Knot Equivalence
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In the mathematical field of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, knot theory is the study of
mathematical 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 ...
s. While inspired by
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
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 In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
in 3-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its
homeomorphism 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 isomorphi ...
s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an
ambient isotopy In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one ...
); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing 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
complexity Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generall ...
. In practice, knots are often distinguished using a ''
knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some i ...
'', a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include
knot polynomials In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. History The first knot polynomial, the Alexander polynomial, was introd ...
,
knot group In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3, :\pi_1(\mathbb^3 \setminus K). Other conventions co ...
s, and hyperbolic invariants. The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links 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 three-dimensional spaces and objects other than circles can be used; see ''
knot (mathematics) 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 ...
''. A higher-dimensional knot is an ''n''-dimensional sphere embedded in (''n''+2)-dimensional Euclidean space.


History

Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and 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 Chinese knotting, also known as () and decorative knots in non-Chinese cultures, is a decorative handcraft art that began as a form of Chinese folk art in the Tang dynasty, Tang and Song dynasty (960–1279 CE) in China. This form of craft or ...
). The
endless knot Endless knot in a Burmese Pali manuscript The endless knot or eternal knot is a symbolic knot and one of the Eight Auspicious Symbols. It is an important symbol in Hinduism, Jainism and Buddhism. It is an important cultural marker in place ...
appears in
Tibetan Buddhism Tibetan Buddhism (also referred to as Indo-Tibetan Buddhism, Lamaism, Lamaistic Buddhism, Himalayan Buddhism, and Northern Buddhism) is the form of Buddhism practiced in Tibet and Bhutan, where it is the dominant religion. It is also in majo ...
, while the
Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
have made repeated appearances in different cultures, often representing strength in unity. The
Celtic Celtic, Celtics or Keltic may refer to: Language and ethnicity *pertaining to Celts, a collection of Indo-European peoples in Europe and Anatolia **Celts (modern) *Celtic languages **Proto-Celtic language * Celtic music *Celtic nations Sports Fo ...
monks who created the
Book of Kells The Book of Kells ( la, Codex Cenannensis; ga, Leabhar Cheanannais; Dublin, Trinity College Library, MS A. I. 8 sometimes known as the Book of Columba) is an illuminated manuscript Gospel book in Latin, containing the four Gospels of the New ...
lavished entire pages with intricate
Celtic knot Celtic knots ( ga, snaidhm Cheilteach, cy, cwlwm Celtaidd, kw, kolm Keltek, gd, snaidhm Ceilteach) are a variety of knots and stylized graphical representations of knots used for decoration, used extensively in the Celtic style of Insular ...
work. A mathematical theory of knots was first developed in 1771 by
Alexandre-Théophile Vandermonde Alexandre-Théophile Vandermonde (28 February 1735 – 1 January 1796) was a French mathematician, musician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was b ...
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 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 ...
, who defined the
linking integral 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 ...
. In the 1860s,
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
's theory that atoms were knots in the aether led to
Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...
'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 The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conje ...
. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. These topologists in the early part of the 20th century—
Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
, J. W. Alexander, and others—studied knots from the point of view of the
knot group In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3, :\pi_1(\mathbb^3 \setminus K). Other conventions co ...
and invariants from
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
theory such as the
Alexander polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
. This would be the main approach to knot theory until a series of breakthroughs transformed the subject. In the late 1970s,
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...
introduced
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
into the study of knots with the
hyperbolization theorem In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture. Statement One form of Thurston's geometrization theore ...
. Many knots were shown to be
hyperbolic knot Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
s, enabling the use of geometry in defining new, powerful
knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some i ...
s. The discovery of the
Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...
by
Vaughan Jones Sir Vaughan Frederick Randal Jones (31 December 19526 September 2020) was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990. Early life Jones was born in Gisb ...
in 1984 , and subsequent contributions from
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, q ...
,
Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
, and others, revealed deep connections between knot theory and mathematical methods in
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and
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 and ...
. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as
quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...
s and
Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...
. In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers. Knot theory can be used to determine if a molecule 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 ...
(has a "handedness") or not .
Tangle Tangle may refer to: Science, Technology, Engineering & Mathematics *''The Tangle'' is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA *Tangle (mathematics), a topological object Natural sciences & medicine ...
s, strings with both ends fixed in place, have been effectively used in studying the action of
topoisomerase DNA topoisomerases (or topoisomerases) are enzymes that catalyze changes in the topological state of DNA, interconverting relaxed and supercoiled forms, linked (catenated) and unlinked species, and knotted and unknotted DNA. Topological issues i ...
on DNA . Knot theory may be crucial in the construction of quantum computers, through the model of
topological quantum computation A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to form ...
.


