Markov braid theorem

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If two closed braids represent the same ambient isotopy class of oriented links (cf. also Braid theory), then one can transform one braid to another by a sequence of Markov moves:

i) (conjugation).

ii) , where is an element of the th braid group

and is the th generator of the th braid group.

Markov's braid theorem is an important ingredient in the construction of the Jones polynomial and its generalizations (e.g. the Jones–Conway polynomial).


[a1] J.S. Birman, "Braids, links and mapping class groups" , Ann. of Math. Stud. , 82 , Princeton Univ. Press (1974)
[a2] A.A. Markov, "Über die freie Aquivalenz der geschlossen Zopfe" Recueil Math. Moscou , 1 (1935) pp. 73–78
[a3] N.M. Weinberg, "On free equivalence of free braids" C.R. (Dokl.) Acad. Sci. USSR , 23 (1939) pp. 215–216 (In Russian)
How to Cite This Entry:
Markov braid theorem. Jozef Przytycki (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098