133 lines
3.2 KiB
TeX
133 lines
3.2 KiB
TeX
\section{Knot Composition}
|
|
|
|
Say we have two knots $A$ and $B$.
|
|
The knot $A \boxplus B$ is created by cutting $A$ and $B$ and joining their ends:
|
|
|
|
\begin{center}
|
|
\hfill
|
|
\begin{minipage}[t]{0.15\textwidth}
|
|
\begin{center}
|
|
\includegraphics[width=\linewidth]{images/composition a.png}
|
|
$A$
|
|
\end{center}
|
|
\end{minipage}
|
|
\hfill
|
|
\begin{minipage}[t]{0.13\textwidth}
|
|
\begin{center}
|
|
\includegraphics[width=\linewidth]{images/composition b.png}
|
|
$B$
|
|
\end{center}
|
|
\end{minipage}
|
|
\hfill
|
|
\begin{minipage}[t]{0.3\textwidth}
|
|
\begin{center}
|
|
\includegraphics[width=\linewidth]{images/composition c.png}
|
|
$A \boxplus B$
|
|
\end{center}
|
|
\end{minipage}
|
|
\hfill~
|
|
\end{center}
|
|
|
|
We must be careful to avoid new crossings when composing knots:
|
|
|
|
\vspace{2mm}
|
|
\begin{center}
|
|
\includegraphics[width=0.45\linewidth]{images/composition d.png}
|
|
\end{center}
|
|
\vspace{2mm}
|
|
|
|
We say a knot is \textit{composite} if it can be obtained by composing two other knots. \par
|
|
We say a knot is \textit{prime} otherwise.
|
|
|
|
\problem{}
|
|
For any knot $K$, what is $K \boxplus \text{unknot}$?
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
Use a pencil or a cord to compose the figure-eight knot with itself.
|
|
|
|
\vfill
|
|
|
|
\vfill
|
|
\pagebreak{}
|
|
|
|
\problem{}
|
|
The following knots are composite. 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}
|
|
\hfill
|
|
\includegraphics[height=30mm]{images/decompose a.png}
|
|
\hfill
|
|
\includegraphics[height=30mm]{images/decompose b.png}
|
|
\hfill~\par
|
|
\vspace{4mm}
|
|
\end{center}
|
|
|
|
\begin{solution}
|
|
The first is easy, it's the trefoil composed with itself. \par
|
|
|
|
\vspace{2mm}
|
|
|
|
The second is knot $5_2$ composed with itself. \par
|
|
Note that the \say{three-crossing figure eight} is another projection of $5_2$. \par
|
|
The figure-eight knot is NOT a part of this composition. Look closely at its crossings.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\definition{}
|
|
When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}.
|
|
|
|
\vspace{2mm}
|
|
|
|
An \textit{orientated knot} is created by defining a \say{direction of travel.} \par
|
|
There are two distinct ways to compose a pair of oriented knots:
|
|
|
|
\begin{center}
|
|
\hfill
|
|
\begin{minipage}[t]{0.25\textwidth}
|
|
\begin{center}
|
|
\includegraphics[width=\linewidth]{images/orientation b.png}
|
|
Matching orientation
|
|
\end{center}
|
|
\end{minipage}
|
|
\hfill
|
|
\begin{minipage}[t]{0.25\textwidth}
|
|
\begin{center}
|
|
\includegraphics[width=\linewidth]{images/orientation c.png}
|
|
Inverse orientation
|
|
\end{center}
|
|
\end{minipage}
|
|
\hfill~
|
|
\end{center}
|
|
|
|
In this example, both compositions happen to give the same result. This is because the trefoil knot is \textit{invertible}: its direction can be reversed by deforming it. This is not true in general, as you will soon see.
|
|
|
|
\problem{}
|
|
Invert a directed trefoil.
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
The smallest non-invertible knot is $8_{17}$, shown below. \par
|
|
Compose $8_{17}$ with itself to obtain two different knots.
|
|
|
|
\begin{center}
|
|
\includegraphics[height=30mm]{knot table/8_17.png} \par
|
|
\vspace{2mm}
|
|
{\large Knot $8_{17}$}
|
|
\end{center}
|
|
|
|
\begin{solution}
|
|
\begin{center}
|
|
\includegraphics[width=0.8\linewidth]{images/noninvertible.png}
|
|
\end{center}
|
|
\end{solution}
|
|
|
|
|
|
\vfill
|
|
\pagebreak |