\section{Knot Composition}

Say we have two knots $A$ and $B$.
The knot $A \boxplus B$ is created by cutting $A$ and $B$ and joining their ends:

\begin{center}
	\hfill
	\begin{minipage}[t]{0.15\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/composition a.png}
		$A$
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.13\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/composition b.png}
		$B$
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.3\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/composition c.png}
		$A \boxplus B$
	\end{center}
	\end{minipage}
	\hfill~
\end{center}

We must be careful to avoid new crossings when composing knots:

\vspace{2mm}
\begin{center}
	\includegraphics[width=0.45\linewidth]{images/composition d.png}
\end{center}
\vspace{2mm}

We say a knot is \textit{composite} if it can be obtained by composing two other knots. \par
We say a knot is \textit{prime} otherwise.

\problem{}
For any knot $K$, what is $K \boxplus \text{unknot}$?

\vfill

\problem{}
Use a pencil or a cord to compose the figure-eight knot with itself.

\vfill

\vfill
\pagebreak{}

\problem{}
The following knots are composite. \par
What are their prime components? \par
Try to make them with a cord. \par
\hint{Use the table at the back of this handout to decompose the second knot.}

\begin{center}
	\hfill
	\includegraphics[height=30mm]{images/decompose a.png}
	\hfill
	\includegraphics[height=30mm]{images/decompose b.png}
	\hfill~\par
	\vspace{4mm}
\end{center}

\begin{solution}
	The first is easy, it's the trefoil composed with itself. \par

	\vspace{2mm}

	The second is knot $5_2$ composed with itself. \par
	Note that the \say{three-crossing figure eight} is another projection of $5_2$. \par
	The figure-eight knot is NOT a part of this composition. Look closely at its crossings.
\end{solution}

\vfill
\pagebreak

\definition{}
When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}.

\vspace{2mm}

An \textit{orientated knot} is created by defining a \say{direction of travel.} \par
There are two distinct ways to compose a pair of oriented knots:

\begin{center}
	\hfill
	\begin{minipage}[t]{0.25\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/orientation b.png}
		Matching orientation
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.25\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/orientation c.png}
		Inverse orientation
	\end{center}
	\end{minipage}
	\hfill~
\end{center}

In this example, both compositions happen to give the same result. This is because the trefoil knot is \textit{invertible}: its direction can be reversed by deforming it. This is not true in general, as you will soon see.

\problem{}
Invert a directed trefoil.

\vfill

\problem{}
The smallest non-invertible knot is $8_{17}$, shown below. \par
Compose $8_{17}$ with itself to obtain two different knots.

\begin{center}
	\includegraphics[height=30mm]{knot table/8_17.png} \par
	\vspace{2mm}
	{\large Knot $8_{17}$}
\end{center}

\begin{solution}
	\begin{center}
		\includegraphics[width=0.8\linewidth]{images/noninvertible.png}
	\end{center}
\end{solution}


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\pagebreak