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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, tex2html_wrap_inline1224 ; 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 tex2html_wrap_inline1228 represent the ith step of the walk. Then the winding direction tex2html_wrap_inline1232 is defined as

  equation310

where tex2html_wrap_inline1234 if the transition from tex2html_wrap_inline1228 to tex2html_wrap_inline1238 is clockwise and -1 otherwise. By clockwise we mean in the frame of reference of the mirror (viz., tex2html_wrap_inline1242 ), which is counter-clockwise in the frame of the shirt. Such distinctions, however, need not concern us. The balance b may then be expressed

  equation317

With tex2html_wrap_inline1246 and tex2html_wrap_inline1232 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