In mathematics Alexander's theorem states that every

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...

or link can be represented as a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

braid A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

; that is, a braid in which the corresponding ends of the strings are connected in pairs. The theorem is named after

James Waddell Alexander II James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. ...

, who published a proof in 1923. Braids were first considered as a tool of

knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...

by Alexander. His theorem gives a positive answer to the question ''Is it always possible to transform a given knot into a closed braid?'' A good construction example is found in Colin Adams's book. However, the correspondence between knots and braids is clearly not one-to-one: a knot may have many braid representations. For example, conjugate braids yield equivalent knots. This leads to a second fundamental question: ''Which closed braids represent the same knot type?'' This question is addressed in Markov's theorem, which gives ‘moves’ relating any two closed braids that represent the same knot.

References

* Knot theory Theorems in algebraic topology {{Knottheory-stub