Markov theorem

The Markov Theorem gives necessary and sufficient conditions for two braids to have closures which are equivalent links. In algebraic topology, Alexander's theorem states that every knot or link in three dimensional space is the closure of a braid. The Markov Theorem, proved by Russian mathematician Andrei Andreevich Markov states that three conditions are necessary and sufficient for two braids to have equivalent closures:

1. They are equivalent braids
2. They are conjugate braids
3. Appending or removing on the right of the braid a strand which crosses strand to its left exactly once.

References

• J. S. Birman, Knots, links, and mapping class groups, Annals of Math Study, no. 82, Princeton University Press (1974)
• Louis H. Kauffman, Knots and Physics, p. 95, World Scientific, (1991)