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:

i) $a \leftrightarrow b a b ^ { - 1 }$ (conjugation).

ii) $a \leftrightarrow a b ^ { \pm 1 }_ { n }$, where $a$ is an element of the $n$th braid group

and $b _ { n }$ is the $n$th generator of the $( n + 1 )$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).

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