diff --git a/Advanced/Knots/images/noninvertible a.png b/Advanced/Knots/images/noninvertible a.png new file mode 100644 index 0000000..7c6f357 Binary files /dev/null and b/Advanced/Knots/images/noninvertible a.png differ diff --git a/Advanced/Knots/images/noninvertible b.png b/Advanced/Knots/images/noninvertible b.png new file mode 100644 index 0000000..d8eb437 Binary files /dev/null and b/Advanced/Knots/images/noninvertible b.png differ diff --git a/Advanced/Knots/images/noninvertible.png b/Advanced/Knots/images/noninvertible.png new file mode 100644 index 0000000..dcba318 Binary files /dev/null and b/Advanced/Knots/images/noninvertible.png differ diff --git a/Advanced/Knots/images/orientation a.png b/Advanced/Knots/images/orientation a.png new file mode 100644 index 0000000..47e6b15 Binary files /dev/null and b/Advanced/Knots/images/orientation a.png differ diff --git a/Advanced/Knots/images/orientation b.png b/Advanced/Knots/images/orientation b.png new file mode 100644 index 0000000..fb87bad Binary files /dev/null and b/Advanced/Knots/images/orientation b.png differ diff --git a/Advanced/Knots/images/orientation c.png b/Advanced/Knots/images/orientation c.png new file mode 100644 index 0000000..6171df2 Binary files /dev/null and b/Advanced/Knots/images/orientation c.png differ diff --git a/Advanced/Knots/knot table/8_17.png b/Advanced/Knots/knot table/8_17.png new file mode 100644 index 0000000..2c38392 Binary files /dev/null and b/Advanced/Knots/knot table/8_17.png differ diff --git a/Advanced/Knots/main.tex b/Advanced/Knots/main.tex index 1650979..d8c2730 100755 --- a/Advanced/Knots/main.tex +++ b/Advanced/Knots/main.tex @@ -5,37 +5,15 @@ 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 +% Set to true to show knot measurements. +% Only used in tikzset.tex \newif{\ifDebugKnot} -\DebugKnottrue +%\DebugKnottrue \DebugKnotfalse - -\ifDebugKnot - \tikzset{ - knot diagram/draft mode = crossings, - knot diagram/only when rendering/.style = { - show curve endpoints, - %show curve controls - } - } -\fi +\input{tikzset.tex} -% 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 @@ -91,251 +34,9 @@ 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/0 intro.tex} + \input{parts/1 composition.tex} \input{parts/table} diff --git a/Advanced/Knots/parts/0 intro.tex b/Advanced/Knots/parts/0 intro.tex new file mode 100644 index 0000000..611cd30 --- /dev/null +++ b/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 \ No newline at end of file diff --git a/Advanced/Knots/parts/1 composition.tex b/Advanced/Knots/parts/1 composition.tex new file mode 100644 index 0000000..578889f --- /dev/null +++ b/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 \ No newline at end of file diff --git a/Advanced/Knots/tikzset.tex b/Advanced/Knots/tikzset.tex new file mode 100644 index 0000000..64fb30b --- /dev/null +++ b/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) {}; + } + } + } +} \ No newline at end of file