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If two closed braids represent the same ambient isotopy class of oriented links (cf. also [[Braid theory|Braid theory]]), then one can transform one braid to another by a sequence of Markov moves:
 
If two closed braids represent the same ambient isotopy class of oriented links (cf. also [[Braid theory|Braid theory]]), then one can transform one braid to another by a sequence of Markov moves:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300501.png" /> (conjugation).
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i) $a \leftrightarrow b a b ^ { - 1 }$ (conjugation).
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300502.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300503.png" /> is an element of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300504.png" />th braid group
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ii) $a \leftrightarrow a b ^ { \pm 1 }_ { n }$, where $a$ is an element of the $n$th braid group
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300505.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300505.png"/></td> </tr></table>
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300506.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300507.png" />th generator of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130050/m1300508.png" />th braid group.
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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|Jones–Conway polynomial]]).
 
Markov's braid theorem is an important ingredient in the construction of the Jones polynomial and its generalizations (e.g. the [[Jones–Conway polynomial|Jones–Conway polynomial]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.S. Birman,  "Braids, links and mapping class groups" , ''Ann. of Math. Stud.'' , '''82''' , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Markov,  "Über die freie Aquivalenz der geschlossen Zopfe"  ''Recueil Math. Moscou'' , '''1'''  (1935)  pp. 73–78</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.M. Weinberg,  "On free equivalence of free braids"  ''C.R. (Dokl.) Acad. Sci. USSR'' , '''23'''  (1939)  pp. 215–216  (In Russian)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  J.S. Birman,  "Braids, links and mapping class groups" , ''Ann. of Math. Stud.'' , '''82''' , Princeton Univ. Press  (1974)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A.A. Markov,  "Über die freie Aquivalenz der geschlossen Zopfe"  ''Recueil Math. Moscou'' , '''1'''  (1935)  pp. 73–78</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  N.M. Weinberg,  "On free equivalence of free braids"  ''C.R. (Dokl.) Acad. Sci. USSR'' , '''23'''  (1939)  pp. 215–216  (In Russian)</td></tr></table>

Revision as of 16:56, 1 July 2020

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).

References

[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)
How to Cite This Entry:
Markov braid theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_braid_theorem&oldid=50123
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article