Finished knot composition

Advanced/Knots/main.tex

@@ -5,37 +5,15 @@
 	singlenumbering
 ]{../../resources/ormc_handout}
 
-\usepackage{ifthen}
+% Set to true to show knot measurements.
-\usetikzlibrary{
+% Only used in tikzset.tex
-	knots,
-	hobby,
-	decorations.pathreplacing,
-	shapes.geometric,
-	calc
-}
-
-\newif{\ifShowKnots}
-\ShowKnotsfalse
-%\ShowKnotstrue
-
-% Knot debugging.
-% Set to true to show knot info
 \newif{\ifDebugKnot}
-\DebugKnottrue
+%\DebugKnottrue
 \DebugKnotfalse
-
+\input{tikzset.tex}
-\ifDebugKnot
-	\tikzset{
-		knot diagram/draft mode = crossings,
-		knot diagram/only when rendering/.style = {
-			show curve endpoints,
-			%show curve controls
-		}
-	}
-\fi
 
 
-% From "Why knot" by
+% Problems from "Why knot"
 %
 % Create largest crossing number with cord
 % Human knot number: how many humans do you need to make the knot?
@@ -44,45 +22,10 @@
 %
 % 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}
 
+\begin{document}
 	\maketitle
 		<Advanced 2>
 		<Spring 2023>
@@ -91,251 +34,9 @@
 			Prepared by Mark on \today
 		}
 
-	\section{Introduction}
 
-	\definition{}
+	\input{parts/0 intro.tex}
-	To form a \textit{knot}, take a string, tie a knot, then join the ends. \par
+	\input{parts/1 composition.tex}
-	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}
 

Advanced/Knots/parts/0 intro.tex

@@ -0,0 +1,163 @@
+\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

Advanced/Knots/parts/1 composition.tex

@@ -0,0 +1,133 @@
+\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
+
+\definition{}
+When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}.
+
+\vspace{2mm}
+
+An \textit{orientated knot} is created by defining a \say{direction of travel.} \par
+There are two distinct ways to compose a pair of oriented knots:
+
+\begin{center}
+	\hfill
+	\begin{minipage}[t]{0.25\textwidth}
+	\begin{center}
+		\includegraphics[width=\linewidth]{images/orientation b.png}
+		Matching orientation
+	\end{center}
+	\end{minipage}
+	\hfill
+	\begin{minipage}[t]{0.25\textwidth}
+	\begin{center}
+		\includegraphics[width=\linewidth]{images/orientation c.png}
+		Inverse orientation
+	\end{center}
+	\end{minipage}
+	\hfill~
+\end{center}
+
+In this example, both compositions happen to give the same result. This is because the trefoil knot is \textit{invertible}: its direction can be reversed by deforming it. This is not true in general, as you will soon see.
+
+\problem{}
+Invert a directed trefoil.
+
+\vfill
+
+\problem{}
+The smallest non-invertible knot is $8_{17}$, shown below. \par
+Compose $8_{17}$ with itself to obtain two different knots.
+
+\begin{center}
+	\includegraphics[height=30mm]{knot table/8_17.png} \par
+	\vspace{2mm}
+	{\large Knot $8_{17}$}
+\end{center}
+
+\begin{solution}
+	\begin{center}
+		\includegraphics[width=0.8\linewidth]{images/noninvertible.png}
+	\end{center}
+\end{solution}
+
+
+\vfill
+\pagebreak

Advanced/Knots/tikzset.tex

@@ -0,0 +1,51 @@
+\usetikzlibrary{
+	knots,
+	hobby,
+	decorations.pathreplacing,
+	shapes.geometric,
+	calc
+}
+
+\ifDebugKnot
+	\tikzset{
+		knot diagram/draft mode = crossings,
+		knot diagram/only when rendering/.style = {
+			show curve endpoints,
+			%show curve controls
+		}
+	}
+\fi
+
+\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) {};
+			}
+		}
+	}
+}