strings with both ends fixed in place, have been effectively used in studying the action of [[topoisomerase]] on DNA {{Harv|Flapan|2000}}.  Knot theory may be crucial in the construction of quantum computers, through the model of [[topological quantum computation]] {{Harv|Collins|2006}}.

==Knot equivalence<!--[[Knot equivalence]] redirects directly here.-->==
{{multiple image
 | align=right
 | total_width=200
 | image1=unknots.svg
 | width1=289 | height1=500
 | image2=Knot Unfolding.gif
 | width2=240 | height2=240
 | footer=On the left, the unknot, and a knot equivalent to it. It can be more difficult to determine whether complex knots, such as the one on the right, are equivalent to the unknot.
}}
A knot is created by beginning with a one-[[dimension]]al line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop {{Harv|Adams|2004}} {{Harv|Sossinsky|2002}}. Simply, we can say a knot <math>K</math> is a "simple closed curve" (see [[Curve]]) — that is: a "nearly" [[injective]] and [[continuous function]] <math>K\colon[0,1]\to \mathbb{R}^3</math>, with the only "non-injectivity" being <math>K(0)=K(1)</math>. Topologists consider knots and other entanglements such as [[link (knot theory)|links]] and [[Braid theory|braid]]s to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.

The idea of '''knot equivalence''' is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. A formal mathematical definition is that two knots <math>K_1, K_2</math> are equivalent if there is an [[orientation-preserving]] [[homeomorphism]] <math>h\colon\R^3\to\R^3</math> with <math>h(K_1)=K_2</math>.

What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms <math>\{ h_t: \mathbb R^3 \rightarrow \mathbb R^3\ \mathrm{for}\ 0 \leq t \leq 1\}</math> of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail:  Two knots <math>K_1</math> and <math>K_2</math> are '''equivalent''' if there exists a continuous mapping <math>H: \mathbb R^3 \times [0,1] \rightarrow \mathbb R^3</math> such that a) for each <math>t \in [0,1]</math> the mapping taking <math>x \in \mathbb R^3</math> to <math>H(x,t) \in \mathbb R^3</math> is a homeomorphism of <math>\mathbb R^3</math> onto itself; b) <math>H(x, 0) = x</math> for all <math>x \in \mathbb R^3</math>; and c) <math>H(K_1,1) = K_2</math>. Such a function <math>H</math> is known as an [[ambient isotopy]].)

These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of <math>\mathbb R^3</math> to itself is the final stage of an ambient isotopy starting from the identity. Conversely, two knots equivalent under the ambient isotopy definition are also equivalent under the orientation-preserving homeomorphism definition, because the <math>t=1</math> (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other.

The basic problem of knot theory, the '''recognition problem''', is determining the equivalence of two knots.  [[Algorithm]]s exist to solve this problem, with the first given by [[Wolfgang Haken]] in the late 1960s {{Harv|Hass|1998}}. Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is {{Harv|Hass|1998}}. The special case of recognizing the [[unknot]], called the [[unknotting problem]], is of particular interest {{Harv|Hoste|2005}}. In February 2021 [[Marc Lackenby]] announced a new unknot recognition algorithm that runs in [[Time complexity|quasi-polynomial time]].<ref>{{citation|url=https://www.maths.ox.ac.uk/node/38304|title=Marc Lackenby announces a new unknot recognition algorithm that runs in quasi-polynomial time|date=2021-02-03|publisher=Mathematical Institute, [[University of Oxford]]|accessdate=2021-02-03}}</ref>

==Knot diagrams<!--[[Knot diagram]] links directly here, [[MOS:HEAD]]-->==
[[File:Tenfold Knottiness, plate IX.png|thumb|Tenfold Knottiness, plate IX, from [[Peter Guthrie Tait]]'s article "On Knots", 1884]]
A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall. A small change in the direction of projection will ensure that it is [[Injective function|one-to-one]] except at the double points, called ''crossings'', where the "shadow" of the knot crosses itself once transversely {{Harv|Rolfsen|1976}}. At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an [[immersed plane curve]] with the additional data of which strand is over and which is under at each crossing. (These diagrams are called '''knot diagrams''' when they represent a [[Knot (mathematics)|knot]] and '''link diagrams''' when they represent a [[Link (knot theory)|link]].) Analogously, knotted surfaces in 4-space can be related to [[immersed surface]]s in 3-space.

