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"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 - - - {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} \ No newline at end of file diff --git a/Advanced/Knots/parts/0 intro.tex b/Advanced/Knots/parts/0 intro.tex deleted file mode 100644 index a543910..0000000 --- a/Advanced/Knots/parts/0 intro.tex +++ /dev/null @@ -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 \ No newline at end of file diff --git a/Advanced/Knots/parts/1 composition.tex b/Advanced/Knots/parts/1 composition.tex deleted file mode 100644 index e20e1db..0000000 --- a/Advanced/Knots/parts/1 composition.tex +++ /dev/null @@ -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 \ No newline at end of file diff --git a/Advanced/Knots/parts/2 links.tex b/Advanced/Knots/parts/2 links.tex deleted file mode 100644 index 916505c..0000000 --- a/Advanced/Knots/parts/2 links.tex +++ /dev/null @@ -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 \ No newline at end of file diff --git a/Advanced/Knots/parts/3 sticks.tex b/Advanced/Knots/parts/3 sticks.tex deleted file mode 100644 index d47a6a2..0000000 --- a/Advanced/Knots/parts/3 sticks.tex +++ /dev/null @@ -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 \ No newline at end of file diff --git a/Advanced/Knots/parts/4 braids.tex b/Advanced/Knots/parts/4 braids.tex deleted file mode 100644 index 6f43f7c..0000000 --- a/Advanced/Knots/parts/4 braids.tex +++ /dev/null @@ -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{} -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 \ No newline at end of file diff --git a/Advanced/Knots/parts/table.tex b/Advanced/Knots/parts/table.tex deleted file mode 100644 index 9d87d54..0000000 --- a/Advanced/Knots/parts/table.tex +++ /dev/null @@ -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 \ No newline at end of file diff --git a/Advanced/Knots/tikzset.tex b/Advanced/Knots/tikzset.tex deleted file mode 100644 index aeabf4a..0000000 --- a/Advanced/Knots/tikzset.tex +++ /dev/null @@ -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) {$\x$}; - } -} \ No newline at end of file