View source for Reidemeister theorem

{{TEX|done}}{{MSC|57K}}

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" />

Figure: r130060a

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 $M=F\times[0,1]$, 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 {{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====
* {{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}}
* {{Ref|a2}} M. Dehn, P. Heegaard, "Analysis situs" , ''Encykl. Math. Wiss.'' , '''III AB3''' , Leipzig  (1907) pp. 153–220 {{ZBL|38.0510.14}}
* {{Ref|a3}} K. Reidemeister, "Elementare Begrundung der Knotentheorie"  ''Abh. Math. Sem. Univ. Hamburg'' , '''5'''  (1927) pp. 24–32 {{ZBL|52.0579.01}}