Markov theorem
In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures which are equivalent links[disambiguation needed]. In algebraic topology, Alexander's theorem states that every knot or link[disambiguation needed] in three-dimensional space is the closure of a braid. The Markov theorem, proved by Russian mathematician Andrei Andreevich Markov Jr.[1] states that three conditions are necessary and sufficient for two braids to have equivalent closures:
- They are equivalent braids
- They are conjugate braids
- Appending or removing on the right of the braid a strand which crosses strand to its left exactly once.
References
- ^ A. A. Markov Jr., Über die freie Äquivalenz der geschlossenen Zöpfe
- 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)
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