On link diagrams that are minimal with respect to Reidemeister moves I and II
Kishin Sasaki
In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram $D$ of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to $D$ is unique. The proof also shows that the number of crossings of such a minimal diagram is unique for any diagram of any link. As the unknot admits infinitely many non-trivial minimal diagrams, we see that every link has infinitely many minimal diagrams, by considering the connected sums with such diagrams. We show that for a link without a trivial split component, an arbitrary Reidemeister move III either does not change the associated minimal diagram or can be reduced to a special type of a move up to Reidemeister moves I and II.
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\title[Minimal link diagrams with respect to Reidemeister moves I and II]{On link diagrams that are minimal with respect to Reidemeister moves I and II}
\author{Kishin Sasaki}
\address{Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.}



\begin{document}
\maketitle
\begin{abstract}
In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram $D$ of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to $D$ is unique. 
The proof also shows that the number of crossings of such a minimal diagram is unique for any diagram of any link.
As the unknot admits infinitely many non-trivial minimal diagrams,
we see that every link has infinitely many minimal diagrams, by considering the 
connected sums with such diagrams. 
We show that for a link without a trivial split component, an arbitrary Reidemeister move III either does not change the associated minimal diagram or can be reduced to a special type of a move up to Reidemeister moves I and II.
\end{abstract}


\begin{comment}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnote[0]{2010 Mathematics Subject Classification. Primary 57M25; Secondary 57M99.}
\footnote[0]{Key Words and Phrases. knot theory, Reidemeister moves.}
\renewcommand{\thefootnote}{\arabic{footnote}}
\end{comment}



\section{Introduction}
It is well known that every pair of diagrams of a given link can be transformed to each other by applying finitely many Reidemeister moves. Furthermore, any two diagrams of links that are transformed to each other by finitely many Reidemeister moves represent equivalent links. For details, see \cite{1} for example. There have been a lot of studies about the Reidemeister moves (see, for example, \cite{2,3,4,5,6}).
In \cite{2}, independence of the Reidemeister moves is studied.
In \cite{3}, an upper bound for the number of Reidemeister moves required to pass between two diagrams of the same link is studied. 
In \cite{4}, an upper bound for the number of Reidemeister moves which turn a diagram of a split link into a disconnected one is studied. 
In \cite{5}, a distance for diagrams of a knot is studied.
In \cite{6}, new moves are constructed from the Reidemeister moves. 

In this paper, we consider smooth unoriented link diagrams in $\mathbb{R}^{2}$ or $S^{2}$. In Section~\ref{preliminaries}, we prepare some terminologies and notions necessary for later sections. Reidemeister moves I and II change the number of crossings, while a Redemeister move III does not change the number of crossings. From this viewpoint, in Section~\ref{section3}, we say that a link diagram is minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings, and show that for an arbitrary diagram $D$ of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to $D$ is unique (Theorem~\ref{theorem1}). Furthermore, the proof also shows that the number of crossings of such a minimal diagram is unique for a diagram of an arbitrary link possibly with a trivial split component (Corollary~\ref{minimal_crossing_number}). The idea of this minimality seems to have appeared first in Theorem 2.2. of \cite{7}, where the unique of the minimality without crossings' informations is proved. As far as the author knows, the unique of the minimality with crossings' informations has not been studied, which is rather surprising.  



By studying the Reidemeister move III from the viewpoint of minimal diagram change, in Section~\ref{section4},
we show that for a link without a trivial split component, an arbitrary Reidemeister move III either does not change the associated minimal diagram or can be reduced to a special type of a move up to Reidemeister moves I and II (Theorem~\ref{theorem2}). As a corollary, we will see that, for every RI-II equivalence class (see Definition~\ref{RI-II equivalence}) of a link without a trivial component, the set of certain RI-II equivalence classes adjacent to the original RI-II equivalence class is the set of RI-II equivalence classes obtained by applying a Reidemeister move III to the minimal diagram in the original RI-II equivalence class. See Theorem~\ref{theorem2} and Corollary~\ref{corollary2} for the details. These results enhance the utility of the minimal diagrams.





\section{Preliminaries}
\label{preliminaries}
In this section, we give definitions and a remark which will be used in Sections~\ref{section3}~and~\ref{section4}. 


\begin{Definition}
\emph{Reidemeister moves} are defined as the local moves of link diagrams as depicted in Figure~\ref{Reidemeister moves}~(1). The moves RI, RII, RIII and RIII* depicted in Figure~\ref{Reidemeister moves}~(1) are called Reidemeister moves I, II, III and III*, respectively. The moves RIII and RIII* can be distinguished by using the orientation of $\mathbb{R}^{2}$ (or $S^{2}$). We need to distinguish the moves RIII and RIII* in this paper, for example, for Theorem~\ref{theorem2}. 
\end{Definition}

The following theorem is well known.

\begin{Theorem}[Reidemeister \cite{1}, 1926]
Every pair of diagrams of a link can be transformed to each other by applying finitely many Reidemeister moves. Furthermore, any two diagrams of links which are transformed to each other by applying finitely many Reidemeister moves represent equivalent links.    
\end{Theorem}