Knot equivalence

A knot is created by beginning with a one-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop . Simply, we can say a knot K is a "simple closed curve" or "(closed) Jordan curve" (see
Curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
) — that is: a "nearly"
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
and
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
K\colon ,1to \mathbb^3, with the only "non-injectivity" being K(0)=K(1). Topologists consider knots and other entanglements such as links and
braid A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...
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 K_1, K_2 are equivalent if there is an
orientation-preserving The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed ...
homeomorphism 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 isomorphi ...
h\colon\R^3\to\R^3 with h(K_1)=K_2. What this definition of knot equivalence means is that two knots are equivalent when there is a continuous family of homeomorphisms \ of space onto itself, such that the last one of them carries the first knot onto the second knot. (In detail: Two knots K_1 and K_2 are equivalent if there exists a continuous mapping H: \mathbb R^3 \times ,1\rightarrow \mathbb R^3 such that a) for each t \in ,1/math> the mapping taking x \in \mathbb R^3 to H(x,t) \in \mathbb R^3 is a homeomorphism of \mathbb R^3 onto itself; b) H(x, 0) = x for all x \in \mathbb R^3; and c) H(K_1,1) = K_2. Such a function H is known as an
ambient isotopy In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one ...
.) 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 \mathbb R^3 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 t=1 (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 In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s exist to solve this problem, with the first given by
Wolfgang Haken Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds. Biography Haken was born in Berlin, Germany. His father was Werner Haken, a physicist who had Max ...
in the late 1960s . Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is . The special case of recognizing the
unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
, called the
unknotting problem In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algorithms. A major unresolved challenge is to d ...
, is of particular interest . In February 2021
Marc Lackenby Marc Lackenby is a professor of mathematics at the University of Oxford whose research concerns knot theory, low-dimensional topology, and group theory. Lackenby studied mathematics at the University of Cambridge beginning in 1990, and earned his ...
announced a new unknot recognition algorithm that runs in
quasi-polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
.


Why not just isotopy?

It might seem tempting to define two knots (embeddings f, g : S1 → R3) as being equivalent precisely when there is any continuous family of embeddings such that f0 = f and f1 = g. But in this case, all knots would be equivalent. That is because a knot can be "pulled tight", that is, made to disappear by causing it to occupy less and less of the t parameter, approaching zero — as well as a smaller and smaller regions of 3-space, approaching a point — until the knot has become the standard unknot. (This occurs only for t = 1.) Since this works for any knot that can be localized, it follows that any two such knots are equivalent. "Knots that can be localized" consist of the knots that can be represented as a closed polygon embedded in 3-space, otherwise known as tame knots.


Knot diagrams

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 one-to-one except at the double points, called ''crossings'', where the "shadow" of the knot crosses itself once transversely . 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 In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, is an immersion if :D_pf : T_p M \to T_N\, is an injective function at ever ...
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 A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
and link diagrams when they represent a link.) Analogously, knotted surfaces in 4-space can be related to immersed surfaces 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. 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".


Reidemeister moves

In 1927, working with this diagrammatic form of knots, J. W. Alexander and
Garland Baird Briggs A garland is a decorative braid, knot or wreath of flowers, leaves, or other material. Garlands can be worn on the head or around the neck, hung on an inanimate object, or laid in a place of cultural or religious importance. Etymology From the ...
, and independently
Kurt Reidemeister Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttinge ...
, 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: 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 .


Knot invariants

A knot invariant is a "quantity" that is the same for equivalent knots . 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 In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two d ...
. "Classical" knot invariants include the
knot group In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3, :\pi_1(\mathbb^3 \setminus K). Other conventions co ...
, which is 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 the
knot complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
, and the
Alexander polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement . In the late 20th century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.


