Markov braid theorem
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:
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)|
Markov braid theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_braid_theorem&oldid=17995