\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{RI,RII,RIII,RI-,RII-.pdf}
\caption{Reidemeister moves}
\label{Reidemeister moves}
\end{figure}

\begin{Definition}
The moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ are defined to be the Reidemeister moves I and II which decrease the number of crossings, respectively. The moves $\mathrm{RI}_+$ and $\mathrm{RII}_+$ are defined to be the Reidemeister moves I and II which increase the number of crossings, respectively. See Figure~\ref{Reidemeister moves}~(2) for details. 
\end{Definition}


\begin{Remark}
\label{remark}
The Reidemeister moves I, II, III (and III*) are related with a monogon, a special bigon, and a special triangle, respectively, which appear in the local disks on the right hand sides of Figure~\ref{Reidemeister moves}~(1) (the triangle also appears in the local disk on the left hand sides for the Reidemeister moves III (and III*)). For a link diagram having a crossing which can be eliminated by applying a move $\mathrm{RI}_-$ or  $\mathrm{RII}_-$, the corresponding monogon or the corresponding special bigon is on a region adjacent to the crossing. See Figure~\ref{Reidemeister moves}~(2) for details. 
\label{notification}
\end{Remark}







\section{Minimal link diagrams with respect to Reidemeister moves I and II}
\label{section3}
In this section, for every diagram of a link without a trivial split component, we prove that all diagrams obtained by applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ until they cannot be applied are equivalent (or ambient isotopic in $\mathbb{R}^2$ or $S^2$). We also show that for a diagram of an arbitrary link, the number of crossings of such a minimal diagram is uniquely determined. 

\begin{Definition}
\label{minimal}
A link diagram is said to be \emph{minimal} if no Reidemeister move $\mathrm{RI}_-$ or $\mathrm{RII}_-$ can be applied to it.
\end{Definition}

\begin{Theorem}\label{theorem1}
For an arbitrary diagram $D$ of a link without a split unknot component, all minimal diagrams which are obtained by applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ to $D$ are equivalent. 
\end{Theorem}

\begin{proof}
Let $L$ be a link without a split unknot component. From a given diagram $D$ of this link $L$, we clearly obtain a minimal diagram
$D_{1}$ by applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$. Note that this minimal diagram $D_{1}$ may depend on the sequence of the moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ applied to the diagram $D$. The minimal diagram $D_{1}$ clearly satisfies condition (\#) below.

\medskip

(\#)\quad All minimal diagrams obtained by applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ to that diagram are equivalent.

\medskip

Let us prove that all diagrams which are obtained by applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ to the diagram $D$ satisfy condition (\#) by checking that every move RI and RII maintain condition (\#).

First, we prove that a move RI maintains condition (\#).

A move $\mathrm{RI}_-$ clearly maintains condition (\#) because of the definition of condition (\#).

Let us now prove the case of a move $\mathrm{RI}_+$. In other words, we prove that, in Figure~\ref{RI+}, the diagram on the right hand side satisfies condition (\#) if we assume that the diagram on the left hand side satisfies condition (\#).


\begin{figure}[h]
\centering
\includegraphics[width=.45\linewidth]{RI+.pdf}
\caption{Applying $\mathrm{RI}_+$}
\label{RI+}
\end{figure}

We consider a process of applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ to the diagram on the right hand side in Figure~\ref{RI+} until we get a minimal diagram. If this process does not eliminate the crossing (say $c_0$) inside the local disk, then the resulting diagram inside this local disk remains the same after this process, which is a contradiction. So this process necessarily eliminates the crossing $c_0$. If crossing(s) eliminated by the first move in this process are outside the local disk, then the same move corresponding to the same position can be applied to the diagram on the left hand side, and the resulting diagram still satisfies condition (\#). This maintains the relation between the diagrams on the left and on the right hand sides in terms of condition (\#). Let us now consider the first move (say $m_1$) in the process that eliminates the crossing $c_0$. Note that whenever the crossing $c_0$ is eliminated by the move $m_1$, the corresponding monogon or the corresponding special bigon is on a region $A$, $B$ or $C$ (see Remark~\ref{notification} for details).

When $m_1$ is a move $\mathrm{RI}_-$ and uses the region $A$ as the monogon for this move, the diagram on the right hand side which is obtained by applying the move $m_1$ is equivalent to the diagram on the left hand side which satisfies condition (\#). See Figure~\ref{RI+1} for details. So in this case, the diagram on the right hand side also satisfies condition (\#).


\begin{figure}[h]
\centering
\includegraphics[width=.45\linewidth]{RI+_proof_example1.pdf}
\caption{Applying $\mathrm{RI}_-$ to region A}
\label{RI+1}
\end{figure}


When $m_1$ is a move $\mathrm{RII}_-$ and uses the region $A$ as the special bigon for this move, the diagram on the right hand side which is obtained by applying the move $m_1$ is obtained by applying a move $\mathrm{RI}_-$ to the diagram on the left hand side which satisfies condition (\#). See Figure~\ref{RI+2} for details.
So in this case, the diagram on the right hand side also satisfies condition (\#).

When $m_1$ is a move $\mathrm{RI}_-$ and uses the region $C$ as the monogon for this move, the diagram on the right hand side also clearly satisfies condition (\#), since the diagram on the right hand side which is obtained by applying the move $m_1$ is equivalent to the diagram on the left hand side which satisfies condition (\#).

