s with seven crossings
or fewer (not including mirror images).
by joining the ends.
, a knot is an embedding
of a topological circle
in 3-dimensional Euclidean space
, (also known as ), considered up to continuous deformations (isotopies
A crucial difference between the standard mathematical and conventional notions of a knot
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
, and has many simple relations to graph theory
A knot is an embedding
of the circle
() into three-dimensional Euclidean space
(). or the 3-sphere
, , since the 3-sphere is compact
. Two knots are defined to be equivalent if there is an ambient isotopy
A knot in (or alternatively in the 3-sphere
, ), can be projected onto a plane (respectively a sphere
). This projection is almost always regular, meaning that it is injective
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 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
is thus a quadrivalent planar graph
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
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
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 the table), the figure-eight knot
() and the cinquefoil knot
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
in is the union
of a finite set
of line segments
A ''tame'' knot is any knot equivalent to a polygonal knot.
Knots which are not tame are called ''wild
and can have pathological
In knot theory and 3-manifold
theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.
A ''framed knot'' is the extension of a tame knot to an embedding of the solid torus in .
The ''framing'' of the knot is the linking number
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
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
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
Given a knot in the 3-sphere, the knot complement
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
The JSJ decomposition
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
. The trefoil complement has the geometry of , while the Borromean rings complement has the geometry of .
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
s with up to seven crossings
represented as knot diagram
s with their medial graph
Another convenient representation of knot diagrams was introduced by Peter Tait
Any knot diagram defines a plane graph
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
to a 2-dimensional disk
. 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
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
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
may be embedded into the Euclidean plane without crossings, but in three dimensions, any undirected graph
may be embedded into space without crossings. However, a spatial analogue of the planar graphs is provided by the graphs with linkless embedding
s and knotless embedding
s. A linkless embedding is an embedding of the graph with the property that any two cycles are unlink
ed; a knotless embedding is an embedding of the graph with the property that any single cycle is unknot
ted. The graphs that have linkless embeddings have a forbidden graph characterization
involving the Petersen family
, 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
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 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
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
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
, Barry Mazur
, and Marston Morse
. The Alexander horned sphere
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
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
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
'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
proved that spheres do not knot when the co-dimension is greater than 2. See a generalization to manifolds
*List of mathematical knots and links
The Manifold Atlas Project