Knot polynomials

A knot polynomial is a
knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some i ...
that is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
. Well-known examples include the
Jones Jones may refer to: People *Jones (surname), a common Welsh and English surname *List of people with surname Jones * Jones (singer), a British singer-songwriter Arts and entertainment * Jones (''Animal Farm''), a human character in George Orwell ...
and
Alexander polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
s. A variant of the Alexander polynomial, the
Alexander–Conway polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ver ...
, is a polynomial in the variable ''z'' with
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 language ...
coefficients . The Alexander–Conway polynomial is actually defined in terms of 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 L_+, L_-, L_0 be the oriented link diagrams resulting from changing the diagram as indicated in the figure: The original diagram might be either L_+ or L_-, depending on the chosen crossing's configuration. Then the Alexander–Conway polynomial, C(z), is recursively defined according to the rules: * C(O) = 1 (where O is any diagram of the
unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
) * C(L_+) = C(L_-) + z C(L_0). The second rule is what is often referred to as a
skein relation Skein relations are a mathematical tool used to study knots. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One way to answer the question is using knot polynomials, which are invaria ...
. 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 In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest kno ...
. The yellow patches indicate where the relation is applied. :''C''() = ''C''() + ''z'' ''C''() gives the unknot and the
Hopf link In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. Geometric realization A concrete model consists of ...
. Applying the relation to the Hopf link where indicated, :''C''() = ''C''() + ''z'' ''C''() gives a link deformable to one with 0 crossings (it is actually the
unlink In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. Properties * An ''n''-component link ''L'' ⊂ S3 is an unlink if and only if ...
of two components) and an unknot. The unlink takes a bit of sneakiness: :''C''() = ''C''() + ''z'' ''C''() 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: :C(\mathrm) = 1 + z(0 + z) = 1 + z^2 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". Image:Trefoil knot left.svg, The left-handed trefoil knot. Image:TrefoilKnot_01.svg, The right-handed trefoil knot. Actually, there are two trefoil knots, called the right and left-handed trefoils, which are
mirror images 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 ...
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 Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
, before the invention of knot polynomials, using group theoretical methods . 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 .


Hyperbolic invariants

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...
proved many knots are
hyperbolic knot Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
s, meaning that the
knot complement In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a ...
(i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant . Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s of the geometry. An example is provided by the picture of the complement of the
Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of
horoball In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of ...
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 .


Higher dimensions

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. 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 A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically or smoothly slice knot, if it is the boundary of an embedded disk in ...
s and
ribbon knot In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ''ribbon singularities''. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part o ...
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
two-dimensional 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 ce ...
(\mathbb^2) embedded in 4-dimensional Euclidean space (\R^4). Such an embedding is knotted if there is no homeomorphism of \R^4 onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. Suspended knots and
spun knot ''Spun'' is a 2002 American black comedy crime drama film directed by Jonas Åkerlund from an original screenplay by William De Los Santos and Creighton Vero, based on three days of De Los Santos's life in the Eugene, Oregon, drug subculture. T ...
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 ''n''-spheres form knots only in (''n'' + 2)-dimensional space , although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted (4k-1)-spheres in 6''k''-dimensional space; e.g., there is a smoothly knotted 3-sphere in \R^6 . 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 \R^n with 2n-3k-3>0 is unknotted. The notion of a knot has further generalisations in mathematics, see:
Knot (mathematics) 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 ...
, isotopy classification of embeddings. Every knot in the ''n''-sphere \mathbb^n is the link of a real-algebraic set with isolated singularity in \R^ . An ''n''-knot is a single \mathbb^n embedded in \R^m. An ''n''-link consists of ''k''-copies of \mathbb^n embedded in \R^m, where ''k'' is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
. Both the m=n+2 and the m>n+2 cases are well studied, and so is the n>1 case.


Adding 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 : 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
. A 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
and composite numbers . 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.


Tabulating knots

Traditionally, knots have been catalogued in terms of 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) . The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult . Tabulation efforts have succeeded in enumerating over 6 billion knots and links . 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, , , ... . While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing . The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the
Dowker notation Dowker is a surname. Notable people with the surname include: * Clifford Hugh Dowker (1912–1982), Canadian mathematician * Fay Dowker (born 1965), British physicist *Felicity Dowker, Australian fantasy writer * Hasted Dowker (1900–1986), Canadi ...
. Different notations have been invented for knots which allow more efficient tabulation . The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings . 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 John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
, who not only developed a new notation but also the
Alexander–Conway polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ver ...
. 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 In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 10 ...
, which would only be noticed in 1974 by
Kenneth Perko In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 1 ...
. 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 Alain may refer to: People * Alain (given name), common given name, including list of persons and fictional characters with the name * Alain (surname) * "Alain", a pseudonym for cartoonist Daniel Brustlein * Alain, a standard author abbreviation u ...
. ee Perko (1982), Primality of certain knots, Topology ProceedingsLess famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. ee 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 . In 2003 Rankin, Flint, and Schermann, tabulated the
alternating knot In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing ...
s through 22 crossings . In 2020 Burton tabulated all
prime knot In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
s with up to 19 crossings .


