\section{Braids}

\definition{}
A \textit{braid} is a set of $n$ strings with fixed ends. Two braids are equivalent if they may be deformed into each other without disconnecting the strings. \par
Two braids are shown below.

\begin{center}
\begin{tikzpicture}
	\pic[rotate=90, name prefix=braid] {
		braid = {s_2^{-1} s_1^{-1} s_2 s_2 s_3^{-1} }
	};

	\braidbars{4}
\end{tikzpicture}
\hfill
\begin{tikzpicture}
	\pic[rotate=90, name prefix=braid] {
		braid = {s_2^{-1} s_3 s_2 s_1 s_1^{-1} s_2^{-1} s_3^{-1} s_1^{-1} s_2 s_2 s_3^{-1} }
	};
	\braidbars{4}
\end{tikzpicture}
\end{center}

\problem{}
Convince yourself that the braids above are equivalent.

\vfill
\pagebreak

\definition{}
A braid can be \textit{closed} by conecting its ends:

\begin{center}
\begin{tikzpicture}
	\pic[rotate=90, name prefix=braid] {
		braid = {s_2^{-1} s_1^{-1} s_2 s_2 s_3^{-1} }
	};

	\closebraid{4}
	\widebraidbars{4}
\end{tikzpicture}
\end{center}

\problem{}
When will a closed braid form a knot? \par
When will a closed braid form a link?

\vfill

\problem{}
Draw a braid that creates a $3$-unlink when closed.

\vfill
\pagebreak

\problem{}<braidify>
Draw the following knots as closed braids.

\begin{center}
	\hfill
	\begin{minipage}[t]{0.13\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/trefoil.png}
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.15\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/closed braid a.png}
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.15\textwidth}
	\begin{center}
		\includegraphics[width=\linewidth]{images/closed braid b.png}
	\end{center}
	\end{minipage}
	\hfill~
\end{center}

\vfill
\pagebreak

\problem{}
We can describe the projection of a braid by listing which strings cross over and under each other as we move along the braid. \par

\vspace{2mm}

For example, consider a three-string braid. If the first string crosses over the second, we'll call that a $1$ crossing. If first string crosses \textbf{under} the second, we'll call that a $-1$ crossing.

\begin{center}
	\hfill
	\begin{minipage}[t]{0.2\textwidth}
	\begin{center}
		\begin{tikzpicture}
			\pic[
				name prefix = braid,
				braid/number of strands = 3
			] {
				braid = {s_1}
			};
		\end{tikzpicture} \par
		\texttt{1} crossing
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.2\textwidth}
	\begin{center}
		\begin{tikzpicture}
			\pic[
				name prefix = braid,
				braid/number of strands = 3
			] {
				braid = {s_1^{-1}}
			};
		\end{tikzpicture} \par
		\texttt{-1} crossing
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.2\textwidth}
	\begin{center}
		\begin{tikzpicture}
			\pic[
				name prefix = braid,
				braid/number of strands = 3
			] {
				braid = {s_2}
			};
		\end{tikzpicture} \par
		\texttt{2} crossing
	\end{center}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{0.2\textwidth}
	\begin{center}
		\begin{tikzpicture}
			\pic[
				name prefix = braid,
				braid/number of strands = 3
			] {
				braid = {s_2^{-2}}
			};
		\end{tikzpicture} \par
		\texttt{-2} crossing
	\end{center}
	\end{minipage}
	\hfill~
\end{center}

\problem{}
Verify that the following is a $[1, 2, 1, -2, 1, 2]$ braid. \par
Read the braid right to left, with the \textbf{bottom} string numbered $1$.

\begin{center}
	\begin{tikzpicture}
		\pic[
			rotate = 90,
			name prefix = braid,
			braid/number of strands = 3
		] {
			% When we rotate a braid
			braid = {s_1 s_2 s_1 s_2^{-1} s_1 s_2}
		};
		\labelbraidstart{3}
	\end{tikzpicture}
\end{center}

\vfill

\problem{}
Draw the five-string braid defined by $[1, 3, 4, -3, 2, 4]$

\begin{solution}
	\begin{center}
	\begin{tikzpicture}
		\pic[
			rotate = 90,
			name prefix = braid,
			braid/number of strands = 5
		] {
			braid = {s_1 s_3 s_4 s_3^{-1} s_2 s_4}
		};

		\labelbraidstart{5}
		\labelbraidend{5}
	\end{tikzpicture}
	\end{center}
\end{solution}

\vfill
\pagebreak

\problem{}
Identify the knot generated by the 4-string braid $[(-1, 2, 3)^2,~(3)^3]$ \par
\hint{$[(1, 2)^2, 3] = [1, 2, 1, 2, 3]$}
\hint{This knot has 6 crossings. Use the knot table.}

\vfill

\problem{}
Show that the closure of the $n$-string braid $[(1, 2, ..., n-1)^m]$ is a knot iff $m$ and $n$ are coprime.

\vfill
\pagebreak