A '''reduced diagram''' is a knot diagram in which there are no '''reducible crossings''' (also '''nugatory''' or '''removable crossings'''), or in which all of the reducible crossings have been removed.{{sfn|Weisstein|2013}}{{sfn|Weisstein|2013a}} A [[petal projection]] is a type of projection in which, instead of forming double points, all strands of the knot meet at a single crossing point, connected to it by loops forming non-nested "petals".{{sfn|Adams|Crawford|DeMeo|Landry|2015}}

===Reidemeister moves===
{{main|Reidemeister move}}
In 1927, working with this diagrammatic form of knots, [[James Waddell Alexander II|J. W. Alexander]] and [[Garland Baird Briggs]], and independently [[Kurt Reidemeister]], demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the ''Reidemeister moves'', are:

{{Ordered list|list-style-type=upper-Roman
|Twist and untwist in either direction.
|Move one strand completely over another.
|Move a strand completely over or under a crossing.
}}

{| align="center" style="text-align:center"
|+ '''Reidemeister moves'''
|- style="padding:1em"
| [[File:Reidemeister move 1.png|130px]] [[File:Frame left.png]] || [[File:Reidemeister move 2.png|210px]]
|-
! Type I !! Type II
|- style="padding:1em"
| colspan="2" | [[File:Reidemeister move 3.png|360px]]
|-
! colspan="2" | Type III
|}

The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another.  The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point.  A close inspection will show that complicated events can be eliminated, leaving only the simplest events: (1) a "kink" forming or being straightened out;  (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point.  These are precisely the Reidemeister moves {{Harv|Sossinsky|2002| loc=ch. 3}} {{Harv|Lickorish|1997| loc=ch. 1}}.

==Knot invariants==
[[File:Figure eight knot complement.jpg|thumb|right| text-bottom |280px|  A 3D print depicting the complement of the figure eight knot<br>by François Guéritaud, Saul Schleimer, and [[Henry Segerman]]]]
{{main|Knot invariant}}
A knot invariant is a "quantity" that is the same for equivalent knots {{Harv|Adams|2004}} {{Harv|Lickorish|1997}} {{Harv|Rolfsen|1976}}.  For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots.  An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is [[tricolorability]].

"Classical" knot invariants include the [[knot group]], which is the [[fundamental group]] of the [[knot complement]], and the [[Alexander polynomial]], which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement {{Harv|Lickorish|1997}}{{Harv|Rolfsen|1976}}. In the late 20th century, invariants such as "quantum" knot polynomials, [[Vassiliev invariant]]s and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.

===Knot polynomials===
{{main|Knot polynomial}}
A knot polynomial is a [[knot invariant]] that is a [[polynomial]]. Well-known examples include the [[Jones polynomial]], the [[Alexander polynomial]], and the [[Kauffman polynomial]]. A variant of the Alexander polynomial, the [[Alexander–Conway polynomial]], is a polynomial in the variable ''z'' with [[integer]] coefficients {{Harv|Lickorish|1997}}.

The Alexander–Conway polynomial is actually defined in terms of [[link (knot theory)|links]], which consist of one or more knots entangled with each other.  The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.

Consider an oriented link diagram, ''i.e.'' one in which every component of the link has a preferred direction indicated by an arrow. For a given crossing of the diagram, let <math>L_+, L_-, L_0</math> be the oriented link diagrams resulting from changing the diagram as indicated in the figure:  [[File:Skein (HOMFLY).svg|200px|center]]

The original diagram might be either <math>L_+</math> or <math>L_-</math>, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial, <math>C(z)</math>, is recursively defined according to the rules:

* <math>C(O) = 1</math> (where <math>O</math> is any diagram of the [[unknot]])
* <math>C(L_+) = C(L_-) + z C(L_0).</math>

The second rule is what is often referred to as a [[skein relation]]. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way.

The following is an example of a typical computation using a skein relation. It computes the Alexander–Conway polynomial of the [[trefoil knot]]. The yellow patches indicate where the relation is applied.

:''C''([[File:skein-relation-trefoil-plus-sm.png]])&nbsp;=&nbsp;''C''([[File:skein-relation-trefoil-minus-sm.png]])&nbsp;+&nbsp;''z'' ''C''([[File:skein-relation-trefoil-zero-sm.png]])

gives the unknot and the [[Hopf link]]. Applying the relation to the Hopf link where indicated,

:''C''([[File:skein-relation-link22-plus-sm.png]]) = ''C''([[File:skein-relation-link22-minus-sm.png]]) + ''z'' ''C''([[File:skein-relation-link22-zero-sm.png]])

gives a link deformable to one with 0 crossings (it is actually the [[unlink]] of two components) and an unknot. The unlink takes a bit of sneakiness:

:''C''([[File:skein-relation-link20-plus-sm.png]]) = ''C''([[File:skein-relation-link20-minus-sm.png]]) + ''z''  ''C''([[File:skein-relation-link20-zero-sm.png]])

which implies that ''C''(unlink of two components) = 0, since the first two polynomials are of the unknot and thus equal.

