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]

This is a typical element of the braid group, which is used in the mathematical field of knot theory.

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.

References
  1. Alexander, James (1923). "A lemma on a system of knotted curves". Proceedings of the National Academy of Sciences of the United States of America. 9 (3): 93–95. Bibcode:1923PNAS....9...93A. doi:10.1073/pnas.9.3.93. PMC 1085274. PMID 16576674.CS1 maint: ref=harv (link)
  2. Adams, Colin C. (2004). The Knot Book. Revised reprint of the 1994 original. Providence, RI: American Mathematical Society. p. 130. ISBN 0-8218-3678-1. MR 2079925.
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