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 *i*th 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 3*i*+2 can have zero balance, where *i* is a positive integer;
half-winding numbers 3*i*+1 have balance at least 1.

Sat Nov 7 16:03:57 GMT 1998