In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (also known as E3), 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 (S1) into three-dimensional Euclidean space (R3). or the 3-sphere, S3, since the 3-sphere is compact. [Note 1] Two knots are defined to be equivalent if there is an ambient isotopy between them.
A knot in R3 (respectively in the 3-sphere, S3), can be projected onto a plane R2 (resp. a sphere S2). 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 4-valent 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 R3. In the ordinary sense of the word, the unknot is not "knotted" at all. The simplest nontrivial knots are the trefoil knot (31 in the table), the figure-eight knot (41) and the cinquefoil knot (51).
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 R3 is the union of a finite set of line segments. A tame knot is any knot equivalent to a polygonal knot.[Note 2] Knots which are not tame are called wild, and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame.
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 .
Applications to Graph Theory
Another interpretation of a knot diagram as a decorated planar graph is more easy to deal with: The projection decomposes the plane into connected components, the graph itself of dimension 1, and an infinite zone and components which are homeomorphic to disks, of dimension 2.
There is a way to attach a color, black or white, to all of these zones of dimension 2 in the following way: Choose the black color for the infinite zone, and at each crossing in turn, color the opposite zone in black. Proceed until every crossing has been taken into account. The Jordan curve theorem implies that this procedure is well defined. It is called the medial graph of the 4-valent original graph.
Then you define a planar graph whose vertices are the white zones and whose edges are associated with every crossing. The over/under patterns associated to the crossings are now decorating the edges with a simple sign +/- or left/right according to the local configuration: viewing an edge from any of its two incident vertices, one of the two threads, whether left or right, goes above and the other one goes below. A left edge can be encoded as a plain edge, a right edge as a dashed edge. Changing the chirality of all edges amounts to reflecting the knot in a mirror.
We have thus constructed, for each knot diagram, a planar graph with "signed" edges associated to every crossing; the type of crossing that takes place at the middle of each edge is determined by the left/right sign of that edge.
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. A full characterization of the graphs with knotless embeddings is not known, but the complete graph K7 is one of the minimal forbidden graphs for knotless embedding: no matter how K7 is embedded, it will contain a cycle that forms a trefoil knot.
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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 circle in the 2-sphere is isotopic to the standard 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 Brown and Mazur. 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?
Haefliger proved that there are no smooth j-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 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, Zeeman proved that spheres do not knot when the co-dimension is larger than two. See a generalization to manifolds.
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- Entrelacs.net tutorial
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- Cromwell, Peter R. (2004). Knots and Links. Cambridge University Press, Cambridge. doi:10.1017/CBO9780511809767. ISBN 0-521-83947-5. MR 2107964.
- Farmer, David W.; Stanford, Theodore B. (1995). Knots and Surfaces: A Guide to Discovering Mathematics.
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|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)