\section{Knots}

A mathematical knot is a knotted loop.   For example,
you might take an extension
cord from a drawer and plug one end into the other: this makes
a mathematical knot.

 It might or might not be possible to unknot
it without unplugging the cord.  A knot which can be unknotted is
called an \dt{unknot}.  

Two knots are
considered equivalent if it is possible to rearrange one to
the form of the other, without cutting the loop and without
allowing it to pass through itself.
The reason for using loops of string in the mathematical
definition is that knots in a length of string can always
be undone, so any two lengths of string are
equivalent in this sense.

If you drop a knotted loop of string on a table, it crosses over
itself in a certain number of places.  Possibly, there are ways to rearrange
it with fewer crossings---the minimum possible number of crossings
is the \dt{crossing number} of the knot.

\fig{knota}{Drawing of a knot}{This is drawing of a knot with $7$
crossings.  Is it possible to rearrange it to have fewer crossings?}

Make drawings and use short lengths of string to investigate the following
problems.

\begin{problems}
\item
Are there any knots with one or two crossings?  Why?
\item
How many inequivalent knots are there with three crossings?
\item
How many knots are there with four crossings?
\item
How many knots can you find with five crossings?
\item
How many knots can you find with six crossings?
\end{problems}
