\section{Introduction}

\definition{}
To form a \textit{knot}, take a string, tie a knot, then join the ends. \par
You can also think of a knot as a path in three-dimensional space that doesn't intersect itself:

\vspace{2mm}

\begin{center}
	\begin{minipage}[t]{0.3\textwidth}
	\begin{center}
		\begin{tikzpicture}[scale = 0.8, baseline=(p)]
		\begin{knot}
			\strand
				(1,2) .. controls +(-45:1) and +(1,0) ..
				(0, 0) .. controls +(-1,0) and +(-90 -45:1) ..
				(-1,2);
		\end{knot}

		\coordinate (p) at (current bounding box.center);
		\end{tikzpicture}
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.3\textwidth}
	\begin{center}
		\begin{tikzpicture}[scale = 0.8, baseline=(p)]

		% Knot is stupid and includes invisible handles in the tikz bounding box. This line crops the image to fix that.
		\clip (-2,-1.7) rectangle + (4, 4);

		\begin{knot}[
			consider self intersections=true,
			flip crossing = 2,
		]
			\strand
				(1,2) .. controls +(-45:1) and +(120:-2.2) ..
				(210:2) .. controls +(120:2.2) and +(60:2.2) ..
				(-30:2) .. controls +(60:-2.2) and +(-90 -45:1) ..
				(-1,2);
		\end{knot}

		\coordinate (p) at (current bounding box.center);


		\end{tikzpicture}
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.3\textwidth}
		\begin{center}
			\begin{tikzpicture}[scale = 0.8, baseline=(p)]

			\clip (-2,-1.7) rectangle + (4, 4);


			\begin{knot}[
				consider self intersections=true,
				flip crossing = 2,
			]
				\strand
					(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
					(210:2) .. controls +(120:2.2) and +(60:2.2) ..
					(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
					(0,2);
			\end{knot}

			\coordinate (p) at (current bounding box.center);

			\end{tikzpicture}
		\end{center}
		\end{minipage}
\end{center}

If two knots may be deformed into each other without cutting, we say they are \textit{isomorphic}. \par
If two knots are isomorphic, they are essentially the same knot.

\definition{}
The simplest knot is the \textit{unknot}. It is show below on the left. \par
The simplest nontrivial knot is the \textit{trefoil} knot, shown to the right.

\begin{center}
	\begin{minipage}[t]{0.48\textwidth}
	\begin{center}
		\begin{tikzpicture}[baseline=(p)]

		\draw[circle] (0,0) circle (1);

		\coordinate (p) at (current bounding box.center);
		\end{tikzpicture}
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.48\textwidth}
	\begin{center}
	\begin{tikzpicture}[baseline=(p), scale = 0.8]
		\clip (-2,-1.7) rectangle + (4, 4);

		\begin{knot}[
			consider self intersections=true,
			flip crossing = 2,
		]
			\strand
				(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
				(210:2) .. controls +(120:2.2) and +(60:2.2) ..
				(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
				(0,2);
		\end{knot}
		\coordinate (p) at (current bounding box.center);

		\end{tikzpicture}
	\end{center}
	\end{minipage}
\end{center}

\vfill
\pagebreak

\problem{}
Below are the only four knots with one crossing. \par
Show that every nontrivial knot more than two crossings. \par
\hint{There are four knots with two crossings. What are they?}

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

\begin{solution}
	Draw them all. Each is isomorphic to the unknot.
\end{solution}

\vfill

\problem{}
Show that this is the unknot. \par
A wire or an extension cord may help.

\begin{center}
	\includegraphics[width=0.35\linewidth]{images/big unknot.png}
\end{center}


\definition{}
As we said before, there are many ways to draw the same knot. \par
We call each drawing a \textit{projection}. Below are four projections of the \textit{figure-eight knot}.


\vspace{2mm}
\begin{center}
	\includegraphics[width=0.8\linewidth]{images/figure eight.png}
\end{center}
\vspace{2mm}

\problem{}
Convince yourself that these are equivalent. \par
Try to deform them into each other with a cord!

\vfill
\pagebreak