It is easy to see that the region $B$ cannot be the monogon associated with a move $\mathrm{RI}_-$ or the special bigon associated with a move $\mathrm{RII}_-$.



Thus, all the cases for a move RI have been checked, so every move RI maintains condition (\#).

\begin{figure}[h!]
\centering
\includegraphics[width=.45\linewidth]{RI+_proof_example2.pdf}
\caption{Applying $\mathrm{RII}_-$ to region A}
\label{RI+2}
\end{figure}

Let us now prove that a move RII maintains condition (\#). Let us check all cases for a move RII as we checked all the cases for a move RI above. A move $\mathrm{RII}_-$ clearly maintains condition (\#) because of the definition of condition (\#).

So let us now prove that a move $\mathrm{RII}_+$ maintains condition (\#).
See Figure~\ref{RII+} which depicts a move $\mathrm{RII}_+$ whose regions adjacent to these two crossings inside the local disk on the right hand side are indicated by alphabets, and we assume that the diagram on the left hand side satisfies condition (\#). 

\begin{figure}[h!]
\centering
\includegraphics[width=.45\linewidth]{RII+.pdf}
\caption{Applying $\mathrm{RII}_+$}
\label{RII+}
\end{figure}

We consider a process of applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ to the diagram on the right hand side in Figure~\ref{RII+} until we get a minimal diagram. If this process does not eliminate a crossing inside the local disk (the upside ( or downside) crossing is said to be $c_1$ (resp. $c_2$)), then the resulting diagram inside this local disk remains the same after this process, which is a contradiction. So this process necessarily eliminates a crossing $c_1$ or $c_2$. If crossing(s) eliminated by the first move in this process are outside the local disk, then the same move corresponding to the same position can be applied to the diagram on the left hand side and the resulting diagram still satisfies condition (\#). This maintains the relation between the diagrams on the left and on the right hand sides in terms of condition (\#). Let us now consider the first move (say $l_1$) in the process that eliminates a crossing $c_1$ or $c_2$.

When $l_1$ is a move $\mathrm{RI}_-$ and uses the region $A$ as the monogon for this move,
the diagram on the right hand side which is obtained by applying the move $l_1$ is transformed to the diagram on the left hand by applying a move RI, which maintains condition (\#) (we have proved this above). See Figure~\ref{RII+1} for details. So in this case, the diagram on the right hand side also satisfies condition (\#).

\begin{figure}[h!]
\centering
\includegraphics[width=.45\linewidth]{RII+_proof_example1.pdf}
\caption{Applying $\mathrm{RI}_-$ to region A}
\label{RII+1}
\end{figure}

The proof of when $l_1$ is a move $\mathrm{RI}_-$ and uses the region $E$ as the monogon for this move is the same of the above proof. 

When $l_1$ is a move $\mathrm{RII}_-$ and uses the region $A$ as the special bigon for this move, the diagram on the right hand side which is obtained by applying the move $l_1$ is equivalent to the diagram on the left hand side which satisfies condition (\#). See Figure~\ref{RII+2} for details. So in this case, the diagram on the right hand side also satisfies condition (\#).

\begin{figure}[h]
\centering
\includegraphics[width=.45\linewidth]{RII+_proof_example2.pdf}
\caption{Applying $\mathrm{RII}_-$ to region $A$}
\label{RII+2}
\end{figure}

The proof of when $l_1$ is a move $\mathrm{RII}_-$ and uses the region $E$ as the special bigon for this move is the same of the above proof.

When $l_1$ is a move $\mathrm{RII}_-$ and uses the region $C$ as the special bigon for this move, the diagram on the right hand side also clearly satisfies condition (\#).

This link $L$ is without a split trivial component, so the case that $l_1$ is a move $\mathrm{RII}_-$ and uses the region $B$ or $C$ as the special bigon for this move cannot occur.

Thus, all the cases for a move RII have been checked. Hence, every move RII maintains condition (\#). We have proved that every move RI and RII maintains condition (\#) above, which completes the proof of Theorem~\ref{theorem1}
\end{proof}

\begin{Example}

Figure~\ref{counter_example} depicts a typical example of a diagram of a link with a trivial split component, which has inequivalent minimal diagrams obtained by applying finitely many moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$.

\begin{figure}
\centering
\includegraphics[width=.6\linewidth,keepaspectratio]{counter_example.pdf}
\caption{A counter example for link-version Theorem~\ref{theorem1}}
\label{counter_example}
\end{figure}

\end{Example}

\begin{Corollary}
\label{minimal_crossing_number}
For every diagram $D$ of a link possibly with trivial split components, the numbers of crossings of all minimal diagrams which are obtained by applying finitely many moves RI and RII to $D$ are the same.
 
