163 lines
3.9 KiB
TeX
163 lines
3.9 KiB
TeX
\section{Introduction}
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\definition{}
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To form a \textit{knot}, take a string, tie a knot, then join the ends. \par
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You can also think of a knot as a path in three-dimensional space that doesn't intersect itself:
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\vspace{2mm}
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\begin{center}
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\begin{minipage}[t]{0.3\textwidth}
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\begin{center}
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\begin{tikzpicture}[scale = 0.8, baseline=(p)]
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\begin{knot}
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\strand
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(1,2) .. controls +(-45:1) and +(1,0) ..
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(0, 0) .. controls +(-1,0) and +(-90 -45:1) ..
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(-1,2);
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\end{knot}
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\coordinate (p) at (current bounding box.center);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.3\textwidth}
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\begin{center}
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\begin{tikzpicture}[scale = 0.8, baseline=(p)]
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% Knot is stupid and includes invisible handles in the tikz bounding box. This line crops the image to fix that.
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\clip (-2,-1.7) rectangle + (4, 4);
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\begin{knot}[
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consider self intersections=true,
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flip crossing = 2,
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]
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\strand
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(1,2) .. controls +(-45:1) and +(120:-2.2) ..
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(210:2) .. controls +(120:2.2) and +(60:2.2) ..
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(-30:2) .. controls +(60:-2.2) and +(-90 -45:1) ..
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(-1,2);
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\end{knot}
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\coordinate (p) at (current bounding box.center);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.3\textwidth}
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\begin{center}
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\begin{tikzpicture}[scale = 0.8, baseline=(p)]
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\clip (-2,-1.7) rectangle + (4, 4);
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\begin{knot}[
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consider self intersections=true,
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flip crossing = 2,
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]
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\strand
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(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
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(210:2) .. controls +(120:2.2) and +(60:2.2) ..
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(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
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(0,2);
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\end{knot}
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\coordinate (p) at (current bounding box.center);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\end{center}
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If two knots may be deformed into each other without cutting, we say they are \textit{isomorphic}. \par
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If two knots are isomorphic, they are essentially the same knot.
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\definition{}
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The simplest knot is the \textit{unknot}. It is show below on the left. \par
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The simplest nontrivial knot is the \textit{trefoil} knot, shown to the right.
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\begin{center}
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\begin{minipage}[t]{0.48\textwidth}
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\begin{center}
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\begin{tikzpicture}[baseline=(p), scale = 0.8]
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\begin{knot}
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\strand
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(0,2) .. controls +(1.5,0) and +(1.5,0) ..
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(0, 0) .. controls +(-1.5,0) and +(-1.5,0) ..
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(0,2);
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\end{knot}
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\coordinate (p) at (current bounding box.center);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\hfill
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\begin{minipage}[t]{0.48\textwidth}
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\begin{center}
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\begin{tikzpicture}[baseline=(p), scale = 0.8]
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\clip (-2,-1.7) rectangle + (4, 4);
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\begin{knot}[
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consider self intersections=true,
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flip crossing = 2,
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]
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\strand
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(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
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(210:2) .. controls +(120:2.2) and +(60:2.2) ..
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(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
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(0,2);
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\end{knot}
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\coordinate (p) at (current bounding box.center);
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\end{center}
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\vfill
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\pagebreak
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\problem{}
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Below are the only four distinct knots with only one crossing. \par
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Show that no nontrivial knot can have has fewer than three crossings. \par
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\hint{There are 4 such knots. What are they?}
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\begin{center}
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\includegraphics[width=0.8\linewidth]{images/one crossing.png}
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\end{center}
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\begin{solution}
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Draw all four. Each is isomorphic to the unknot.
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\end{solution}
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\vfill
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\problem{}
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Show that this is the unknot. \par
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A wire or an extension cord may help.
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\begin{center}
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\includegraphics[width=0.35\linewidth]{images/big unknot.png}
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\end{center}
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\definition{}
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As we said before, there are many ways to draw the same knot. \par
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We call each drawing a \textit{projection}. Below are four projections of the \textit{figure-eight} knot.
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\vspace{2mm}
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\begin{center}
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\includegraphics[width=0.8\linewidth]{images/figure eight.png}
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\end{center}
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\vspace{2mm}
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\problem{}
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Convince yourself that these are equivalent.
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\vfill
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\pagebreak |