2023-05-04 10:50:36 -07:00
\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 { }
2023-05-04 11:48:07 -07:00
The following knots are composite. \par
What are their prime components? \par
Try to make them with a cord. \par
2023-05-04 10:50:36 -07:00
\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
