Added knot draft

Advanced/Knots/main.tex

@@ -0,0 +1,343 @@
+% use [nosolutions] flag to hide solutions.
+% use [solutions] flag to show solutions.
+\documentclass[
+	solutions,
+	singlenumbering
+]{../../resources/ormc_handout}
+
+\usepackage{ifthen}
+\usetikzlibrary{
+	knots,
+	hobby,
+	decorations.pathreplacing,
+	shapes.geometric,
+	calc
+}
+
+\newif{\ifShowKnots}
+\ShowKnotsfalse
+%\ShowKnotstrue
+
+% Knot debugging.
+% Set to true to show knot info
+\newif{\ifDebugKnot}
+\DebugKnottrue
+\DebugKnotfalse
+
+\ifDebugKnot
+	\tikzset{
+		knot diagram/draft mode = crossings,
+		knot diagram/only when rendering/.style = {
+			show curve endpoints,
+			%show curve controls
+		}
+	}
+\fi
+
+
+% From "Why knot" by
+%
+% Create largest crossing number with cord
+% Human knot number: how many humans do you need to make the knot?
+% Human knot number for trefoil composition?
+%	(looks like a wrap around center string)
+%
+% Figure-8 knot: mirror without letting go
+
+
+\tikzset{
+	knot diagram/every strand/.append style={
+		line width = 0.8mm,
+		black
+	},
+	show curve controls/.style={
+		postaction=decorate,
+		decoration={
+			show path construction,
+			curveto code={
+				\draw[blue, dashed]
+					(\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
+					node [at end, draw, solid, red, inner sep=2pt]{}
+				;
+
+				\draw[blue, dashed]
+					(\tikzinputsegmentsupportb) -- (\tikzinputsegmentlast)
+					node [at start, draw, solid, red, inner sep=2pt]{}
+					node [at end, fill, red, ellipse, inner sep=2pt]{}
+				;
+			}
+		}
+	},
+	show curve endpoints/.style={
+		postaction=decorate,
+		decoration={
+			show path construction,
+			curveto code={
+				\node [fill, blue, ellipse, inner sep=2pt] at (\tikzinputsegmentlast) {};
+			}
+		}
+	}
+}
+
+%\usepackage{lua-visual-debug}
+
+\begin{document}
+
+	\maketitle
+		<Advanced 2>
+		<Spring 2023>
+		{Knots}
+		{
+			Prepared by Mark on \today
+		}
+
+	\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), scale = 0.8]
+			\begin{knot}
+				\strand
+					(0,2) .. controls +(1.5,0) and +(1.5,0) ..
+					(0, 0) .. controls +(-1.5,0) and +(-1.5,0) ..
+					(0,2);
+			\end{knot}
+
+			\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 distinct knots with only one crossing. \par
+	Show that no nontrivial knot can have has fewer than three crossings. \par
+	\hint{There are 4 such knots. What are they?}
+
+	\begin{center}
+		\includegraphics[width=0.8\linewidth]{images/one crossing.png}
+	\end{center}
+
+	\begin{solution}
+		Draw all four. 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.
+
+	\vfill
+	\pagebreak
+
+
+	\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. 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
+
+	\input{parts/table}
+
+
+\end{document}

Advanced/Knots/parts/table.tex

@@ -0,0 +1,50 @@
+\section{Table of Prime Knots}
+This table contains the 15 smallest prime knots, ordered by crossing number. \par
+Mirror images are not accounted for, even if the mirror image produces a nonisomorphic knot.
+
+\vspace{5mm}
+
+% Images are from the appendix of the Knot book.
+
+\vfill
+
+{
+	\def\w{25mm}
+	\foreach \l/\c/\r in {%
+		{3_1}/{4_1}/{5_1},%
+		{5_2}/{6_1}/{6_2},%
+		{6_3}/{7_1}/{7_2},%
+		{7_3}/{7_4}/{7_5},%
+		{7_6}/{7_7}/{8_1}%
+	}{
+		\hfill
+		\begin{minipage}{\w}
+		\begin{center}
+			\includegraphics[width=\linewidth]{knot table/\l.png} \par
+			\vspace{2mm}
+			{\huge $\l$}
+		\end{center}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{\w}
+		\begin{center}
+			\includegraphics[width=\linewidth]{knot table/\c.png} \par
+			\vspace{2mm}
+			{\huge $\c$}
+		\end{center}
+		\end{minipage}
+		\hfill
+		\begin{minipage}{\w}
+		\begin{center}
+			\includegraphics[width=\linewidth]{knot table/\r.png} \par
+			\vspace{2mm}
+			{\huge $\r$}
+		\end{center}
+		\end{minipage}
+		\hfill~\par
+		\vspace{4mm}
+	}
+}
+
+\vfill
+\pagebreak