Putting all this together will show:

:<math>C(\mathrm{trefoil}) = 1 + z(0 + z) = 1 + z^2</math>

Since the Alexander–Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot. So the trefoil really is "knotted".

<gallery widths="80px" heights="80px" align="right">
Image:Trefoil knot left.svg|The left-handed trefoil knot.
Image:TrefoilKnot_01.svg|The right-handed trefoil knot.
</gallery>
Actually, there are two trefoil knots, called the right and left-handed trefoils, which are [[chiral knot|mirror images]] of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other, meaning that they are not amphichiral.  This was shown by [[Max Dehn]], before the invention of knot polynomials, using group theoretical methods {{Harv|Dehn|1914}}. But the Alexander–Conway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The ''Jones'' polynomial can in fact distinguish between the left- and right-handed trefoil knots {{Harv|Lickorish|1997}}.

===Hyperbolic invariants===
[[William Thurston]] proved many knots are [[hyperbolic knot]]s, meaning that the [[knot complement]] (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of [[hyperbolic geometry]]. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant {{Harv|Adams|2004}}.

{{multiple image
 | align = right
 | total_width = 320
 | image1 = BorromeanRings.svg
 | width1 = 626 | height1 = 600
 | caption1 = The [[Borromean rings]] are a link with the property that removing one ring unlinks the others.
 | image2 = SnapPea-horocusp_view.png
 | width2 = 560 | height2 = 416
 | caption2 = [[SnapPea]]'s cusp view: the [[Borromean rings]] complement from the perspective of an inhabitant living near the red component.
}}

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the [[geodesic]]s of the geometry. An example is provided by the picture of the complement of the [[Borromean rings]]. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of [[horoball]] neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color)  as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task {{Harv|Adams|Hildebrand|Weeks|1991}}.

==Higher dimensions==
<!--{{Main|Higher-dimensional knots}}-->
A knot in three dimensions can be untied when placed in four-dimensional space. This is done by changing crossings. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.<!-- can't really find a source for this...Adams misleadingly doesn't mention the first step. This step more or less follows from very basic, well-known, general position arguments, e.g. the PL-case is essentially contained in Zeeman's paper although not explicitly described. Comment by [[User: C S]] -->

In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.

Four-dimensional space occurs in classical knot theory, however, and an important topic is the study of [[slice knot]]s and [[ribbon knot]]s. A notorious open problem asks whether every slice knot is also ribbon.

===Knotting spheres of higher dimension===
Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a [[2-sphere|two-dimensional sphere]] (<math>\mathbb{S}^2</math>) embedded in 4-dimensional Euclidean space (<math>\R^4</math>). Such an embedding is knotted if there is no homeomorphism of <math>\R^4</math> onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere.  [[Suspended knot]]s and [[spun knot]]s are two typical families of such 2-sphere knots.

The mathematical technique called "general position" implies that for a given ''n''-sphere in ''m''-dimensional Euclidean space, if ''m'' is large enough (depending on ''n''), the sphere should be unknotted. In general, [[piecewise linear manifold|piecewise-linear]] [[n-sphere|''n''-sphere]]s form knots only in (''n''&nbsp;+&nbsp;2)-dimensional space {{Harv|Zeeman|1963}}, although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted <math>(4k-1)</math>-spheres in 6''k''-dimensional space; e.g.,  there is a smoothly knotted 3-sphere in <math>\R^6</math> {{Harv|Haefliger|1962}} {{Harv|Levine|1965}}. Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth ''k''-sphere embedded in <math>\R^n</math> with <math>2n-3k-3>0</math> is unknotted. The notion of a knot has further generalisations in mathematics, see: [[Knot (mathematics)]], [[Whitney embedding theorem#Isotopy versions|isotopy classification of embeddings]].

Every knot in the [[n-sphere|''n''-sphere]] <math>\mathbb{S}^n</math> is the link of a [[real algebraic set|real-algebraic set]] with isolated singularity in <math>\R^{n+1}</math> {{Harv|Akbulut|King|1981}}.

