Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R³, 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 $S^j$ in $S^n$ , especially in the case $j=n-2$ . The branch of mathematics that studies knots is known as knot theory.

1 Formal definition
- 1.1 Tame vs. wild knots
2 Types of knots
3 Generalization
4 See also
5 Notes
6 References
7 Further reading
8 External links

Formal definition

A knot is an embedding of the circle (S₁) into three-dimensional Euclidean space (E³)^[1]. Two knots are defined to be equivalent if there is an ambient isotopy between them.

Tame vs. wild knots

A polygonal knot is a knot whose image in E³ is the union of a finite set of line segments^[2]. A tame knot is any knot equivalent to a polygonal knot^[2]. Knots which are not tame are called wild^[3].

Types of knots

The simplest knot, called the unknot, 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, possibly tangled together, are called links. Knots are links with a single component.

Often mathematicians prefer to consider knots embedded into the 3-sphere, S³, rather than R³ since the 3-sphere is compact. The 3-sphere is equivalent to R³ with a single point added at infinity (see one-point compactification).

A knot is tame if it can be "thickened up", that is, if there exists an extension to an embedding of the solid torus, $S^1 \times D^2$ , into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. Knots that 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 $H^2 \times R$ , while the Borromean rings complement has the geometry of $H^3$ .

Generalization

In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold $M$ with a submanifold $N$ , one sometimes says $N$ can be knotted in $M$ if there exists an embedding of $N$ in $M$ which is not isotopic to $N$ . Traditional knots form the case where $N=S^1$ and $M=\mathbb R^3$ or $M=S^3$ .

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 $n$ -sphere does not knot in the $n%2B1$ -sphere for all $n$ . 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 $n$ -sphere is known not to knot in the $n%2B1$ -sphere provided $n \neq 3$ . The case $n=3$ 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 $S^n$ provided $2n-3j-3>0$ , and gave further examples of knotted spheres for all $n > j \geq 1$ such that $2n-3j-3=0$ . $n-j$ is called the codimension of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of $S^j$ in $S^n$ 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.