\section{Knots and Sticks}

\definition{}
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.

\begin{center}
	\includegraphics[width=30mm]{images/sticks.png}
\end{center}

\problem{}
Make the trefoil knot with sticks. \par
How many do you need?

\begin{solution}
	\begin{center}
		\includegraphics[width=20mm]{images/stick trefoil.png}
	\end{center}
\end{solution}

\vfill

\problem{}
How many sticks will you need to make a figure-eight knot?

\begin{solution}
	The figure-eight knot has stick number 7. \par
	In fact, this is the \textit{only} knot with stick number 7.
\end{solution}

\vfill
\pagebreak

\problem{}
Make the knot $5_1$ (refer to the knot table) with eight sticks.

\vfill

\problem{}
Show that the only nontrivial knot you can make with six sticks is the trefoil.

\vfill

\problem{}
Let $S(k)$ be the stick number of a knot $k$. \par
Show that $S(j \boxplus k) \leq s(j) + s(k) - 1$


\vfill

\problem{}
What is the stick number of $(\text{trefoil} \boxplus \text{trefoil})$?

\begin{solution}
	You can make $(\text{trefoil} \boxplus \text{trefoil})$ with 8 sticks.

	\begin{center}
		\includegraphics[angle=90, width=40mm]{images/stick trefoil composition.png}
	\end{center}
\end{solution}

\vfill
\pagebreak