An ''n''-knot is a single <math>\mathbb{S}^n</math> embedded in  <math>\R^m</math>. An ''n''-link consists of ''k''-copies of <math>\mathbb{S}^n</math> embedded in <math>\R^m</math>, where ''k'' is a [[natural number]]. Both the <math>m=n+2</math> and the <math>m>n+2</math> cases are well studied, and so is the <math>n>1</math> case.<ref>{{citation |first1=J. |last1=Levine |first2=K |last2=Orr |chapter=A survey of applications of surgery to knot and link theory |citeseerx = 10.1.1.64.4359 |title=Surveys on Surgery Theory: Papers Dedicated to C.T.C. Wall |volume=1 |series=Annals of mathematics studies |publisher=[[Princeton University Press]] |year=2000 |isbn=978-0691049380 }} — An introductory article to high dimensional knots and links for the advanced readers</ref><ref>{{citation |first=Eiji |last=Ogasa |arxiv=1304.6053 |title=Introduction to high dimensional knots |bibcode=2013arXiv1304.6053O |year=2013 }} — An introductory article to high dimensional knots and links for beginners</ref>

==Adding knots==
{{main|Knot sum}}
[[File:Sum of knots3.svg|thumb|Adding two knots]]
Two knots can be added by cutting both knots and joining the pairs of ends.  The operation is called the ''knot sum'', or sometimes the ''connected sum'' or ''composition'' of two knots. This can be formally defined as follows {{Harv|Adams|2004}}: consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is a sum of the original knots.  Depending on how this is done, two different knots (but no more) may result.  This ambiguity in the sum can be eliminated regarding the knots as ''oriented'', i.e. having a preferred direction of travel along the knot, and requiring the arcs of the knots in the sum are oriented consistently with the oriented boundary of the rectangle.

The knot sum of oriented knots is [[commutative]] and [[associative]]. A [[prime knot|knot is ''prime'']] if it is non-trivial and cannot be written as the knot sum of two non-trivial knots. A knot that can be written as such a sum is ''composite''. There is a prime decomposition for knots, analogous to [[prime number|prime]] and composite numbers {{Harv|Schubert|1949}}.  For oriented knots, this decomposition is also unique. Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers ''smooth'' knots in codimension at least 3.

Knots can also be constructed using the [[circuit topology]] approach. This is done by combining basic units called soft contacts using five operations (Parallel, Series, Cross, Concerted, and Sub).<ref>{{cite journal |last1=Golovnev |first1=Anatoly |last2=Mashaghi |first2=Alireza |title=Circuit Topology for Bottom-Up Engineering of Molecular Knots |journal=Symmetry |date=7 December 2021 |volume=13 |issue=12 |pages=2353 |doi=10.3390/sym13122353 |arxiv=2106.03925 |bibcode=2021Symm...13.2353G |doi-access=free }}</ref><ref>{{cite journal |last1=Flapan |first1=Erica |last2=Mashaghi |first2=Alireza |last3=Wong |first3=Helen |title=A tile model of circuit topology for self-entangled biopolymers |journal=Scientific Reports |date=1 June 2023 |volume=13 |issue=1 |pages=8889 |doi=10.1038/s41598-023-35771-8 |pmid=37264056 |pmc=10235088 |bibcode=2023NatSR..13.8889F }}</ref> The approach is applicable to open chains as well and can also be extended to include the so-called hard contacts.

==Tabulating knots==
<!-- {{Main|knot tabulation}} more can be said about tabulation.... -->
{{see also|List of prime knots|Knot tabulation}}
[[File:Knot table.svg|thumb|upright=1.5|A table of prime knots up to seven crossings. The knots are labeled with Alexander–Briggs notation]]
Traditionally, knots have been catalogued in terms of [[crossing number (knot theory)|crossing number]].  Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) {{Harv|Hoste|Thistlethwaite|Weeks|1998}}. The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult {{Harv|Hoste|2005|p=20}}. Tabulation efforts have succeeded in enumerating over 6 billion knots and links {{Harv|Hoste|2005|p=28}}. The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, {{val|46972}}, {{val|253293}}, {{val|1388705}}... {{OEIS|id=A002863}}. While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing {{Harv|Adams|2004}}.

The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the [[Dowker notation]]. Different notations have been invented for knots which allow more efficient tabulation {{Harv|Hoste|2005}}.

The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings {{Harv|Hoste|Thistlethwaite|Weeks|1998}}. The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander–Briggs and Reidemeister in the late 1920s.

