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b/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 + + + {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} \ No newline at end of file diff --git a/Advanced/Knots/parts/table.tex b/Advanced/Knots/parts/table.tex new file mode 100644 index 0000000..c7dbb06 --- /dev/null +++ b/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 \ No newline at end of file