\end{Corollary}

\begin{proof}
By Theorem~\ref{theorem1}, for every diagram of a link without a trivial split component, the number of crossings of the minimal diagram is clearly unique. For every diagram of a link with trivial split components, by the proof of Theorem~\ref{theorem1}, all the differences in the minimal diagrams are the locations of trivial split components which contain no crossing. So the numbers of crossings of the minimal diagrams are the same. This completes the proof.
\end{proof}

\begin{Remark}
Figure~\ref{minimal_diagram} depicts an example of a minimal diagram of the trivial knot. It is easy to check that this diagram is minimal, since there is no monogon and no special bigon in this diagram. By considering connected sums of copies of this diagram and a minimal diagram, we see that an arbitrary link has infinitely many minimal diagrams.

\begin{figure}
\includegraphics[width=.3\linewidth,keepaspectratio]{minimal_diagram.pdf}
\caption{A minimal diagram of a trivial knot}
\label{minimal_diagram}
\end{figure}
\end{Remark}

\section{Reidemeister move III on minimal diagrams' relation}
\label{section4}

In the above sections, we have considered minimal diagrams by using only the Reidemeister moves I and II. Then we may ask what type of Reidemeister move III (and III*) changes the associated minimal diagram.

In this section, by studying the Reidemeister moves III and III* from the viewpoint of minimal diagram change, we construct special types of Reidemeister moves III and III* for an arbitrary link without a trivial split component. As a corollary, we can see that for every RI-II equivalence class (see Definition~\ref{RI-II equivalence}) of a link without a trivial component, the set of certain RI-II equivalence classes adjacent to the original RI-II equivalence class is the set of RI-II equivalence classes obtained by applying a Reidemeister move III or III* to the minimal diagram in the original RI-II equivalence class.


\begin{Definition}
\label{RI-II equivalence}
Two link diagrams are said to be \emph{RI-II equivalent} if these two diagrams can be transformed to each other by applying finitely many moves RI and RII, without using moves RIII or RIII*.
\end{Definition}




\begin{Definition}
Let $D$ be a diagram of an arbitrary link, $\mathcal{D}$ the RI-II equivalence class containing this diagram $D$. By Corollary~\ref{minimal_crossing_number}, the numbers of crossings of the minimal diagrams in the RI-II equivalence class $\mathcal{D}$ are the same. The number of crossings in the minimal diagrams in the RI-II equivalence class $\mathcal{D}$ is called the \emph{minimal crossing number} and is denoted by $c(\mathcal{D})$.
\end{Definition}


\begin{Definition}
We say that two RI-II equivalence classes are \emph{adjacent} if a diagram exists in each of these two RI-II equivalence classes such that they are transformed to each other by applying a single move RIII or RIII*. Moreover, we say that a RI-II equivalence class $\mathcal{D}_{1}$ is
$(-)$-adjacent (or $(+)$-adjacent) to a RI-II equivalence class $\mathcal{D}_{2}$ if they are adjacent and satisfy $c(\mathcal{D}_{1}) \leq c(\mathcal{D}_{2})$ (resp. $c(\mathcal{D}_{1}) \geq c(\mathcal{D}_{2})$).
\end{Definition}


\begin{Theorem}
\label{theorem2}
Assume that two diagrams $D_1$ and $D_2$ of a link without a trivial split component are transformed to each other by applying a single move RIII or RIII*. Then there exist two link diagrams $D_{1}'$ and $D_{2}'$ which are RI-II equivalent to $D_{1}$ and $D_{2}$, respectively, such that they are transformed to each other by applying exactly one of the local moves in Figures~\ref{RIII1}~and~\ref{RIII2}.
\end{Theorem}


\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{RIII_summary.pdf}
\caption{RIII's that may change the RI-II equivalence classes}
\label{RIII1}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{RIII_summary_upsidedown.pdf}
\caption{RIII*'s that may change the RI-II equivalence classes}
\label{RIII2}
\end{figure}


\begin{Remark}
The diagrams which are indicated as minimal in Figures~\ref{RIII1}~and~\ref{RIII2} are all minimal (see Definition~\ref{minimal}). Note that this minimality is not local, but global.
\end{Remark}

\begin{Remark}
Every RI-II equivalence class containing the diagram on the left hand side in Figures~\ref{RIII1}~and~\ref{RIII2} has a same or lower minimal crossing number compared with the RI-II equivalence class containing the corresponding diagram on the right hand side, since the diagrams on the right hand sides are all minimal and all the local moves do not change the number of crossings.
Moreover, moves (1)--(4) in Figures~\ref{RIII1}~and~\ref{RIII2} do change the associated minimal diagrams, since we can apply a move $\mathrm{RI}_-$ or $\mathrm{RII}_-$ to every diagram on the left hand side (which suggests that these RI-II equivalence classes have a strictly lower minimal crossing number). 
\end{Remark}


\begin{proof}[Proof of Theorem~\ref{theorem2}]
Let us first prove the case of a move RIII. Assume that the two diagrams $D_{1}$ and $D_{2}$ are transformed to each other by applying a single move RIII, not a move RIII*.
When we consider applying finitely many same moves $\mathrm{RI}_-$ and $\mathrm{RII}_-$ to the two diagrams $D_{1}$ and $D_{2}$ which change the corresponding positions outside the two local disks for the move RIII until a move $\mathrm{RI}$ or $\mathrm{RII}$ cannot be applied to the outside of the two local disks, we get a 'special RIII'. Note that only the regions adjacent to the two triangles for the special RIII can be the monogons or the special bigons. Figure~\ref{RIII3} depicts this special RIII, whose every region around the two triangles is indicated by one alphabet.


\begin{figure}
\centering
\includegraphics[width=.6\linewidth]{RIII_colored.pdf}
\caption{Special RIII}
\label{RIII3}
\end{figure}

Let us consider every region adjacent to the two triangles for the special RIII in Figure~\ref{RIII3} on whether it is the monogon or the special bigon or not. Assume now that every monogon and every special bigon in regions $A$--$L$ does cover a single region in the two local disks in Figure~\ref{RIII3}.

