handouts/Advanced/Knots/parts/2 links.tex

\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}
	One of many possible solutions:
	\begin{center}
		\includegraphics[width=40mm]{images/brunnian.png}
	\end{center}
\end{solution}


\vfill
\pagebreak