In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 the
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 ...
into three-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 ...
, (also known as ). Often two knots are considered equivalent if they are
ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other.
A crucial difference between the standard mathematical and conventional notions of 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 ' ...
is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
and has many relations to
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
.
Formal definition
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 the
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 ...
() into
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
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 ...
(), or the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
(), since the 3-sphere is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
. Two knots are defined to be equivalent if there is 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 ...
between them.
Projection
A knot in (or alternatively in the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
, ), can be projected onto a plane (respectively a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
). This projection is almost always regular, meaning that it is
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 ...
everywhere, except at a ''finite number'' of crossing points, which are the projections of ''only two points'' of the knot, and these points are not
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
. In this case, by choosing a projection side, one can completely encode the
isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a regular projection of a knot, or
knot diagram
In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
is thus a quadrivalent
planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...
with over/under-decorated vertices. The local modifications of this graph which allow to go from one diagram to any other diagram of the same knot (up to ambient
isotopy of the plane) are called
Reidemeister moves
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), German Empire, Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg ...
.
Image:Reidemeister_move_1.png, Reidemeister move 1
Image:Reidemeister_move_2.png, Reidemeister move 2
Image:Reidemeister_move_3.png, Reidemeister move 3
Types of knots
The simplest knot, called 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 ...
or trivial knot, is a round circle embedded in . In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are 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 ...
( in the table), the
figure-eight knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...
() and the
cinquefoil knot
In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, a ...
().
Several knots, linked or tangled together, are called
links. Knots are links with a single component.
Tame vs. wild knots
A ''polygonal'' knot is a knot whose
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
in is the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of a
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
of
line segments
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
.
A ''tame'' knot is any knot equivalent to a polygonal knot.
Knots which are not tame are called ''
wild
Wild, wild, wilds or wild may refer to:
Common meanings
* Wild animal
* Wilderness, a wild natural environment
* Wildness, the quality of being wild or untamed
Art, media and entertainment Film and television
* ''Wild'' (2014 film), a 2014 A ...
'',
and can have
pathological
Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
behavior.
In knot theory and
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...
theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.
Framed knot
A ''framed knot'' is the extension of a tame knot to an embedding of the
solid torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology.
A standard way to visuali ...
in .
The ''framing'' of the knot is the
linking number
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 Eu ...
of the image of the ribbon with the knot. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. This definition generalizes to an analogous one for ''framed links''. Framed links are said to be ''equivalent'' if their extensions to solid tori are ambient isotopic.
Framed link ''diagrams'' are link diagrams with each component marked, to indicate framing, by an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
representing a slope with respect to the meridian and preferred longitude. A standard way to view a link diagram without markings as representing a framed link is to use the ''blackboard framing''. This framing is obtained by converting each component to a ribbon lying flat on the plane. A type I
Reidemeister move
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 Götting ...
clearly changes the blackboard framing (it changes the number of twists in a ribbon), but the other two moves do not. Replacing the type I move by a modified type I move gives a result for link diagrams with blackboard framing similar to the Reidemeister theorem: Link diagrams, with blackboard framing, represent equivalent framed links if and only if they are connected by a sequence of (modified) type I, II, and III moves.
Given a knot, one can define infinitely many framings on it. Suppose that
we are given a knot with a fixed framing. One may obtain a new framing from the existing one by cutting
a ribbon and twisting it an integer multiple of 2π around the knot and then glue back again in the place
we did the cut. In this way one obtains a new framing from an old one, up to the equivalence relation
for framed knots„ leaving the knot fixed. The framing in this sense is associated to the number of twists
the vector field performs around the knot. Knowing how many times the vector field is twisted around
the knot allows one to determine the vector field up to diffeomorphism, and the equivalence class of the
framing is determined completely by this integer called the framing integer.
Knot complement
Given a knot in the 3-sphere, 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 ...
is all the points of the 3-sphere not contained in the knot. A major
theorem of Gordon and Luecke states that at most two knots have homeomorphic complements (the original knot and its mirror reflection). This in effect turns the study of knots into the study of their complements, and in turn into
3-manifold theory.
JSJ decomposition
The
JSJ decomposition In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
:Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy ...
and
Thurston's hyperbolization theorem reduces the study of knots in the 3-sphere to the study of various geometric manifolds via ''splicing'' or ''
satellite operations''. In the pictured knot, the JSJ-decomposition splits the complement into the union of three manifolds: two
trefoil complements and 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 trefoil complement has the geometry of , while the Borromean rings complement has the geometry of .
Harmonic knots
Parametric representations of knots are called harmonic knots. Aaron Trautwein compiled parametric representations for all knots up to and including those with a crossing number of 8 in his PhD thesis.
