diff --git a/Advanced/Knots/images/stick trefoil composition.png b/Advanced/Knots/images/stick trefoil composition.png new file mode 100644 index 0000000..b8dbda6 Binary files /dev/null and b/Advanced/Knots/images/stick trefoil composition.png differ diff --git a/Advanced/Knots/images/stick trefoil.png b/Advanced/Knots/images/stick trefoil.png new file mode 100644 index 0000000..77f6f50 Binary files /dev/null and b/Advanced/Knots/images/stick trefoil.png differ diff --git a/Advanced/Knots/images/sticks.png b/Advanced/Knots/images/sticks.png new file mode 100644 index 0000000..6f678a7 Binary files /dev/null and b/Advanced/Knots/images/sticks.png differ diff --git a/Advanced/Knots/main.tex b/Advanced/Knots/main.tex index bda14e5..b42a354 100755 --- a/Advanced/Knots/main.tex +++ b/Advanced/Knots/main.tex @@ -38,6 +38,7 @@ \input{parts/0 intro.tex} \input{parts/1 composition.tex} \input{parts/2 links.tex} + \input{parts/3 sticks.tex} % Make sure the knot table is on an odd page diff --git a/Advanced/Knots/parts/2 links.tex b/Advanced/Knots/parts/2 links.tex index 5ab3745..916505c 100644 --- a/Advanced/Knots/parts/2 links.tex +++ b/Advanced/Knots/parts/2 links.tex @@ -66,6 +66,7 @@ Find a Brunnian link with four components. Find a Brunnian link with $n$ components. \begin{solution} + One of many possible solutions: \begin{center} \includegraphics[width=40mm]{images/brunnian.png} \end{center} diff --git a/Advanced/Knots/parts/3 sticks.tex b/Advanced/Knots/parts/3 sticks.tex new file mode 100644 index 0000000..ff62b9c --- /dev/null +++ b/Advanced/Knots/parts/3 sticks.tex @@ -0,0 +1,62 @@ +\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 \ No newline at end of file