\section{Notation for some knots}

It is a hard mathematical question to completely codify
all possible knots.  Given two knots, it is hard to tell whether
they are the same. It is harder still to tell for sure
that they are different.  

Many simple knots can be arranged in a certain form, as illustrated
below, which is described by a string of positive integers along
with a sign. 

\fig{knotnotation}{Notation for certain knots}{Here are drawings of
some examples of knots that Conway `names' by a string of positive integers.
The drawings use the convention that when one strand crosses under
another strand, it is broken.
Notice that as you
run along the knot, the strand alternates going over and under at
its crossings.  Knots with this property are called \dt{alternating
knots}.  Can you find any examples of knots with more than one
name of this type?}

\fig{sixcrossings}{Prime knots up to six crossings}{Here are the prime
knots with
up to six crossings.  The names follow an old system, used widely
in knot tables, where the $k$th knot with $n$ crossings is called
$n-k$.
  Mirror images are not included: some of these
knots are equivalent to their mirror images, and some are not.
Can you tell which are which?}

\section{Knots diagrams and maps}

A knot diagram gives a map on the plane, where there are four edges
coming together at each vertex.  Actually, it is better to think
of the diagram as a map on the sphere, with a polygon on the outside.
It sometimes helps in recognizing when diagrams are topologically
identical to label the regions with how many edges they have.

