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2023-05-04 11:48:07 -07:00
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Advanced/Knots/parts/0 intro.tex
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@ -82,13 +82,9 @@ 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}
\begin{tikzpicture}[baseline=(p)]
\draw[circle] (0,0) circle (1);
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
@ -98,7 +94,6 @@ The simplest nontrivial knot is the \textit{trefoil} knot, shown to the right.
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[baseline=(p), scale = 0.8]
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
@ -122,16 +117,16 @@ The simplest nontrivial knot is the \textit{trefoil} knot, shown to the right.
\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?}
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 all four. Each is isomorphic to the unknot.
Draw them all. Each is isomorphic to the unknot.
\end{solution}
\vfill
@ -147,7 +142,7 @@ A wire or an extension cord may help.
\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.
We call each drawing a \textit{projection}. Below are four projections of the \textit{figure-eight knot}.
\vspace{2mm}
@ -157,7 +152,8 @@ We call each drawing a \textit{projection}. Below are four projections of the \t
\vspace{2mm}
\problem{}
Convince yourself that these are equivalent.
Convince yourself that these are equivalent. \par
Try to deform them into each other with a cord!
\vfill
\pagebreak