Alexander's theorem

In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; 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, who published a proof in 1923.^[1]

Braids were first considered as a tool of knot theory 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.^[2]

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.

• Sossinsky, Alexei B. (2002). Knots: Mathematics with a Twist. Cambridge, MA: Harvard University Press. p. 17. ISBN 9780674009448. MR 1941191.

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