The first major verification of this work was done in the 1960s by [[John Horton Conway]], who not only developed a new notation but also the [[Alexander–Conway polynomial]] {{Harv|Conway|1970}} {{Harv|Doll|Hoste|1991}}. This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only one duplication in the Tait–Little tables; however he missed the duplicates called the [[Perko pair]], which would only be noticed in 1974 by [[Kenneth Perko]] {{Harv|Perko|1974}}. This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work. Conway's 1970 paper on knot theory also contains a typographical duplication on its non-alternating 11-crossing knots page and omits 4 examples — 2 previously listed in D. Lombardero's 1968 Princeton senior thesis and 2 more subsequently discovered by [[Alain Caudron]]. [see Perko (1982),  Primality of certain knots, Topology Proceedings] Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. [See Perko (2016), Historical highlights of non-cyclic knot theory, J. Knot Theory Ramifications].

In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings {{Harv|Hoste|Thistlethwaite|Weeks|1998}}.  In 2003 Rankin, Flint, and Schermann, tabulated the [[alternating knot]]s through 22 crossings {{Harv|Hoste|2005}}. In 2020 Burton tabulated all [[prime knot]]s with up to 19 crossings {{Harv|Burton|2020}}.

===Alexander–Briggs notation===<!--this section is linked to from locations including [[Template:Infobox knot theory]]-->
This is the most traditional notation, due to the 1927 paper of [[James Waddell Alexander II|James W. Alexander]] and [[Garland Baird Briggs|Garland B. Briggs]] and later extended by [[Dale Rolfsen]] in his knot table (see image above and [[List of prime knots]]). The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance (though in each number of crossings the [[twist knot]] comes after the [[torus knot]]). Links are written by the crossing number with a superscript to denote the number of components and a subscript to denote its order within the links with the same number of components and crossings. Thus the trefoil knot is notated 3<sub>1</sub> and the Hopf link is 2{{sup sub|2|1}}.  Alexander–Briggs names in the range 10<sub>162</sub> to 10<sub>166</sub> are ambiguous, due to the discovery of the [[Perko pair]] in [[Charles Newton Little]]'s original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.<ref>"[http://richardelwes.co.uk/2013/08/14/the-revenge-of-the-perko-pair/ The Revenge of the Perko Pair]", ''RichardElwes.co.uk''. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.</ref>

===Dowker–Thistlethwaite notation===
{{main|Dowker–Thistlethwaite notation}}
[[File:Dowker-notation-example.svg|thumb|upright=.85|A knot diagram with crossings labelled for a Dowker sequence]]
The [[Dowker–Thistlethwaite notation]], also called the Dowker notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in this figure the knot diagram has crossings labelled with the pairs (1,6) (3,&minus;12) (5,2) (7,8) (9,&minus;4) and (11,&minus;10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6, &minus;12, 2, 8, &minus;4, &minus;10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker–Thistlethwaite notation.

===Conway notation===
{{main|Conway notation (knot theory)}}

The [[Conway notation (knot theory)|Conway notation]] for knots and links, named after [[John Horton Conway]], is based on the theory of [[tangle (mathematics)|tangles]] {{Harv|Conway|1970}}. The advantage of this notation is that it reflects some properties of the knot or link.

The notation describes how to construct a particular link diagram of the link. Start with a ''basic polyhedron'', a 4-valent connected planar graph with no [[digon]] regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedra. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.

Each vertex then has an [[algebraic tangle]] substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or &minus; signs.

An example is 1*2 &minus;3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 &minus;3 2 is a sequence describing the continued fraction associated to a [[rational tangle]]. One inserts this tangle at the vertex of the basic polyhedron&nbsp;1*.

A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where the ones are omitted and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.

Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.

===Gauss code===
{{main|Gauss code}}
[[Gauss code]], similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number. For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3

Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the [[extended Gauss code]].

==See also==
*[[Arithmetic rope]]
*[[Circuit topology]]
*[[Lamp cord trick]]
*[[Contact geometry#Legendrian submanifolds and knots|Legendrian submanifolds and knots]]
*[[List of knot theory topics]]
*[[Molecular knot]]
*{{section link|Necktie|Knots}}
*[[Quantum topology]]
*[[Ribbon theory]]

==References==
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=== Footnotes ===
{{reflist}}

==Further reading==

===Introductory textbooks===
There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is {{Harv|Rolfsen|1976}}. Other good texts from the references are {{Harv|Adams|2004}} and {{Harv|Lickorish|1997}}. Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics. {{Harv|Cromwell|2004}} is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required.

