# Reidemeister theorem

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.

How to Cite This Entry:
Reidemeister theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reidemeister_theorem&oldid=52732
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article