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\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 \ No newline at end of file diff --git a/Advanced/Knots/tikzset.tex b/Advanced/Knots/tikzset.tex index 3b65f8f..aeabf4a 100644 --- a/Advanced/Knots/tikzset.tex +++ b/Advanced/Knots/tikzset.tex @@ -1,6 +1,7 @@ \usetikzlibrary{ knots, hobby, + braids, decorations.pathreplacing, shapes.geometric, calc @@ -50,5 +51,56 @@ \node [fill, blue, ellipse, inner sep=2pt] at (\tikzinputsegmentlast) {}; } } + }, + braid/gap = 0.2, + braid/width = 5mm, + braid/height = -8mm, + braid/control factor = 0.75, + braid/nudge factor = 0.05, + braid/every strand/.style = { + line width = 0.7mm + }, + bar/.style = { + % Should match braid line width + line width = 0.7mm, + fill = black + } +} + +% Braid bar macro +% Argument: number of strands +\newcommand{\braidbars}[1]{ + \draw[bar] ([xshift=-0.7mm]braid-1-s) rectangle (braid-#1-s); + \draw[bar] ([xshift=-0.7mm]braid-rev-1-e) rectangle (braid-rev-#1-e); +} +\newcommand{\widebraidbars}[1]{ + \draw[bar] ([xshift=-0.7mm,yshift=-2mm]braid-1-s) rectangle ([yshift=2mm]braid-#1-s); + \draw[bar] ([xshift=-0.7mm,yshift=-2mm]braid-rev-1-e) rectangle ([yshift=2mm]braid-rev-#1-e); +} + +% Closed braid loops +% Argument: number of strands +\newcommand{\closebraid}[1]{ + \foreach \x in {1, ..., #1} { + \draw[braid/every strand, rounded corners = 4mm] + (braid-\x-s) -- + ([shift=(west:5*\x mm)]braid-\x-s) -- + ([shift=(west:5*\x mm),shift=(south:10*\x mm)]braid-\x-s) -- + ([shift=(east:5*\x mm),shift=(south:10*\x mm)]braid-rev-\x-e) -- + ([shift=(east:5*\x mm)]braid-rev-\x-e) -- + (braid-rev-\x-e) + ; + } +} + +\newcommand{\labelbraidstart}[1]{ + \foreach \x in {1, ..., #1} { + \node at ([xshift=-2mm]braid-\x-s) {$\x$}; + } +} + +\newcommand{\labelbraidend}[1]{ + \foreach \x in {1, ..., #1} { + \node at ([xshift=2mm]braid-\x-e) {$\x$}; } } \ No newline at end of file

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