Namespaces
Variants
Actions

Difference between revisions of "Markov braid theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Automatically changed introduction)
Line 20: Line 20:
  
 
====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>
+
<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 08:02, 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 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=52946
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article