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Shape of Knots

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

Next: Symmetry Up: Tie Knots and Random Previous: Size of Knots

Yong Mao
Sat Nov 7 16:03:57 GMT 1998