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.

**^**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.**^**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.

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