Markov theorem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> the Markov theorem gives necessary and sufficient conditions for two <a href="/facts/Braid_(mathematics)/rn5323Kx">braids</a> to have closures that are equivalent <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a> or <a href="/facts/Link_(knot_theory)/hoFDYmVj">links</a>. The conditions are stated in terms of the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> structures on braids. 
Braids are algebraic objects described by diagrams; the relation to topology is given by <a href="/facts/Alexander%27s_theorem/9cMJHqLu">Alexander's theorem</a> which states that every <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a> or <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a> in three-dimensional Euclidean space is the closure of a <a href="/facts/Braid_(mathematics)/rn5323Kx">braid</a>. The Markov theorem, proved by Russian mathematician <a href="/facts/Andrey_Markov_Jr./3s1VYSdq">Andrei Andreevich Markov Jr.</a> describes the elementary moves generating the equivalence relation on braids given by the equivalence of their closures. 
More precisely Markov's theorem can be stated as follows: given two braids represented by elements 
 
 
 
 
 β
 
 n
 
 
 ,
 
 β
 
 m
 
 ′
 
 
 
 {\displaystyle \beta _{n},\beta _{m}'}
 
 in the braid groups 
 
 
 
 
 B
 
 n
 
 
 ,
 
 B
 
 m
 
 
 
 
 {\displaystyle B_{n},B_{m}}
 
, their closures are equivalent links if and only if 
 
 
 
 
 β
 
 m
 
 ′
 
 
 
 {\displaystyle \beta _{m}'}
 
 can be obtained from applying to 
 
 
 
 
 β
 
 n
 
 
 
 
 {\displaystyle \beta _{n}}
 
 a sequence of the following operations:

<ol><li><a href="/facts/Conjugacy_class/AFuPIFNW">conjugating</a> 
 
 
 
 
 β
 
 n
 
 
 
 
 {\displaystyle \beta _{n}}
 
 in 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
;</li>
<li>replacing 
 
 
 
 
 β
 
 n
 
 
 
 
 {\displaystyle \beta _{n}}
 
 by 
 
 
 
 
 β
 
 n
 
 
 
 σ
 
 n
 +
 1
 
 
 ±
 1
 
 
 ∈
 
 B
 
 n
 +
 1
 
 
 
 
 {\displaystyle \beta _{n}\sigma _{n+1}^{\pm 1}\in B_{n+1}}
 
 (here 
 
 
 
 
 σ
 
 i
 
 
 
 
 {\displaystyle \sigma _{i}}
 
 are the standard generators of the braid groups; geometrically this amounts to adding a strand to the right of the braid diagram and twisting it once with the (previously) last strand);</li>
<li>the inverse of the previous operation (if 
 
 
 
 
 β
 
 n
 
 
 =
 
 β
 
 n
 −
 1
 
 
 
 σ
 
 n
 
 
 ±
 1
 
 
 
 
 {\displaystyle \beta _{n}=\beta _{n-1}\sigma _{n}^{\pm 1}}
 
 with 
 
 
 
 
 β
 
 n
 −
 1
 
 
 ∈
 
 B
 
 n
 −
 1
 
 
 
 
 {\displaystyle \beta _{n-1}\in B_{n-1}}
 
 replace with 
 
 
 
 
 β
 
 n
 −
 1
 
 
 
 
 {\displaystyle \beta _{n-1}}
 
).</li></ol>
In 1974 American mathematician <a href="/facts/Joan_Birman/ceIIseYm">Joan Birman</a> published a <a href="/facts/Monograph/o8VRGpVc">monograph</a>, <a href="/facts/Braids,_Links,_and_Mapping_Class_Groups/1mAVt3BJ">Braids, Links, and Mapping Class Groups</a>, based on a graduate course she taught as a visiting professor at <a href="/facts/Princeton_University/n3EPGHBM">Princeton University</a> in 1971–72; this book contains the first complete proof of the Markov theorem.

Markov theorem open-in-new

Markov theorem