2-bridge knot
Encyclopedia
In the mathematical field of knot theory
, a 2-bridge knot is a knot
which can be isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points.
Other names for 2-bridge knots are rational knots, 4-plats, and Viergeflechte. 2-bridge links are defined similarly as above, but each component will have one min and max. 2-bridge knots were classified by Herbert Seifert
, using the fact that the 2-sheeted branched cover of the 3-sphere
over the knot is a lens space
.
The names rational knot and rational link were coined by John Conway
who defined them as arising from numerator closures of rational tangles.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a 2-bridge knot is a knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . 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...
which can be isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points.
Other names for 2-bridge knots are rational knots, 4-plats, and Viergeflechte. 2-bridge links are defined similarly as above, but each component will have one min and max. 2-bridge knots were classified by Herbert Seifert
Herbert Seifert
Herbert Karl Johannes Seifert was a German mathematician known for his work in topology....
, using the fact that the 2-sheeted branched cover of the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
over the knot is a lens space
Lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions....
.
The names rational knot and rational link were coined by John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
who defined them as arising from numerator closures of rational tangles.