Namespaces
Variants
Actions

Difference between revisions of "Markov braid theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (link)
 
(One intermediate revision by the same user not shown)
Line 11: Line 11:
 
i) $a \leftrightarrow b a b ^ { - 1 }$ (conjugation).
 
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
+
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 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>
 
<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>
Line 22: Line 22:
 
<table>
 
<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">[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">[a2]</td> <td valign="top"> A.A. Markov, "Über die freie Äquivalenz der geschlossenen Zöpfe", ''Recueil Math. Moscou'', '''1''' (1935)  pp. 73–78 {{ZBL|0014.04202}}</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>
 
<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>
 
</table>

Latest revision as of 08:04, 19 March 2023

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 Äquivalenz der geschlossenen Zöpfe", Recueil Math. Moscou, 1 (1935) pp. 73–78 Zbl 0014.04202
[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=52946
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article