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