\bigskip

\textbf{In the case of there existing no monogone and no special bigon in all the regions adjacent to the two triangles in Figure~\ref{RIII3}}

The diagrams on the left and on the right hand sides in Figure~\ref{RIII3} are both minimal, so the special RIII is a move (5) in Figure~\ref{RIII1}.

\medskip

\textbf{In the case of there existing no special bigon in all the regions adjacent to the two triangles in Figure~\ref{RIII3}}


In this case, the region which can be the monogon is only region $C$ or $J$. 
This reason is explained below.

First, every region in regions $B$ and $D$ and $F$ and $G$ and $I$ and $K$ contains two or more crossings, so every these region cannot be the monogon.

Second, when a region $A$ or $E$ or $H$ or $L$ is the monogon, automatically a region becomes the special bigon. For instance, when region $A$ is the monogon, automatically region $G$ becomes the special bigon. The other cases are the same as this. So, every these case is not included in this case which is no special bigon in regions $A$--$L$.

Let us consider this case on the number of the monogons existing in regions $A$--$L$.

\medskip

\uline{CASE1: When the number of the monogons in all regions adjacent to the two triangles is one.}


When the monogon exists on region $C$, the diagram on the right hand side is minimal. Hence, the special RIII is a move (1) in Figure~\ref{RIII1}.

When the monogon exists on region $J$, the proof is the same of the above proof, since the position of region $J$ is the position of region $C$ in the local disk of the special RIII.


\medskip

\uline{CASE2: When the number of the monogons in all regions adjacent to the two triangles is two or more.}


When the monogons exist on regions $C$ and $J$, the special RIII can be expressed by applying finitely many moves RI and a move (6). See Figure~\ref{RIIIex1} for the details. Note that the diagrams at the lower left and the lower right in Figure~\ref{RIIIex1} are both minimal, since all the regions which can be seen in the two local disks of these two diagrams cannot be the monogons or the special bigons. 


\begin{figure}[h!]
\centering 
\includegraphics[width=.7\linewidth]{RIII_proof_example1.pdf}
\caption{The monogons on regions $C$ and $J$}
\label{RIIIex1}
\end{figure}


\bigskip

\textbf{In the case of there existing no monogon in all the regions adjacent to the two triangles in Figure~\ref{RIII3}} 

The region which can be the special bigon is a region $A$ or $C$ or $E$ or $H$ or $J$ or $L$. 
This reason is explained below.

First, the region $D$ or $I$ cannot be the special bigon, respectively, since the crossings that these regions contain have different up and down informations compared with the special bigon. 

Second, when a region $B$ or $F$ or $G$ or $K$ is the special bigon, a region automatically becomes the monogon. For example, when region $B$ is the special bigon, region $H$ automatically becomes the monogon. The other cases are same as this. So, every these case is not included in this case which is no monogon in regions $A$--$L$.  

Let us consider this case on the number of the special bigons existing in regions $A$--$J$.

\medskip

\uline{CASE1: When the number of the special bigons in all the regions adjacent to the two triangles is one.}

When the special bigon exists on region $A$, the diagram on the right hand side is minimal. Hence, the special RIII is a move (3) in Figure~\ref{RIII1}.

The proof of when the special bigon exists on a region $C$ or $E$ or $H$ or $J$ or $L$ is clearly the same as the above proof. The special RIII in every these case is a move (2) or (3) or (4) in Figure~\ref{RIII1}. 

\medskip

\uline{CASE2: When the number of the special bigons in all the regions adjacent to the two triangles is two.}

The region which can be the special bigon is a region $A$ or $C$ or $E$ or $H$ or $J$ or $L$, whose reason has been explained above. Let us consider the pairs of these regions being the special bigons for the proof of this case.



First cases that we consider in CASE2 are the pairs which exist on the diagram of the left or the right hand side (in other words, the pairs $A$$C$ and $A$$E$ and $C$$E$ and $H$$J$ and $J$$L$ and $H$$L$ being the special bigons ($6$ cases)). Note that in every these case, the diagram on the other hand side is minimal.

Second cases that we consider in CASE2 are the pairs which exist on the diagrams of both the right and the left hand sides (in other words, the pairs $A$$H$ and $A$$J$ and $A$$L$ and $C$$H$ and $C$$J$ and $C$$L$ and $E$$H$ and $E$$J$ and $E$$L$ being the special bigons ($9$ cases)).

