Alexander's theorem

In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid. The theorem is named after James Waddell Alexander II, who published its proof in 1923.

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 on page 130 of Adams's The Knot book (see ref. below). 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?

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