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Two link diagrams represent the same ambient isotopy class of a link in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130060/r1300601.png" /> if and only if they are related by a finite number of Reidemeister moves (see Fig.a1) and a plane isotopy.
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Two link diagrams represent the same ambient isotopy class of a link in $S^3$ if and only if they are related by a finite number of Reidemeister moves (see Fig. a1) and a plane isotopy.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r130060a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r130060a.gif" />
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Figure: r130060a
 
Figure: r130060a
  
Proofs of the theorem were published in 1927 by K. Reidemeister [[#References|[a3]]], and by J.W. Alexander and G.B. Briggs [[#References|[a1]]].
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Proofs of the theorem were published in 1927 by K. Reidemeister {{Cite|a3}}, and by J.W. Alexander and G.B. Briggs {{Cite|a1}}.
  
The theorem also holds for oriented links and oriented diagrams, provided that Reidemeister moves observe the orientation of diagrams. It holds also for links in a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130060/r1300602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130060/r1300603.png" /> is a surface.
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The theorem also holds for oriented links and oriented diagrams, provided that Reidemeister moves observe the orientation of diagrams. It holds also for links in a manifold $M=F\times[0,1]$, where $F$ is a surface.
  
The first formalization of knot theory was obtained by M. Dehn and P. Heegaard by introducing lattice knots and lattice moves [[#References|[a2]]]. Every knot has a lattice knot representation and two knots are lattice equivalent if and only if they are ambient isotopic. The Reidemeister approach was to consider polygonal knots up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130060/r1300604.png" />-moves. (A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130060/r1300606.png" />-move replaces one side of a triangle by two other sides or vice versa. A regular projection of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130060/r1300607.png" />-move can be decomposed into Reidemeister moves.) This approach was taken by Reidemeister to prove his theorem.
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The first formalization of knot theory was obtained by M. Dehn and P. Heegaard by introducing lattice knots and lattice moves {{Cite|a2}}. Every knot has a lattice knot representation and two knots are lattice equivalent if and only if they are ambient isotopic. The Reidemeister approach was to consider polygonal knots up to $\Delta$-moves. (A $\Delta$-move replaces one side of a triangle by two other sides or vice versa. A regular projection of a $\Delta$-move can be decomposed into Reidemeister moves.) This approach was taken by Reidemeister to prove his theorem.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Alexander,   G.B. Briggs,   "On types of knotted curves"  ''Ann. of Math.'' , '''28''' :  2  (1927/28) pp. 563–586</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Dehn,   P. Heegaard,   "Analysis situs" , ''Encykl. Math. Wiss.'' , '''III AB3''' , Leipzig  (1907) pp. 153–220</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Reidemeister,   "Elementare Begrundung der Knotentheorie"  ''Abh. Math. Sem. Univ. Hamburg'' , '''5'''  (1927) pp. 24–32</TD></TR></table>
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* {{Ref|a1}} J.W. Alexander, G.B. Briggs, "On types of knotted curves"  ''Ann. of Math.'' , '''28''' :  2  (1927/28) pp. 563–586 {{ZBL|53.0549.02}}
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* {{Ref|a2}} M. Dehn, P. Heegaard, "Analysis situs" , ''Encykl. Math. Wiss.'' , '''III AB3''' , Leipzig  (1907) pp. 153–220 {{ZBL|38.0510.14}}
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* {{Ref|a3}} K. Reidemeister, "Elementare Begrundung der Knotentheorie"  ''Abh. Math. Sem. Univ. Hamburg'' , '''5'''  (1927) pp. 24–32 {{ZBL|52.0579.01}}

Latest revision as of 20:23, 16 March 2023

2020 Mathematics Subject Classification: Primary: 57K [MSN][ZBL]

Two link diagrams represent the same ambient isotopy class of a link in $S^3$ if and only if they are related by a finite number of Reidemeister moves (see Fig. a1) and a plane isotopy.

Figure: r130060a

Proofs of the theorem were published in 1927 by K. Reidemeister [a3], and by J.W. Alexander and G.B. Briggs [a1].

The theorem also holds for oriented links and oriented diagrams, provided that Reidemeister moves observe the orientation of diagrams. It holds also for links in a manifold $M=F\times[0,1]$, where $F$ is a surface.

The first formalization of knot theory was obtained by M. Dehn and P. Heegaard by introducing lattice knots and lattice moves [a2]. Every knot has a lattice knot representation and two knots are lattice equivalent if and only if they are ambient isotopic. The Reidemeister approach was to consider polygonal knots up to $\Delta$-moves. (A $\Delta$-move replaces one side of a triangle by two other sides or vice versa. A regular projection of a $\Delta$-move can be decomposed into Reidemeister moves.) This approach was taken by Reidemeister to prove his theorem.

References

  • [a1] J.W. Alexander, G.B. Briggs, "On types of knotted curves" Ann. of Math. , 28 : 2 (1927/28) pp. 563–586 Zbl 53.0549.02
  • [a2] M. Dehn, P. Heegaard, "Analysis situs" , Encykl. Math. Wiss. , III AB3 , Leipzig (1907) pp. 153–220 Zbl 38.0510.14
  • [a3] K. Reidemeister, "Elementare Begrundung der Knotentheorie" Abh. Math. Sem. Univ. Hamburg , 5 (1927) pp. 24–32 Zbl 52.0579.01
How to Cite This Entry:
Reidemeister theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reidemeister_theorem&oldid=16693
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article