Alexander theorem on braids

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Every link has a closed braid presentation (cf. also Braid theory; Link).

This result, published by J.W. Alexander in 1923, allows one to study knots and links using the theory of braids, [a1] (cf. also Knot theory). Alexander's theorem has its roots in Brunn's result (1897) that every knot has a projection with only one multiple point (it is usually not a regular projection) [a2].

The smallest number of braid strings used in the presentation is called the braid index of the link.


[a1] J.W. Alexander, "A lemma on systems of knotted curves" Proc. Nat. Acad. Sci. USA , 9 (1923) pp. 93–95
[a2] H.K. Brunn, "Über verknotete Kurven" , Verh. Math. Kongr. Zürich (1897) pp. 256–259
How to Cite This Entry:
Alexander theorem on braids. Jozef Przytycki (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098