In the mathematical field of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 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 cons ...
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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
,
(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 isom ...
s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
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 ...
); 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 interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generally used to ch ...
. 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 groups, 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)''. 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). The
endless knot 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 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 Foo ...
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 ...
lavished entire pages with intricate
Celtic knotwork.
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
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 E ...
. In the 1860s,
Lord Kelvin's
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
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 conjec ...
. 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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.
These topologists in the early part of the 20th century—
Max Dehn,
J. W. Alexander, and others—studied knots from the point of view of the
knot group and invariants from
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
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 the ...
. Many knots were shown to be
hyperbolic knots, enabling the use of geometry in defining new, powerful
knot invariants. The discovery of the
Jones polynomial by
Vaughan Jones in 1984 , 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
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 algebr ...
s and
Floer homology.
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 (has a "handedness") or not .
Tangles, strings with both ends fixed in place, have been effectively used in studying the action of
topoisomerase on DNA . Knot theory may be crucial in the construction of quantum computers, through the model of
topological quantum computation .
Knot equivalence
A knot is created by beginning with a one-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
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
is a "simple closed curve" or "(closed) Jordan curve" (see
Curve) — 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 contrapositi ...
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 val ...
, with the only "non-injectivity" being
. 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
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 isom ...
with
.
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
and
are equivalent if there exists a continuous mapping
such that a) for each