Added braids

Advanced/Knots/parts/4 braids.tex

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+\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