* {{citation |first1=Gerhard |last1=Burde |author-link=Gerhard Burde |author2-link=Heiner Zieschang |first2=Heiner |last2=Zieschang |title=Knots |publisher=Walter de Gruyter |series=De Gruyter Studies in Mathematics |volume=5 |year=1985 |isbn=978-3-11-008675-1 |url=https://books.google.com/books?id=DJHI7DpgIbIC}}
* {{cite book |first1=Richard H. |last1=Crowell |author-link=Richard H. Crowell |first2=Ralph |last2=Fox |author-link2=Ralph Fox| title=Introduction to Knot Theory |year=1977 |publisher=Springer |isbn=978-0-387-90272-2 }}
* {{citation |first=Louis H. |last=Kauffman |author-link=Louis H. Kauffman |title=On Knots |year=1987 |publisher=Princeton University Press |isbn=978-0-691-08435-0 |url=https://books.google.com/books?id=BLvGkIY8YzwC }}
* {{citation |first=Louis H. |last=Kauffman |author-link=Louis H. Kauffman |title=Knots and Physics |publisher=World Scientific |year=2013 |isbn=978-981-4383-00-4 |edition=4th |url=https://books.google.com/books?id=fSKrRQ77FMkC }}
* {{citation |first=Peter R. |last=Cromwell |title=Knots and Links |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-54831-1 |url=https://books.google.com/books?id=djvbTNR2dCwC }}
<!-- To keep this section under control, please add only well-known texts that do not substantially overlap an already listed text -->

===Surveys===
*{{citation |editor-first=William W. |editor-last=Menasco |editor2-first=Morwen |editor2-last=Thistlethwaite |editor2-link=Morwen Thistlethwaite |title=Handbook of Knot Theory |publisher=Elsevier |year=2005 |isbn=978-0-444-51452-3 }}
**Menasco and Thistlethwaite's handbook surveys a mix of topics relevant to current research trends in a manner accessible to advanced undergraduates but of interest to professional researchers.
*{{citation |last=Livio |first=Mario |chapter=Ch. 8: Unreasonable Effectiveness? |chapter-url=https://books.google.com/books?id=ebd6QofqY6QC&pg=PA203 |title=Is God a Mathematician? |publisher=Simon & Schuster |year=2009 |isbn=978-0-7432-9405-8 |pages=203–218 }}

==External links==
{{commons category|Knot theory}}
{{Wiktionary}}

*[http://www.groupoids.org.uk/popmath/cpm/exhib/knotexhib.html "Mathematics and Knots"] This is an online version of an exhibition developed for the 1989 Royal Society "PopMath RoadShow". Its aim was to use knots to present methods of mathematics to the general public.

===History===
*{{citation |last=Thomson |first=Sir William  |author-link=Lord Kelvin |title=On Vortex Atoms |journal=Proceedings of the Royal Society of Edinburgh |volume=VI |pages=94–105 |year=1867 |url=http://zapatopi.net/kelvin/papers/on_vortex_atoms.html}}
*{{citation |last=Silliman |first=Robert H. |title=William Thomson: Smoke Rings and Nineteenth-Century Atomism |journal=Isis |volume=54 |issue=4 |pages=461–474 |date=December 1963 |jstor=228151 |doi=10.1086/349764|s2cid=144988108 }}
*[https://jagworks.southalabama.edu/knot-theory-hist/4/ Movie] of a modern recreation of Tait's smoke ring experiment
*[http://www.maths.ed.ac.uk/~aar/knots History of knot theory] (on the home page of [[Andrew Ranicki]])

===Knot tables and software===
* [https://knotinfo.math.indiana.edu/ '''KnotInfo''': ''Table of Knot Invariants and Knot Theory Resources'']
*[http://katlas.math.toronto.edu/wiki/Main_Page The Knot Atlas] — detailed info on individual knots in knot tables
* [http://knotplot.com/ KnotPlot] — software to investigate geometric properties of knots
*[http://www.math.utk.edu/~morwen/knotscape.html Knotscape] — software to create images of knots
*[http://knotilus.math.uwo.ca/ Knoutilus] — online database and image generator of knots
*[https://reference.wolfram.com/language/ref/KnotData.html KnotData.html] — [[Wolfram Mathematica]] function for investigating knots
*[https://regina-normal.github.io/index.html/ Regina] — software for low-dimensional topology with native support for knots and links. [https://regina-normal.github.io/data.html#knots Tables] of prime knots with up to 19 crossings

{{Knot theory|state=collapsed}}

[[Category:Knot theory| ]]
[[Category:Low-dimensional topology]]