Alexander's theorem

{{short description|Every knot or link can be represented as a closed braid}}

File:An Element of the Braid Group.svg

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.{{cite journal|first = James|last = Alexander| author-link=James Waddell Alexander II| title = A lemma on a system of knotted curves | journal = Proceedings of the National Academy of Sciences of the United States of America |volume = 9|issue=3|year = 1923|pages = 93–95|doi=10.1073/pnas.9.3.93|bibcode = 1923PNAS....9...93A|pmc = 1085274 | pmid=16576674|doi-access = free}}

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.{{cite book|title=The Knot Book. Revised reprint of the 1994 original.|first=Colin C.|last = Adams|author-link=Colin Adams (mathematician)| publisher = American Mathematical Society|year=2004| location=Providence, RI| page = 130|isbn=0-8218-3678-1|mr=2079925}}

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.