Renamed folder

Advanced/Knots/main.tex

@@ -1,63 +0,0 @@
-% use [nosolutions] flag to hide solutions.
-% use [solutions] flag to show solutions.
-\documentclass[
-	solutions,
-	singlenumbering
-]{../../resources/ormc_handout}
-
-% Set to true to show knot measurements.
-% Only used in tikzset.tex
-\newif{\ifDebugKnot}
-%\DebugKnottrue
-\DebugKnotfalse
-\input{tikzset.tex}
-
-
-% Problems from "Why knot"
-%
-% 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
-
-%\usepackage{lua-visual-debug}
-
-
-\begin{document}
-	\maketitle
-		<Advanced 2>
-		<Spring 2023>
-		{Knots and Braids}
-		{
-			Prepared by Mark on \today
-		}
-
-
-	\input{parts/0 intro.tex}
-	\input{parts/1 composition.tex}
-	\input{parts/2 links.tex}
-	\input{parts/3 sticks.tex}
-	\input{parts/4 braids.tex}
-
-
-	% Make sure the knot table is on an odd page
-	% so it may be removed in a double-sided
-	% handout.
-	\checkoddpage
-	\ifoddpage\else
-		\vspace*{\fill}
-		\begin{center}
-		{
-			\Large
-			\textbf{This page isn't empty.}
-		}
-		\end{center}
-		\vspace{\fill}
-		\pagebreak
-	\fi
-
-	\input{parts/table}
-
-\end{document}

Advanced/Knots/parts/0 intro.tex

@@ -1,159 +0,0 @@
-\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)]
-
-		\draw[circle] (0,0) circle (1);
-
-		\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 knots with one crossing. \par
-Show that every nontrivial knot more than two crossings. \par
-\hint{There are four knots with two crossings. What are they?}
-
-\begin{center}
-	\includegraphics[width=0.8\linewidth]{images/one crossing.png}
-\end{center}
-
-\begin{solution}
-	Draw them all. 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 many.png}
-\end{center}
-\vspace{2mm}
-
-\problem{}
-Convince yourself that these are equivalent. \par
-Try to deform them into each other with a cord!
-
-\vfill
-\pagebreak

Advanced/Knots/parts/1 composition.tex

@@ -1,135 +0,0 @@
-\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/figure eight.png}
-		$A$
-	\end{center}
-	\end{minipage}
-	\hfill
-	\begin{minipage}[t]{0.13\textwidth}
-	\begin{center}
-		\includegraphics[width=\linewidth]{images/trefoil.png}
-		$B$
-	\end{center}
-	\end{minipage}
-	\hfill
-	\begin{minipage}[t]{0.3\textwidth}
-	\begin{center}
-		\includegraphics[width=\linewidth]{images/composition a.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 b.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. \par
-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{oriented 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/parts/2 links.tex

@@ -1,77 +0,0 @@
-\section{Links}
-
-\definition{}
-A \textit{link} is a set of knots intertwined with each other. \par
-Just as with knots, we say that two links are \textit{isomorphic} if one can be deformed into the other.
-
-\vspace{2mm}
-
-The \textit{Whitehead link} is one of the simplest links we can produce. \par
-It consists of two knots, so we say it is a \textit{link of two components}.
-Two projections of the Whitehead link are shown below.
-
-
-\begin{center}
-	\hfill
-	\begin{minipage}[t]{0.27\textwidth}
-	\begin{center}
-		\includegraphics[width=\linewidth]{images/whitehead a.png}
-	\end{center}
-	\end{minipage}
-	\hfill
-	\begin{minipage}[t]{0.25\textwidth}
-	\begin{center}
-		\includegraphics[width=\linewidth]{images/whitehead b.png}
-	\end{center}
-	\end{minipage}
-	\hfill~
-\end{center}
-
-
-\definition{}
-The \textit{$n$-unlink} is the link that consists of $n$ disjoint unknots. \par
-The 3-unlink is shown below:
-
-\begin{center}
-	\begin{tikzpicture}
-
-	\draw[circle] (0,0) circle (0.7);
-	\draw[circle] (2,0) circle (0.7);
-	\draw[circle] (4,0) circle (0.7);
-
-	\end{tikzpicture}
-\end{center}
-
-\definition{}
-We say a nontrivial link is \textit{Brunnian} if we get an $n$-unlink after removing any component.
-
-\vspace{2mm}
-
-The \textit{Borromean Rings} are a common example of this. If we were to cut any of the three rings, the other two would fall apart.
-
-\begin{center}
-	\includegraphics[height=3cm]{images/borromean.png}
-\end{center}
-
-\vfill
-\pagebreak
-
-
-\problem{}
-Find a Brunnian link with four components.
-
-\vfill
-
-\problem{}
-Find a Brunnian link with $n$ components.
-
-\begin{solution}
-	One of many possible solutions:
-	\begin{center}
-		\includegraphics[width=40mm]{images/brunnian.png}
-	\end{center}
-\end{solution}
-
-
-\vfill
-\pagebreak

