\section{Unicursal curves and knot diagrams}

A unicursal curve in the plane is a curve that you get when you put down
your pencil, and draw until you get back to the starting point.  As you
draw, your pencil mark can intersect itself, but you're not supposed to
have any triple intersections.  You could say that you pencil is allowed to
pass over an point of the plane at most twice.
This property of not having any triple intersections is \dt{generic}:
If you scribble the curve with your eyes closed (and somehow magically
manage to make the curve finish off exactly where it began),
the curve won't have any triple intersections.

A unicursal curve differs from the curves shown in knot diagrams in that
there is no sense of the curve's crossing over or under itself at an
intersection.  You can convert a unicursal curve into a knot diagram by
indicating (probably with the aid of an eraser), which strand crosses over and
which strand crosses under at each of the intersections.

A unicursal curve with $5$ intersections can be converted into a knot diagram
in $2^5$ ways, because each intersection can be converted into a crossing
in two ways.
These 32 diagrams will not represent 32 different knots, however.

\begin{problems}
\item
Draw the 32 knot diagrams that arise from the unicursal curve underlying
the diagram of knot 5-2, and identify
the knots that these diagrams represent.
\item
Show that any unicursal curve can be converted into a diagram of the
unknot.
\item
Show that any unicursal curve can be converted into the diagram of an
alternating knot in precisely two ways.
These two diagrams may or may not represent different knots.
Give an example where the two knots are the same, and another where
the two knots are different.
\item
Show that any unicursal curve gives a map of the plane whose regions
can be colored black and white 
in such a way that adjacent regions have different colors.
In how many ways can this coloring be done?
Give examples.
\end{problems}
