nLab Reidemeister move

Idea

The Reidemeister moves are local changes to link diagrams which can be realised as isotopies between the link diagrams from which they arise. Because of Reidemeister’s theorem, Theorem , they are of fundamental importance in knot theory, allowing knots and links to be treated purely combinatorially and diagrammatically.

The moves were introduced in the book Reidemeister.

Definition

The convention below is that part of a link diagram is shown with the ‘move’ indicated, but that outside that, the diagram of the link is unchanged. As usual in knot theory, everything is up to planar isotopy?.

Definition

The first Reidemeister move, or R1 move, allows either of the following replacements of a fragment of a link diagram to be made.

category: svg

Definition

The second Reidemeister move, or R2 move, allows either of the following replacements of a fragment of a link diagram to be made.

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Definition

The third Reidemeister move, or R3 move, allows the following replacement of a fragment of a link diagram to be made.

category: svg

Definition

A pair of link diagrams $D$ and $D'$ are isotopic, or isotopic by moves, or equivalent if there is a finite sequence of the moves R1, R2 and R3 taking $D$ to $D'$ up to planar isotopy.

Definition

A pair of link diagrams $D$ and $D'$ are regularly isotopic if there is a finite sequence of the moves R2 and R3 taking $D$ to $D'$ up to planar isotopy?.

Remark

Both isotopy and and regular isotopy define equivalence relations on link diagrams.

Remark

There are a number of variants of the R3 move. All of them can be obtained as a finite sequence of the Reidemeister moves as defined above.

Reidemeister’s theorem

The following crucial theorem establishes that the Reidemeister moves are of central significance in knot theory.

Theorem

Let $L_{1}$ and $L_{2}$ be links, and let $D_{1}$ and $D_{2}$ be link diagrams of $L_{1}$ and $L_{2}$ respectively. Then $D_{1}$ and $D_{2}$ are equivalent if and only if $L_{1}$ is isotopic to $L_{2}$.

The key idea of the proof is that of subdivision. Working piecewise-linearly, one can reduce to the consideration of a few ‘minimal’ moves on knot diagrams, which one shows can be obtained as a finite sequence of Reidemeister moves. See for example section 2.1 of Kauffman.

Consequences of Reidemeister’s theorem

Because of Theorem , we can use the Reidemeister moves to verify invariance of a potential link invariant. For instance, as none of the moves alters the number of components of a link diagram, the number of components is an isotopy invariant (which is not at all surprising!). We can conclude that the Hopf link and the Borromean rings are not isotopic.

A slightly deeper observation is that the linking number is a link invariant, for which see that entry.

3-colourability is a knot invariant as is easy to check. This shows, for instance, that the trefoil knot is not equivalent to the unknot, and hence shows, very simply, that non-trivial knots exist. It also shows that the figure eight knot is not equivalent to the trefoil.

References

The original source is the following.

• K. Reidemeister, Knotentheorie, Ergebnisse der Mathematik (1932)

Simple examples of invariants under the Reidemeister Moves are given in, for instance, the following.

• Nick Gilbert, Tim Porter, Knots and surfaces, Oxford University Press (1994) [ISBN:9780198514909]

• Louis H. Kauffman, Knot diagrammatics, Handbook of knot theory, Elsevier, 233 - 318, 2005 arXiv:math/0410329

Most texts on Knot Theory contain discussions of the Reidemeister Moves.

category: knot theory