Advanced/Knots/parts/3 sticks.tex

@@ -1,62 +0,0 @@
-\section{Knots and Sticks}
-
-\definition{}
-The \textit{stick number} of a knot is the smallest number of \say{sticks} you must glue together to make the knot. An example of this is below.
-
-\begin{center}
-	\includegraphics[width=30mm]{images/sticks.png}
-\end{center}
-
-\problem{}
-Make a trefoil knot with sticks. \par
-How many do you need?
-
-\begin{solution}
-	\begin{center}
-		\includegraphics[width=20mm]{images/stick trefoil.png}
-	\end{center}
-\end{solution}
-
-\vfill
-
-\problem{}
-How many sticks do you need to make a figure-eight knot?
-
-\begin{solution}
-	The figure-eight knot has stick number 7. \par
-	In fact, it is the only knot with stick number 7.
-\end{solution}
-
-\vfill
-\pagebreak
-
-\problem{}
-Make the knot $5_1$ (refer to the knot table) with eight sticks.
-
-\vfill
-
-\problem{}
-Show that the only nontrivial knot you can make with six sticks is the trefoil.
-
-\vfill
-
-\problem{}
-Let $S(k)$ be the stick number of a knot $k$. \par
-Show that $S(j \boxplus k) \leq s(j) + s(k) - 1$
-
-
-\vfill
-
-\problem{}
-What is the stick number of $(\text{trefoil} \boxplus \text{trefoil})$?
-
-\begin{solution}
-	You can make $(\text{trefoil} \boxplus \text{trefoil})$ with 8 sticks.
-
-	\begin{center}
-		\includegraphics[angle=90, width=40mm]{images/stick trefoil composition.png}
-	\end{center}
-\end{solution}
-
-\vfill
-\pagebreak

Advanced/Knots/parts/4 braids.tex

@@ -1,204 +0,0 @@
-\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. 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^{-1}}
-			};
-		\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 left to right, with the 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 $n$-string braid $[(1, 2, ..., n-1)^m]$ forms a knot iff $m$ and $n$ are coprime.
-
-\vfill
-\pagebreak

Advanced/Knots/parts/table.tex

@@ -1,42 +0,0 @@
-\section{Table of Prime Knots}
-A knot's \textit{crossing number} is the minimal number of crossings its projection must contain. \par
-Finding a knot's crossing number is a fairly difficult problem.
-
-\vspace{1mm}
-
-This table contains the a 20 smallest prime knots, ordered by crossing number. \par
-Mirror images are not included, even if the mirror image produces a nonisomorphic knot.
-
-\vfill
-
-% Images are from the appendix of the Knot book.
-{
-	\def\w{24mm}
-	\newcounter{knotcounter}
-
-	\foreach \a 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},{8_2},%
-		{8_3},{8_4},{8_5},{8_6}%
-	}{
-		\stepcounter{knotcounter}
-		\hfill
-		\begin{minipage}{\w}
-		\begin{center}
-			\includegraphics[width=\linewidth]{knot table/\a.png} \par
-			\vspace{2mm}
-			{\huge $\a$}
-		\end{center}
-		\end{minipage}
-		\ifnum\value{knotcounter}=4
-			\hfill~\par
-			\vspace{4mm}
-			\setcounter{knotcounter}{0}
-		\fi
-	}
-}
-
-\vfill
-\pagebreak

Advanced/Knots/tikzset.tex

@@ -1,106 +0,0 @@
-\usetikzlibrary{
-	knots,
-	hobby,
-	braids,
-	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{
-	circle/.style = {
-		line width = 0.8mm,
-	},
-	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) {};
-			}
-		}
-	},
-	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) {$\x$};
-	}
-}