Knot (mathematics) - Types of Knots

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, 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, while the Borromean rings complement has the geometry of .