62 lines
1.3 KiB
TeX
62 lines
1.3 KiB
TeX
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\section{Knots and Sticks}
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\definition{}
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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.
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\begin{center}
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\includegraphics[width=30mm]{images/sticks.png}
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\end{center}
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\problem{}
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Make the trefoil knot with sticks. \par
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How many do you need?
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\begin{solution}
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\begin{center}
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\includegraphics[width=20mm]{images/stick trefoil.png}
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\end{center}
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\end{solution}
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\vfill
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\problem{}
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How many sticks will you need to make a figure-eight knot?
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\begin{solution}
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The figure-eight knot has stick number 7. \par
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In fact, this is the \textit{only} knot with stick number 7.
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\end{solution}
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\vfill
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\pagebreak
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\problem{}
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Make the knot $5_1$ (refer to the knot table) with eight sticks.
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\vfill
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\problem{}
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Show that the only nontrivial knot you can make with six sticks is the trefoil.
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\vfill
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\problem{}
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Let $S(k)$ be the stick number of a knot $k$. \par
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Show that $S(j \boxplus k) \leq s(j) + s(k) - 1$
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\vfill
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\problem{}
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What is the stick number of $(\text{trefoil} \boxplus \text{trefoil})$?
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\begin{solution}
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You can make $(\text{trefoil} \boxplus \text{trefoil})$ with 8 sticks.
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\begin{center}
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\includegraphics[angle=90, width=40mm]{images/stick trefoil composition.png}
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\end{center}
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\end{solution}
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\vfill
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\pagebreak
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