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\begin{document}

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\section*{Reidemeister move}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{knot_theory}{}\paragraph*{{Knot theory}}\label{knot_theory}

[[!include knot theory - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{reidemeisters_theorem}{Reidemeister's theorem}\dotfill \pageref*{reidemeisters_theorem} \linebreak
\noindent\hyperlink{consequences_of_reidemeisters_theorem}{Consequences of Reidemeister's theorem}\dotfill \pageref*{consequences_of_reidemeisters_theorem} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Reidemeister moves} are local changes to [[link diagrams]] which can be realised as [[isotopy|isotopies]] between the link diagrams from which they arise. Because of Reidemeister's theorem, Theorem \ref{TheoremReidemeister}, 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 \hyperlink{Reidemeister}{Reidemeister}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{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]].

$\backslash$begin\{defn\} The \emph{first Reidemeister move}, or \emph{R1 move}, allows either of the following replacements of a fragment of a link diagram to be made.

[[!include Reidemeister move 1 - SVG]]

$\backslash$end\{defn\}

$\backslash$begin\{defn\} The \emph{second Reidemeister move}, or \emph{R2 move}, allows either of the following replacements of a fragment of a link diagram to be made.

[[!include Reidemeister move 2 - SVG]]

$\backslash$end\{defn\}

$\backslash$begin\{defn\} The \emph{third Reidemeister move}, or \emph{R3 move}, allows the following replacement of a fragment of a link diagram to be made.

[[!include Reidemeister move 3 - SVG]]

$\backslash$end\{defn\}

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

$\backslash$begin\{defn\} A pair of link diagrams $D$ and $D'$ are \emph{regularly isotopic} if there is a finite sequence of the moves R2 and R3 taking $D$ to $D'$ up to [[planar isotopy]]. $\backslash$end\{defn\}

$\backslash$begin\{rmk\} Both isotopy and and regular isotopy define equivalence relations on link diagrams. $\backslash$end\{rmk\}

$\backslash$begin\{rmk\} 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. $\backslash$end\{rmk\}

\hypertarget{reidemeisters_theorem}{}\subsection*{{Reidemeister's theorem}}\label{reidemeisters_theorem}

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

$\backslash$begin\{thm\} $\backslash$label\{TheoremReidemeister\} 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 [[isotopy|isotopic]] to $L_{2}$. $\backslash$end\{thm\}

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 \hyperlink{KauffmanKnotDiagrammatics}{Kauffman}.

\hypertarget{consequences_of_reidemeisters_theorem}{}\subsection*{{Consequences of Reidemeister's theorem}}\label{consequences_of_reidemeisters_theorem}

Because of Theorem \ref{TheoremReidemeister}, 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.

[[colourable knot|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.

\hypertarget{references}{}\subsection*{{References}}\label{references}

The original source is the following.

\begin{itemize}%
\item [[K. Reidemeister]], \emph{Knotentheorie,} Ergebnisse der Mathematik (1932)

\end{itemize}
Simple examples of invariants under the Reidemeister Moves are given in, for instance, the following.

\begin{itemize}%
\item [[N. D. Gilbert]] and [[T. Porter]], \emph{Knots and Surfaces}, Oxford U.P., 1994.


\item [[Louis Kauffman | Louis H. Kauffman]], \emph{Knot diagrammatics}, Handbook of knot theory, Elsevier, 233 - 318, 2005 \href{https://arxiv.org/abs/math/0410329}{arXiv:math/0410329}



\end{itemize}
Most texts on [[Knot Theory]] contain discussions of the Reidemeister Moves.

category: knot theory

[[!redirects Reidemeister moves]] [[!redirects Reidemeister's theorem]] [[!redirects first Reidemeister move]] [[!redirects R1 move]] [[!redirects second Reidemeister move]] [[!redirects R2 move]] [[!redirects third Reidemeister move]] [[!redirects R3 move]]



\end{document}