Let us start the first cases. When the special bigons exist in the regions $A$$C$, the diagram on the right hand side is minimal, so the special RIII is a move (2) or (3) in Figure~\ref{RIII1}.
The proof of when the special bigons exist in the regions $A$$E$ or $C$$E$ or $H$$J$ or $H$$L$ or $J$$L$ is clearly the same as the above proof and every these special RIII is a move (2) or (3) or (4) in Figure~\ref{RIII1}. We have checked the $6$ cases above which we should consider in the first cases.

Let us now check the second cases. When the special bigons exist on the regions $A$$H$ or $E$$L$, every these case cannot occur because of up and down informations of a crossing. See Figure~\ref{RIIIex2}~and~Figure~\ref{RIII_proof_example5}, respectively, for details. In Figure~\ref{RIIIex2} (or \ref{RIII_proof_example5}), region $H$ (res. $L$) cannot be the special bigons when region $A$ (res. $E$) is the special bigon.


\begin{figure}
\centering
\includegraphics[width=.6\linewidth]{RIII_proof_example2.pdf}
\caption{The special bigon on region $A$, region $H$ has two crossings}
\label{RIIIex2}
\end{figure}

\begin{figure}
\includegraphics[width=.6\linewidth,keepaspectratio]{RIII_proof_example5.pdf}
\caption{The special bigon on region $E$, region $L$ has two crossings}
\label{RIII_proof_example5}
\end{figure}

When the special bigons exist on regions $A$$J$ or $C$$H$ or $C$$L$ or $E$$J$, every these special RIII can be expressed by applying finitely many moves RI and RII. See Figures~\ref{RIII_proof_example8}~and~\ref{RIII_proof_example7}~and~\ref{RIIIex3}~and~\ref{RIII_proof_example6} respectively, for details. So every these case is not included in Theorem~\ref{theorem2}.

\begin{figure}
\includegraphics[width=.5\linewidth]{RIII_proof_example8.pdf}
\caption{The special bigons on regions $A$$J$}
\label{RIII_proof_example8}
\end{figure}

\begin{figure}
\includegraphics[width=.5\linewidth,keepaspectratio]{RIII_proof_example7.pdf}
\caption{The special bigons on regions $C$$H$}
\label{RIII_proof_example7}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=.8\linewidth]{RIII_proof_example3.pdf}
\caption{The special bigons on the regions $C$$L$}
\label{RIIIex3}
\end{figure}

\begin{figure}
\includegraphics[width=.5\linewidth,keepaspectratio]{RIII_proof_example6.pdf}
\caption{The special bigons on regions $E$$J$}
\label{RIII_proof_example6}
\end{figure}

When the special bigons exist on the regions $A$$L$ or $C$$J$ or $E$$H$, every these special RIII can be expressed by applying finitely many moves RII and a single move RIII*. See Figure~\ref{RIII_proof_example9}~and~Figures~\ref{RIII_proof_example10}~and~\ref{RIII_proof_example11}, respectively, for details. So every these case results in a case of the move RIII*. Note that every these process which changes the move RIII into the move RIII* decreases the number of crossings. We deal with this kind of processes later.


\begin{figure}
\includegraphics[width=.6\linewidth,keepaspectratio]{RIII_proof_example9.pdf}
\caption{The special bigons on regions $A$$L$}
\label{RIII_proof_example9}
\end{figure}

\begin{figure}
\includegraphics[width=.6\linewidth,keepaspectratio]{RIII_proof_example10.pdf}
\caption{The special bigons on regions $C$$J$}
\label{RIII_proof_example10}
\end{figure}

\begin{figure}
\includegraphics[width=.5\linewidth,keepaspectratio]{RIII_proof_example11.pdf}
\caption{The special bigons on regions $E$$H$}
\label{RIII_proof_example11}
\end{figure}

We have checked the $9$ second cases above. All cases in CASE2 have been checked. Note here that in every these second case proof, we used only the information of the positions of the special bigons existing in regions $A$, $C$, $E$, $H$, $J$, $L$. We will use this fact in next Case3.

\medskip

\uline{CASE3: When the number of the special bigons on all regions adjacent to the two triangles is three or more.}

The region which can be the special bigon is a region $A$ or $C$ or $E$ or $H$ or $J$ or $L$, whose proof has been explained above. From here, we consider the combinations (the triples or more) of these regions being the special bigons for the proof of CASE3.

First cases which we consider in CASE3 are the combinations existing on the diagram of the left or the right hand side (in other words, the triples $A$$C$$E$ and $H$$J$$L$ being the special bigons). Note that the diagram on the other hand side is minimal.

Second cases which we consider in CASE3 are the combinations existing on the diagrams of both the left and the right hand sides. We can recall here that in CASE2, we used only the information of the position of the special bigons, which exist the diagrams on both the left and the right hand sides. Hence, the second cases result in CASE2. By this fact, we can see that in every these second case, it cannot occur or the special RIII can be expressed by applying finitely many moves RI and RII. See CASE2 for the details.





When the special bigons exist on the regions $A$$C$$E$, the diagram on the right hand side is minimal. So the special RIII is a move (2) or (3) or (4) in Figure~\ref{RIII1}.

The proof of when the special bigons exist on the regions $H$$J$$L$ is clearly the same of the above proof. The special RIII is a move (2) or (3) or (4) in Figure~\ref{RIII1}.

