In mathematics, a knot is an embedding of a circle in 3dimensional Euclidean space, R^{3} (also known as E^{3}), 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.
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Contents
Formal definition
A knot is an embedding of the circle (S^{1}) into threedimensional Euclidean space (R^{3}).^{[1]} or the 3sphere, S^{3}, since the 3sphere is compact.^{[2]} ^{[Note 1]} Two knots are defined to be equivalent if there is an ambient isotopy between them.^{[3]}
Projection
A knot in R^{3} (respectively in the 3sphere, S^{3}), can be projected onto a plane R^{2} (resp. a sphere S^{2}). 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 4valent 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.
Types of knots
The simplest knot, called the unknot or trivial knot, is a round circle embedded in R^{3}.^{[4]} In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot (3_{1} in the table), the figureeight knot (4_{1}) and the cinquefoil knot (5_{1}).^{[5]}
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 R^{3} is the union of a finite set of line segments.^{[6]} A tame knot is any knot equivalent to a polygonal knot.^{[6]}^{[Note 2]} Knots which are not tame are called wild,^{[7]} and can have pathological behavior.^{[7]} In knot theory and 3manifold 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 D^{2} × S^{1} in S^{3}.
The framing of the knot is the linking number of the image of the ribbon I × S^{1} with the knot. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists.^{[8]} 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) I, II, and III moves.
Knot complement
Given a knot in the 3sphere, the knot complement is all the points of the 3sphere 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 3manifold theory.^{[9]}
JSJ decomposition
The JSJ decomposition and Thurston's hyperbolization theorem reduces the study of knots in the 3sphere to the study of various geometric manifolds via splicing or satellite operations. In the pictured knot, the JSJdecomposition 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 .
Applications to Graph Theory
Medial graph
Another convenient representation of knot diagrams ^{[10]}^{[11]} was introduced by Peter Tait in 1877.^{[12]}^{[13]}
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 2dimensional 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 embeddings and knotless embeddings. A linkless embedding is an embedding of the graph with the property that any two cycles are unlinked; a knotless embedding is an embedding of the graph with the property that any single cycle is unknotted. 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.^{[14]} A full characterization of the graphs with knotless embeddings is not known, but the complete graph K_{7} is one of the minimal forbidden graphs for knotless embedding: no matter how K_{7} is embedded, it will contain a cycle that forms a trefoil knot.^{[15]}
Generalization
In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings.^{[citation needed]} 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 2sphere—every circle in the 2sphere is isotopic to the standard circle. Alexander's theorem states that the 2sphere does not smoothly (or PL or tame topologically) knot in the 3sphere. In the tame topological category, it's known that the sphere does not knot in the sphere for all . This is a theorem of Brown and Mazur. The Alexander horned sphere is an example of a knotted 2sphere in the 3sphere which is not tame. In the smooth category, the sphere is known not to knot in the sphere provided . The case is a longoutstanding problem closely related to the question: does the 4ball admit an exotic smooth structure?
Haefliger proved that there are no smooth jdimensional 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 codimension is greater than two. Haefliger based his work on Smale's hcobordism theorem. One of Smale's theorems is that when one deals with knots in codimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than codimension 2 knot theory. If one allows topological or PLisotopies, Zeeman proved that spheres do not knot when the codimension is larger than two. See a generalization to manifolds.
See also
Notes
 ^ Note that the 3sphere is equivalent to R^{3} with a single point added at infinity (see onepoint compactification).
 ^ A knot is tame if and only if it can be represented as a finite closed polygonal chain
References
 ^ Armstrong (1983), p. 213.
 ^ Cromwell (2004), p. 33; Adams (1994), pp. 246–250.
 ^ Cromwell (2004), p. 5.
 ^ Adams (1994), p. 2.
 ^ Adams (1994), Table 1.1, p. 280; Livingstone (1996), Appendix A: Knot Table, p. 221.
 ^ ^{a} ^{b} Armstrong (1983), p. 215.
 ^ ^{a} ^{b} Livingstone (1996), Section 2.1 Wild Knots and Unknottings, pp. 11–14.
 ^ L. H. Kauffman: "An invariant of regular isotopy", Transactions of the American Mathematical Society 318(2), 1990, pp. 417–471.
 ^ Adams (1994), pp. 261–262.
 ^ Adams, Colin C. (2001). The Knot Book. American Mathematical Society. pp. 52–55.
 ^ Entrelacs.net tutorial
 ^ Tait, Peter G. (1876–1877). "On Knots I". Proceedings of the Royal Society of Edinburgh. 28: 145–190.
Revised May 11, 1877.
 ^ Tait, Peter G. (1876–1877). "On Links (Abstract)". Proceedings of the Royal Society of Edinburgh. 9 (98): 321–332.
 ^ Robertson, Neil; Seymour, Paul; Thomas, Robin (1993), "A survey of linkless embeddings", in Robertson, Neil; Seymour, Paul, Graph Structure Theory: Proc. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors (PDF), Contemporary Mathematics, 147, American Mathematical Society, pp. 125–136.
 ^ Ramirez Alfonsin, J. L. (1999), "Spatial graphs and oriented matroids: the trefoil", Discrete and Computational Geometry, 22 (1): 149–158, doi:10.1007/PL00009446.
 Adams, Colin C. (1994). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. W. H. Freeman & Company.
 Armstrong, M. A. (1983) [1979]. Basic Topology. Undergraduate Texts in Mathematics. New York: SpringerVerlag. ISBN 0387908390.
 Cromwell, Peter R. (2004). Knots and Links. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511809767. ISBN 0521839475. MR 2107964.
 Farmer, David W.; Stanford, Theodore B. (1995). Knots and Surfaces: A Guide to Discovering Mathematics.
 Livingstone, Charles (1996). Knot Theory. The Mathematical Association of America.
External links
Wikimedia Commons has media related to Knots (knot theory). 
 "Main_Page", The Knot Atlas.
 "Classification of embeddings", The Manifold Atlas Project at the Wayback Machine (archived March 4, 2012)