Added link section

Advanced/Knots/parts/0 intro.tex

@@ -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

Advanced/Knots/parts/1 composition.tex

@@ -53,8 +53,9 @@ Use a pencil or a cord to compose the figure-eight knot with itself.
 \pagebreak{}
 
 \problem{}
-The following knots are composite. What are their prime components? \par
-Try to make them with a cord! \par
+The following knots are composite. \par
+What are their prime components? \par
+Try to make them with a cord. \par
 \hint{Use the table at the back of this handout to decompose the second knot.}
 
 \begin{center}

Advanced/Knots/parts/2 links.tex

@@ -0,0 +1,76 @@
+\section{Links}
+
+\definition{}
+A \textit{link} is a set of knots intertwined with each other. \par
+Just as with knots, we say that two links are \textit{isomorphic} if one can be deformed into the other.
+
+\vspace{2mm}
+
+The \textit{Whitehead link} is one of the simplest links we can produce. \par
+It consists of two knots, so we say it is a \textit{link of two components}.
+Two projections of the Whitehead link are shown below.
+
+
+\begin{center}
+	\hfill
+	\begin{minipage}[t]{0.27\textwidth}
+	\begin{center}
+		\includegraphics[width=\linewidth]{images/whitehead a.png}
+	\end{center}
+	\end{minipage}
+	\hfill
+	\begin{minipage}[t]{0.25\textwidth}
+	\begin{center}
+		\includegraphics[width=\linewidth]{images/whitehead b.png}
+	\end{center}
+	\end{minipage}
+	\hfill~
+\end{center}
+
+
+\definition{}
+The \textit{$n$-unlink} is the link that consists of $n$ disjoint unknots. \par
+The 3-unlink is shown below:
+
+\begin{center}
+	\begin{tikzpicture}
+
+	\draw[circle] (0,0) circle (0.7);
+	\draw[circle] (2,0) circle (0.7);
+	\draw[circle] (4,0) circle (0.7);
+
+	\end{tikzpicture}
+\end{center}
+
+\definition{}
+We say a nontrivial link is \textit{Brunnian} if we get an $n$-unlink after removing any component.
+
+\vspace{2mm}
+
+The \textit{Borromean Rings} are a common example of this. If we were to cut any of the three rings, the other two would fall apart.
+
+\begin{center}
+	\includegraphics[height=3cm]{images/borromean.png}
+\end{center}
+
+\vfill
+\pagebreak
+
+
+\problem{}
+Find a Brunnian link with four components.
+
+\vfill
+
+\problem{}
+Find a Brunnian link with $n$ components.
+
+\begin{solution}
+	\begin{center}
+		\includegraphics[width=40mm]{images/brunnian.png}
+	\end{center}
+\end{solution}
+
+
+\vfill
+\pagebreak

Advanced/Knots/parts/table.tex

@@ -1,15 +1,16 @@
 \section{Table of Prime Knots}
+A knot's \textit{crossing number} is the minimal number of crossings its projection must contain. In general, it is very difficult to determine a knot's crossing number.
+
+\vspace{1mm}
+
 This table contains the 15 smallest prime knots, ordered by crossing number. \par
-Mirror images are not accounted for, even if the mirror image produces a nonisomorphic knot.
-
-\vspace{5mm}
-
-% Images are from the appendix of the Knot book.
+Mirror images are not included, even if the mirror image produces a nonisomorphic knot.
 
 \vfill
 
+% Images are from the appendix of the Knot book.
 {
-	\def\w{25mm}
+	\def\w{24mm}
 	\foreach \l/\c/\r in {%
 		{3_1}/{4_1}/{5_1},%
 		{5_2}/{6_1}/{6_2},%