Above are the first cases that we should consider in CASE3. The second cases have already been checked above, so all cases in CASE3 have been checked.

\bigskip

\textbf{In the case of there existing both the monogon and the special bigon in all the regions adjacent to the two triangles in Figure~\ref{RIII3}.}

The region that the monogon can exist on is a region $A$ or $C$ or $E$ or $H$ or $J$ or $L$. We consider this case on which regions among the regions $A$ and $C$ and $E$ and $H$ and $J$ and $L$ being the monogons.

When the monogon exists on region $A$ (automatically region $G$ becomes the special bigon), the special RIII can be expressed by applying finitely many moves RI and RII. See Figure~\ref{RIIIex4} for the details. So this case is not included in Theorem~\ref{theorem2}. 

\begin{figure}[h]
\centering
\includegraphics[width=.7\linewidth]{RIII_proof_example4.pdf}
\caption{The monogon on region $A$ (the special bigon on region $G$)}
\label{RIIIex4}
\end{figure}

The proof of when the monogon exists on a region $E$ or $H$ or $L$  is the same as the above proof and these special RIII can be expressed by applying finitely many moves RI and RII. Hence, every these case is not included in Theorem~\ref{theorem2}.

By the above proofs, we can disregard a region $A$ or $E$ or $H$ or $L$ being the monogon. Hence, we regard only the region $C$ or $J$ being the monogon.

Assume now that the monogon exists on region $C$ and no monogon exists on another region in Figure~\ref{RIII3}. The special bigon cannot exists on a region $H$ or $L$ or $B$ or $D$ or $F$ in Figure~\ref{RIII3} since the numbers or up and down informations of the crossings that these regions contain are different from the ones of the special bigon. When the special bigon exists on the region $G$ or $K$, the special RIII can be expressed by applying finitely many moves RI and RII. See the case of Figure~\ref{RIIIex4}, for the details. So the region which can be the special bigon here is a region $A$ or $E$ or $J$. Let us consider the combinations of the regions $A$ and $E$ and $J$ being the special bigons. (in other words, $A$ and $E$ and $J$ and $A$$E$ and $A$$J$ and $E$$J$ and $A$$E$$J$) We can recall that when the regions $A$$J$ or $E$$J$ or $A$$E$$J$ are the special bigons, the special RIII can be expressed by applying finitely many moves RI and RII. See Figures~\ref{RIII_proof_example6}~and~\ref{RIII_proof_example8} for the details. So it is good to check the cases that the special bigons exist on the regions $A$ or $E$ or $J$ or $A$$E$ in the combinations. When the regions $A$ or $E$ or $A$$E$ are the special bigons, the diagrams on the right hand sides are all minimal, so every these special RIII
is a move (1) in Figure~\ref{RIII1}. When the special bigon exists on the region $J$, the special RIII can be expressed by applying a single RIII* and finitely many moves RI and RII. See Figure~\ref{RIII_proof_example12} for the details. This case result in a case of the move RIII*. Note that this process which change the move RIII into the move RIII* decreases the number of crossings.

\begin{figure}
\includegraphics[width=.6\linewidth,keepaspectratio]{RIII_proof_example12.pdf}
\caption{The monogon on region $C$, the special bigon on region $J$}
\label{RIII_proof_example12}
\end{figure}



The proof of when the region $J$ is the monogon and no another region is the monogon, is the same of the above proof, since the position of the region $J$ is the same position of the region $C$ in the special RIII.
 
The last one that we should check here is when both the regions $C$ and $J$ are the monogons and no another region is the monogon. Assume now that the regions $C$ and $J$ are the monogons and no another region is the monogon. Every another region (in other words, a region $A$ or $B$ or $E$ or $F$ or $G$ or $H$ or $K$ or $L$) cannot be the special bigon since number or up and down informations of the crossings that every these regions contain is different from the ones of the special bigon. So this case is not included in the case which is both the monogon and the bigon.

\medskip

Above are all the cases that every monogon and every special bigon in the regions $A$--$L$ does cover a single region in the two local disks in Figure~\ref{RIII3}. What we should check after that is the cases that the monogon and the special bigon in the regions $A$--$J$ that do not cover a single region in the two local disks in Figure~\ref{RIII3} exist. The monogon contains one crossing, so the monogon cannot cover two regions in the two local disks in Figure~\ref{RIII3}. By considering that the special bigon contains two crossings, what we should check is the case that regions $A$$C$ or $C$$E$ or $H$$J$ or $J$$L$ are the single special bigon. (Note here that regions $A$$E$ or $H$$L$ cannot be the single special bigon since up and down informations of the crossings that the regions $A$$E$ and $H$$L$ contain are different from the ones of the special bigon.) In fact, every these case cannot occur since in every these case, a trivial split component appears in the local disk. See Figure~\ref{RIII_proof_example13}, for instance.   
In Figure~\ref{RIII_proof_example13}, we can see the trivial component is split by considering that a $3$-ball that contains only the trivial component exists.

\begin{figure}
\centering
\includegraphics[width=.6\linewidth]{RIII_proof_example13.pdf}
\caption{The single bigon on regions $A$ and $C$}
\label{RIII_proof_example13}
\end{figure}

 
Above are all cases of the move RIII. Note that every process appeared above which changes the move RIII into the move RIII* decreases the number of crossings. 

