In knot theory

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...

, the Whitehead link, discovered by J.H.C. Whitehead, is one of the most basic links

Link (knot theory)

In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...

.

J.H.C. Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture

Poincaré conjecture

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...

. In 1934, the Whitehead link was used as part of his construction of the now-named Whitehead manifold

Whitehead manifold

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3. Henry Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture....

, which refuted his previous purported proof of the conjecture.

Structure

The link is created with two projections of the unknot

Unknot

The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...

: one circular loop and one figure eight-shaped loop (i.e., a loop with a Reidemeister Type I move applied) intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings. Because each underhand crossing has a paired upperhand crossing, its linking number

Linking number

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...

 is 0. It is not isotopic

Isotopic

The word isotopic has a number of different meanings, including:* In the physical sciences, to do with chemical isotopes;* In mathematics, to do with a relation called isotopy.* In geometry, isotopic refers to facet-transitivity....

 to the unlink

Unlink

In the mathematical field of knot theory, the unlink is a link that is equivalent to finitely many disjoint circles in the plane.- Properties :...

, but it is link homotopic to the unlink.

In braid theory

Braid theory

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a...

 notation, the link is written


Its Jones polynomial is

External links

• L5a1 knot-theoretic link on Knot atlas site