Alexander–Briggs notation

This is the most traditional notation, due to the 1927 paper of James W. Alexander and Garland B. Briggs and later extended by
Dale Rolfsen In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes. Table of prime kn ...
in his knot table (see image above and
List of prime knots In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes. Table of prime kn ...
). 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 In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite f ...
comes after the
torus knot In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of cop ...
). 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 31 and the Hopf link is 2. Alexander–Briggs names in the range 10162 to 10166 are ambiguous, due to the discovery of the
Perko pair In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 10 ...
in
Charles Newton Little Charles Newton Little (1858–1923) was an American mathematician and civil engineer. He was known for his expertise in knot theory, including the construction of a table of knots with ten or fewer crossings... Little's father was a missionary ...
's original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.The Revenge of the Perko Pair
, ''RichardElwes.co.uk''. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.


Dowker–Thistlethwaite notation

The
Dowker–Thistlethwaite notation In the mathematical field of knot theory, the Dowker–Thistlethwaite (DT) notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, who refined a notation origi ...
, 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,−12) (5,2) (7,8) (9,−4) and (11,−10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6, −12, 2, 8, −4, −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

The Conway notation for knots and links, named after
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
, is based on the theory of tangles . 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 In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visua ...
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 − signs. An example is 1*2 −3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 −3 2 is a sequence describing the continued fraction associated to a
rational tangle In mathematics, a tangle is generally one of two related concepts: * In John Conway's definition, an ''n''-tangle is a proper embedding of the disjoint union of ''n'' arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2''n ...
. One inserts this tangle at the vertex of the basic polyhedron 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

Gauss code Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane. It is named for the mathematician Carl Friedrich Gaus ...
, 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 Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane. It is named for the mathematician Carl Friedrich Gau ...
.


Circuit topology

Circuit topology The circuit topology of a folded linear polymer refers to the arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. Proteins fold via formation of contacts of variou ...
is an alternative framework to knot theory and can describe and categorise knots and tangles. Knot theory considers any entangled chain as a connected sum of prime knots.
Prime knot In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
s cannot be divided and they are undecomposable. Circuit topology, in turn, splits any entangled chains (including prime knots) into basic structural units called soft contacts (s-contacts), and lists simple rules how s-contacts can be put together. These rules can be considered as binary operations defined on s-contacts. There are 3 main operations which put two s-contacts in series (S), in parallel (P), or in cross (X); and two supplementary operations, which make s-contacts concerted (C), or add subscripts (Sub). Circuit topology is a non-polynomial approach but comes with a string notation, which can be put in correspondence with Alexander polynomials in knot theory.


See also

*
Circuit topology The circuit topology of a folded linear polymer refers to the arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. Proteins fold via formation of contacts of variou ...
* Contact geometry#Legendrian submanifolds and knots *
Knots and graphs 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 ...
*
List of knot theory topics Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical lang ...
*
Molecular knot In chemistry, a molecular knot is a mechanically interlocked molecular architecture that is analogous to a macroscopic knot. Naturally-forming molecular knots are found in organic molecules like DNA, RNA, and proteins. It is not certain that nat ...
*
Quantum topology Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology. Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associat ...
*
Ribbon theory In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by (X,U) includes a curve X given by a three-dimensional vector X(s), depending continuou ...
*


References


Sources

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *


Footnotes


Further reading


Introductory textbooks

There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is . Other good texts from the references are and . 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. is suitable for undergraduates who know point-set topology; knowledge of algebraic topology is not required. * * * * *


Surveys

* **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. *


External links


"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

* *
Movie
of a modern recreation of Tait's smoke ring experiment
History of knot theory
(on the home page of
Andrew Ranicki Andrew Alexander Ranicki (born Andrzej Aleksander Ranicki; 30 December 1948 – 21 February 2018) was a British mathematician who worked on algebraic topology. He was a professor of mathematics at the University of Edinburgh. Life Ranicki was ...
)


Knot tables and software


KnotInfo: ''Table of Knot Invariants and Knot Theory Resources''The Knot Atlas
— detailed info on individual knots in knot tables
KnotPlot
— software to investigate geometric properties of knots

— software to create images of knots
Knoutilus
— online database and image generator of knots

Wolfram Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimizat ...
function for investigating knots
Regina
— software for low-dimensional topology with native support for knots and links

of prime knots with up to 19 crossings {{Knot theory, state=collapsed