Next: Knotted or Unknotted Up: Tie Knots and Random Previous: Symmetry

Balance

Balance relates to the extent that the moves of a knot sequence are well distributed. Let a binary subsequence designate any subsequence which contains only two moves types; since no adjacent moves may be the same, the subsequence must be alternating. The corresponding subknot is a coil, wound, for example, around the centre leg of the triagonal basis, ; it is loosely bound and apt to telescope out. We define a well distributed knot sequence, and hence a balanced knot, as one whose maximum length binary subsequence is minimal.

The extent to which the corresponding walk is balanced may be expressed by the balance function b. Let represent the ith step of the walk. Then the winding direction is defined as

where if the transition from to is clockwise and -1 otherwise. By clockwise we mean in the frame of reference of the mirror (viz., ), which is counter-clockwise in the frame of the shirt. Such distinctions, however, need not concern us. The balance b may then be expressed

With and the analogue of angular position and velocity, respectively, the balance b may be considered the sum over the magnitude of the angular acceleration.

The Pratt, Windsor and Half-Windsor all have balance 0, while the Four-in-Hand has balance 1. More generally, only knots with half-winding number 3 i and 3i+2 can have zero balance, where i is a positive integer; half-winding numbers 3i+1 have balance at least 1.

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