\bigskip

Let us now prove the case of a move RIII* as the proof of the move RIII. The move RIII* depicted in Figure~\ref{RIII*} is obtained from a move RIII* by applying same finitely many moves $\mathrm{RI}_-$, $\mathrm{RII}_-$ to the outside of the two local disks for a move RIII* until a move $\mathrm{RI}$ or $\mathrm{RII}$ cannot be applied to the outside of the two local disks, whose every region adjacent to the two triangles is indicated by one alphabet. We say that this RIII* depicted in Figure~\ref{RIII*} is a 'special RIII*'. Note that Figure~\ref{RIII*} is the mirror image of Figure~\ref{RIII3} including alphabets.

\begin{figure}
\centering
\includegraphics[width=.6\linewidth]{RIII_upsidedown_colored.pdf}
\caption{Special RIII* - mirror image of Figure \ref{RIII3}}
\label{RIII*}
\end{figure}  

In fact, every case of the special RIII* corresponds to the mirror image of a case of the special RIII. For instance, Figure~\ref{RIII*proofex1} is the mirror image of Figure~\ref{RIIIex3}, and the same moves can be expressed by applying the moves RI and RII as in Figures~\ref{RIII*proofex1}~and~\ref{RIIIex3}. All RIII* cases can be proved as this, since moves RI and RII which change same positions also can be applied to the mirror images. Hence, the result of all cases of the move RIII* is the mirror image of Figure~\ref{RIII1}, in other words, Figure~\ref{RIII2}.

The processes which change the moves RIII* into the moves RIII also appear in the cases of the special RIII*, and necessarily decrease the number of crossings. Every these process which changes the move RIII into the move RIII* or the move RIII* into the move RIII decreases the number of crossings, so we can see that every these case results in the other cases, since the number of crossings of every link diagram is finite. This means that we can disregard these cases, which completes the proof.
\begin{figure}
\centering
\includegraphics[width=.7\linewidth]{RIII_upsidedown_proof_example.pdf}
\caption{Mirror image of Figure \ref{RIIIex3}}
\label{RIII*proofex1}
\end{figure}
\end{proof}




\begin{Corollary}
\label{corollary2}
Let $\mathcal{R}$ be an arbitrary RI-II equivalence class of a link without a trivial component. If we denote the set of all RI-II equivalence classes $(-)$-adjacent to the RI-II equivalence class $\mathcal{R}$ by $\textbf{R}$, and the set of RI-II equivalence classes containing all diagrams obtained by applying a Reidemeister move III or III* to the minimal diagram in the RI-II equivalence class $\mathcal{R}$ by $\textbf{P}$, then
$\textbf{R}$ $=$ $\textbf{P}$.
\end{Corollary}



\begin{proof}
To prove $\textbf{R}$ $\supset$ $\textbf{P}$ is a trivial matter. By Theorem~\ref{theorem2}, we can see that every RI-II equivalence class in the set $\textbf{R}$ is contained in the set $\textbf{P}$, which proves $\textbf{R}$ $\subset$ $\textbf{P}$.   
\end{proof}


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\bibitem{2}Z.~Cheng, H.~Gao: A note on the independence of Reidemeister moves,
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\bibitem{3}A.~Coward, M.~Lackenby: An upper bound on Reidemeister moves, Amer. J. Math. 136 (2014), no.~4, 1023--1066.
  

\bibitem{4}C.~Hayashi: The number of Reidemeister moves for splitting a link, Math. Ann. 332 (2005), no.~2, 239--252. 

\bibitem{7}M.~Khovanov:
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\bibitem{5}Y.~Miyazawa: A distance for diagrams of a knot, Topology Appl. 159 (2012), no.~4, 1122--1131.


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\end{thebibliography}

\end{document}
1612.00764.extracted_bib
{1G.~Burde, H.~Zieschang: Knots, Second edition, de Gruyter Studies in Mathematics 5, Walter de Gruyter \& Co, Berlin, 2003.
1612.00764.extracted_bib
{2Z.~Cheng, H.~Gao: A note on the independence of Reidemeister moves, J. Knot Theory Ramifications 21 (2012), no.~9, 1220001, 7 pp.
1612.00764.extracted_bib
{3A.~Coward, M.~Lackenby: An upper bound on Reidemeister moves, Amer. J. Math. 136 (2014), no.~4, 1023--1066.
1612.00764.extracted_bib
{4C.~Hayashi: The number of Reidemeister moves for splitting a link, Math. Ann. 332 (2005), no.~2, 239--252.
1612.00764.extracted_bib
{7M.~Khovanov: Doodle groups, Trans. Amer. Math. Soc. 349 (1997), no.~6, 2297--2315.
1612.00764.extracted_bib
{5Y.~Miyazawa: A distance for diagrams of a knot, Topology Appl. 159 (2012), no.~4, 1122--1131.
1612.00764.extracted_bib
{6C.~Petronio, A.~Zanellati: Algorithmic simplification of knot diagrams: new moves and experiments, J. Knot Theory Ramifications 25 (2016), no.~10, 1650059, 30~pp.