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Shape of Knots

Key to a knot's shape is the centre fraction, tex2html_wrap_inline802 , 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 tex2html_wrap_inline800 is an integer bounded as tex2html_wrap_inline1048 . Accordingly, for large h, the range of the centre fraction tex2html_wrap_inline802 tends toward tex2html_wrap_inline1054 . However, not all centre fractions allow for any aesthetic knots; knots with tex2html_wrap_inline802 lower than tex2html_wrap_inline1058 are rather cylindrical in shape and afford little balance. We consequently limit our attention to centre fractions tex2html_wrap_inline1060 , in particular, tex2html_wrap_inline1062 . It follows that, for knots of size h, the number of knot classes is the smallest integer less than or equal to tex2html_wrap_inline1066 .

For tex2html_wrap_inline890 , the classes tex2html_wrap_inline898 of interest are

  eqnarray180

The number of possible knots in a given class, tex2html_wrap_inline804 , corresponds to the number of walks of length h containing tex2html_wrap_inline800 steps tex2html_wrap_inline994 , beginning with tex2html_wrap_inline948 and ending with tex2html_wrap_inline950 or tex2html_wrap_inline952 . The sequence of steps may be considered a coarser sequence of tex2html_wrap_inline800 groups, each group composed of tex2html_wrap_inline988 s and tex2html_wrap_inline948 s and separated by a tex2html_wrap_inline994 on the right; the Windsor knot, for example, contains three groups, tex2html_wrap_inline1094 , of lengths 1, 2, 2, respectively. We refer to a particular spacing of the centre steps as a centre structure.

Let tex2html_wrap_inline1096 be the number of groups of length 1 in a given sequence, tex2html_wrap_inline1098 the number of length 2, ..., tex2html_wrap_inline1100 the number of length tex2html_wrap_inline1102 . These group numbers satisfy

   eqnarray184

We desire the number of ordered non-negative integer solutions tex2html_wrap_inline1104 to (10, 11), that is, the number of ordered ways of partitioning the integer tex2html_wrap_inline1106 into tex2html_wrap_inline800 positive integers. Call this function tex2html_wrap_inline1110 ; it is given by

  equation193

The number of centre structures is equivalent to tex2html_wrap_inline1110 subject to the terminal condition -- this simply says that the final group cannot be of length one, reducing the possible centre structures by tex2html_wrap_inline1114 .

The steps within each group must alternate between tex2html_wrap_inline988 and tex2html_wrap_inline948 , since no two consecutive steps may be identical. It follows that the steps of each group may be ordered in two ways, beginning with tex2html_wrap_inline988 or tex2html_wrap_inline948 , except for the first, which necessarily begins with tex2html_wrap_inline948 . For a given centre structure, the number of acceptable walks is then tex2html_wrap_inline1126 .

It follows that the number of possible knots in the class tex2html_wrap_inline898 is

  equation199


next up previous
Next: Symmetry Up: Tie Knots and Random Previous: Size of Knots

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