Applications to graph theory
Medial graph
Another convenient representation of knot diagrams was introduced by
Peter Tait Peter Tait may refer to:
* Peter Tait (physicist) (1831–1901), Scottish mathematical physicist
* Peter Tait (footballer) (1936–1990), English professional footballer
* Peter Tait (mayor) (1915–1996), New Zealand politician
* Peter Tait (radio ...
in 1877.
Any knot diagram defines a
plane graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...
whose vertices are the crossings and whose edges are paths in between successive crossings. Exactly one face of this planar graph is unbounded; each of the others is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to a 2-dimensional
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
. Color these faces black or white so that the unbounded face is black and any two faces that share a boundary edge have opposite colors. The
Jordan curve theorem
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
implies that there is exactly one such coloring.
We construct a new plane graph whose vertices are the white faces and whose edges correspond to crossings. We can label each edge in this graph as a left edge or a right edge, depending on which thread appears to go over the other as we view the corresponding crossing from one of the endpoints of the edge. Left and right edges are typically indicated by labeling left edges + and right edges –, or by drawing left edges with solid lines and right edges with dashed lines.
The original knot diagram is the
medial graph
In the mathematical discipline of graph theory, the medial graph of plane graph ''G'' is another graph ''M(G)'' that represents the adjacencies between edges in the faces of ''G''. Medial graphs were introduced in 1922 by Ernst Steinitz to study ...
of this new plane graph, with the type of each crossing determined by the sign of the corresponding edge. Changing the sign of ''every'' edge corresponds to reflecting
the knot in a mirror.
Linkless and knotless embedding
In two dimensions, only the
planar graphs
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...
may be embedded into the Euclidean plane without crossings, but in three dimensions, any
undirected graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...
may be embedded into space without crossings. However, a spatial analogue of the planar graphs is provided by the graphs with
linkless embedding
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is ...
s and
knotless embedding
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into three-dimensional Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is ...
s. A linkless embedding is an embedding of the graph with the property that any two cycles are
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 ...
ed; a knotless embedding is an embedding of the graph with the property that any single cycle is
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 ...
ted. The graphs that have linkless embeddings have a
forbidden graph characterization
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden ...
involving the
Petersen family
In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph . The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph.
Any o ...
, a set of seven graphs that are intrinsically linked: no matter how they are embedded, some two cycles will be linked with each other. A full characterization of the graphs with knotless embeddings is not known, but the
complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...
is one of the minimal forbidden graphs for knotless embedding: no matter how is embedded, it will contain a cycle that forms a
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 ...
.
Generalization
In contemporary mathematics the term ''knot'' is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold with a submanifold , one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to . Traditional knots form the case where and or .
The
Schoenflies theorem
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology.
Schoenflies ...
states that the circle does not knot in the 2-sphere: every topological circle in the 2-sphere is isotopic to a geometric circle.
Alexander's theorem
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after James Waddell Alexander ...
states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere. In the tame topological category, it's known that the -sphere does not knot in the -sphere for all . This is a theorem of
Morton Brown
Morton Brown (born August 12, 1931, in New York City, New York) is an American mathematician, who specializes in geometric topology.
In 1958 Brown earned his Ph.D. from the University of Wisconsin-Madison under R. H. Bing. From 1960 to 1962 he ...
,
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...
, and
Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
. The
Alexander horned sphere
The Alexander horned sphere is a pathological object in topology discovered by .
Construction
The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...
is an example of a knotted 2-sphere in the 3-sphere which is not tame.
In the smooth category, the -sphere is known not to knot in the -sphere provided . The case is a long-outstanding problem closely related to the question: does the 4-ball admit an
exotic smooth structure?
André Haefliger
André Haefliger (born 22 May 1929 in Nyon, Switzerland) is a Swiss mathematician who works primarily on topology.
Education and career
Haefliger went to school in Nyon and then attended his final years at Collège Calvin in Geneva. He studied ...
proved that there are no smooth -dimensional knots in provided , and gave further examples of knotted spheres for all such that . is called the
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the ...
of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of in form a group, with group operation given by the connect sum, provided the co-dimension is greater than two. Haefliger based his work on
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
's
''h''-cobordism theorem. One of Smale's theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory. If one allows topological or PL-isotopies,
Christopher Zeeman
Sir Erik Christopher Zeeman FRS (4 February 1925 – 13 February 2016), was a British mathematician, known for his work in geometric topology and singularity theory.
Overview
Zeeman's main contributions to mathematics were in topology, partic ...
proved that spheres do not knot when the co-dimension is greater than 2. See a
generalization to manifolds.
See also
*
*
*
Notes
References
*
*
*
*
*
External links
*
The Manifold Atlas Project
{{Knot theory, state=collapsed