Key to a knot's shape is the centre fraction, , which
serves to specify the aspect ratio of the knot. A knot with many
centre half-turns, *e.g.*, the Windsor, resembles a broad triangle,
while the more sparsely centred Four-in-Hand has a narrow shape.

For a knot of half-winding number *h*,
the number of centres is an integer bounded as
.
Accordingly, for large *h*, the range of the centre
fraction tends toward . However, not all
centre fractions allow for any aesthetic knots;
knots with lower than are rather cylindrical
in shape and afford little balance.
We consequently limit our attention to centre fractions
, in particular,
.
It follows that, for knots of size *h*, the number of knot classes
is the smallest integer less than or equal to
.

For , the classes of interest are

The number of possible knots in a given class, , corresponds to
the number of walks of length *h* containing steps ,
beginning with and ending with or .
The sequence of steps may be considered a coarser sequence of
groups, each group
composed of s and s and separated by a on the right; the
Windsor knot, for example, contains three groups,
, of lengths 1, 2, 2,
respectively. We refer to a particular spacing of the centre steps as a
centre structure.

Let be the number of groups of length 1 in a given sequence, the number of length 2, ..., the number of length . These group numbers satisfy

We desire the number of ordered non-negative integer solutions to (10, 11), that is, the number of ordered ways of partitioning the integer into positive integers. Call this function ; it is given by

The number of centre structures is equivalent to subject to the terminal condition -- this simply says that the final group cannot be of length one, reducing the possible centre structures by .

The steps within each group must alternate between and , since no two consecutive steps may be identical. It follows that the steps of each group may be ordered in two ways, beginning with or , except for the first, which necessarily begins with . For a given centre structure, the number of acceptable walks is then .

It follows that the number of possible knots in the class is

Sat Nov 7 16:03:57 GMT 1998