Reidemeister theorem

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