knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R³, considered up to continuous deformations (isotopies). This is basically equivalent to a conventional knot with the ends of the string joined together to prevent it from becoming undone. The study of such objects forms a branch of mathematics called knot theory. The simplest knot, called the unknot, is just a standard 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 and the figure-eight knot. 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. Knots that are not tame are called wild and can have pathological behavior. In knot theory and 3-manifold theory, oftentimes the adjective "tame" is omitted. Since a knot itself is just a tangled piece of metaphorical circular twine, the interesting part about studying it is looking at how its tangling affects the shape of the space it is embedded in. Knot theory primarily concerns itself with studying what's left when the knot is removed — that is R³ minus the knot, or the knot complement. This leads more generally into the study of 3-manifolds. In higher dimensions, circles are always unknotted, so one considers embeddings of spheres, always in codimension 2.

Knot (Mathematics)

See also